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+# Towards Instance-Optimal Offline Reinforcement Learning with Pessimism
+
+Ming Yin 1,2 and Yu-Xiang Wang1
+
+1Department of Computer Science, UC Santa Barbara 2Department of Statistics and Applied Probability, UC Santa Barbara ming_yin@ucsb.edu yuxiangw@cs.ucsb.edu
+
+# Abstract
+
+We study the offline reinforcement learning (offline RL) problem, where the goal is to learn a reward-maximizing policy in an unknown Markov Decision Process (MDP) using the data coming from a policy $\mu$ . In particular, we consider the sample complexity problems of offline RL for finite-horizon MDPs. Prior works study this problem based on different data-coverage assumptions, and their learning guarantees are expressed by the covering coefficients which lack the explicit characterization of system quantities. In this work, we analyze the Adaptive Pessimistic Value Iteration (APVI) algorithm and derive the suboptimality upper bound that nearly matches
+
+$$
+O \left( \sum _ { h = 1 } ^ { H } \sum _ { s _ { h } , a _ { h } } { d _ { h } ^ { \pi ^ { \star } } ( s _ { h } , a _ { h } ) \sqrt { \frac { \mathrm { V a r } _ { P _ { s _ { h } , a _ { h } } } ( V _ { h + 1 } ^ { \star } + r _ { h } ) } { d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) } } \sqrt { \frac { 1 } { n } } } \right) .
+$$
+
+In complementary, we also prove a per-instance information-theoretical lower bound under the weak assumption that $d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) > 0$ if $d _ { h } ^ { \pi ^ { \star } } ( s _ { h } , a _ { h } ) > 0$ . Different from the previous minimax lower bounds, the per-instance lower bound (via local minimaxity) is a much stronger criterion as it applies to individual instances separately. Here $\pi ^ { \star }$ is a optimal policy, $\mu$ is the behavior policy and $d _ { h } ^ { \mu }$ is the marginal state-action probability. We call (1) the intrinsic offline reinforcement learning bound since it directly implies all the existing optimal results: minimax rate under uniform data-coverage assumption, horizon-free setting, single policy concentrability, and the tight problem-dependent results. Later, we extend the result to the assumption-free regime (where we make no assumption on $\mu$ ) and obtain the assumption-free intrinsic bound. Due to its generic form, we believe the intrinsic bound could help illuminate what makes a specific problem hard and reveal the fundamental challenges in offline RL.
+
+# 1 Introduction
+
+In offline reinforcement learning (offline RL Levine et al. [2020], Lange et al. [2012]), the goal is to learn a reward-maximizing policy in an unknown environment (Markov Decision Process or MDP) using the historical data coming from a (fixed) behavior policy $\mu$ . Unlike online RL, where the agent can keep interacting with the environment and gain new feedback by exploring unvisited state-action space, offline RL usually populates when such online interplays are expensive or even unethical. Due to its nature of without the access to interact with the MDP model (which causes the distributional mismatches), most of the literature that study the sample complexity / provable efficiency of offline RL (e.g. Le et al. [2019], Chen and Jiang [2019], Xie and Jiang [2020, 2021],
+
+Yin et al. [2021a,b], Ren et al. [2021], Rashidinejad et al. [2021], Xie et al. [2021b]) rely on making different data-coverage assumptions for making the problem learnable and provide the near-optimal worst-case performance bounds that depend on their data-coverage coefficients. Those results are valuable in general as they do not depend on the structure of the particular problem, therefore, remain valid even for pathological MDPs. But is this good enough?
+
+In practice, the empirical performances of offline reinforcement learning (e.g. Gulcehre et al. [2020], Fu et al. [2020, 2021], Janner et al. [2021]) are often far better than what those non-adaptive / problem-independent bounds would indicate. Although empirical evidence can help explain why we may observe better or worse performances on different MDPs, a systematic understanding of what types of decision processes and what kinds of behavior policies are inherently easier or more challenging for offline RL is lacking. Besides, despite the fact that a non-adaptive bound can learn even the pathological examples within the assumption family, there is no guarantee for the instances outside the family. However, practical offline reinforcement learning problems are usually beyond the scope of certain data-coverage assumptions, which limits the applicability of those results. Can we make as few assumptions as possible? Or even more, what can we guarantee when no assumption is made about offline learning?
+
+Those motivate us to derive the provably efficient bounds that are adaptive to the individual instances but only require minimal assumptions so they can be widely applied in most cases. Ideally, such bounds should characterize the system structures of the specific problems, hold even for peculiar instances that do not satisfy the standard data-coverage assumptions, and recover the worst-case guarantees when the assumptions are satisfied. As mentioned in Zanette and Brunskill [2019], a fully adaptive characterization in RL is important as it might bring considerable saving in the time spent designing domain-specific RL solutions and in training a human expert to judge and recognize the complexity of different problems.
+
+# 1.1 Our contribution
+
+In this work, we provide the analysis for the adaptive pessimistic value iteration (APVI) (Algorithm 1) with finite horizon time-inhomogeneous (non-stationary) MDPs and derive a strong adaptive bound that is near-optimal under the weak assumption $d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) > 0$ if $d _ { h } ^ { \pi ^ { \star } } ( s _ { h } , a _ { h } ) > \bar { 0 }$ (Theorem 4.1). Specifically, our bound (quantity (1)) explicitly depends on the marginal importance ratios (between the optimal policy $\pi ^ { \star }$ and the behavior policy $\mu _ { . }$ ) and the per-step conditional variances. In addition, we provide an instance-dependent (local minimax) lower bound (Theorem 4.3) to certify (1) is nearly optimal at the instance level for offline learning and call it the intrinsic offline learning bound. The intrinsic bound has the following consequences.
+
+• In the non-adaptive / worst-case regime (4.1-4.3), the intrinsic bound implies $\widetilde { O } ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } )$ complexity under the uniform data-coverage 2.1, $\tilde { O } ( H ^ { 3 } S C ^ { \star } / \epsilon ^ { 2 } )$ complexity under the single policy concentrability assumption 2.3 and $\widetilde { O } ( H / d _ { m } \epsilon ^ { 2 } )$ complexity when the sum of rewards is bounded by 1. All of those are optimal in their respectively regimes [Yin et al., 2021a, Rashidinejad et al., 2021, Xie et al., 2021b, Ren et al., 2021];
+
+• In the adaptive domain (4.4), the intrinsic bound implies the tight problem-dependent counterpart of Zanette and Brunskill [2019], yields ${ \tilde { O } } ( H ^ { 3 } / n d _ { m } )$ fast convergence in the deterministic systems, has improved complexity in the partially deterministic systems and a family of highly mixing problems, and remains optimal when reducing to the tabular contextual bandits.
+
+Beyond the above, due to the generic form of the intrinsic bound, we could come up with as many problem instances (that are of our interests) as possible and study their properties. In this sense, the intrinsic bound helps illuminate the fundamental nature of offline RL.
+
+Furthermore, as a step towards assumption-free offline reinforcement learning, we build a modified AVPI and obtain an adaptive bound that could characterize the suboptimality gap in the state-action space that is agnostic to the behavior policy (Theorem 5.1). To the best of our knowledge, all of these results are the first of its kinds.
+
+# 1.2 Related work
+
+Finite sample analysis for offline reinforcement learning can be traced back to Szepesvári and Munos [2005], Antos et al. [2008a,b] for the infinite horizon discounted setting via Fitted Q-Iteration (FQI) type function approximation algorithms. [Chen and Jiang, 2019, Le et al., 2019, Xie and Jiang, 2021, 2020] follow this line of research and derive the information-theoretical bounds. Recently, Xie and Jiang [2021] considers the offline RL with only the realizability assumption, Liu et al. [2020], Chang et al. [2021] considers the offline RL without sufficient coverage and Kidambi et al. [2020], Uehara and Sun [2021] uses the model-based approach for addressing offline RL. Under those weak coverage assumption, their finite sample analysis are suboptimal (e.g. in terms of the effective horizon $( 1 - \gamma ) ^ { \bar { - } 1 } .$ ). Recently, Yin et al. [2021a,b], Ren et al. [2021] study the finite horizon case. In the linear MDP case, Jin et al. [2020] studies the pessimistic algorithm for offline policy learning under only the compliance assumption, and, concurrently, Xie et al. [2021a] proposes the general pessimistic function approximation framework with instantiation in linear MDP and Zanette et al. [2021] shows actor-critic style algorithm is near-optimal for linear Bellman complete model. In addition, Wang et al. [2021], Zanette [2021] prove some exponential lower bounds under their linear function approximation assumptions.
+
+Among them, there are a few works that achieve the sample optimality under their respective assumptions. Under the uniform data coverage (minimal state-action probability $d _ { m } > 0$ ), Yin et al. [2021a] first proves the optimal ${ \cal \tilde { O } } ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } )$ complexity in the time-inhomogeneous MDP. Recently, Yin et al. [2021b] designs the offline variance reduction algorithm to achieve the optimal $\tilde { O } ( H ^ { 2 } / d _ { m } \epsilon ^ { 2 } )$ rate for the time-homogeneous case. Under the setting where the total cumulative reward is bounded by 1, Ren et al. [2021] obtains the horizon-free result with $\tilde { O } ( 1 / d _ { m } )$ . More recently, Rashidinejad et al. [2021] considers the single concentrability coefficient $\begin{array} { r } { C ^ { \star } : = \operatorname* { m a x } _ { s , a } d ^ { \pi ^ { \star } } ( s , a ) / d ^ { \mu } ( s , a ) } \end{array}$ and derives the upper bound $\tilde { O } [ ( 1 - \gamma ) ^ { - 5 } S C ^ { \star } / \epsilon ^ { 2 } ]$ in the infinite horizon setting which is recently improved by the concurrent work Xie et al. [2021b]. While those worst-case guarantees are desirable, none of them can explain the hardness of the individual problems.1
+
+# 2 Preliminaries
+
+Episodic non-stationary (time-varying) reinforcement learning. A finite-horizon Markov Decision Process (MDP) is denoted by a tuple $M = ( S , A , P , r , H , d _ { 1 } )$ [Sutton and Barto, 2018], where $s$ is the finite state space and $\mathcal { A }$ is the finite action space with $S : = | S | < \infty , A : = | A | < \infty$ . A nonstationary transition kernel $P _ { h } : S \times \mathcal { A } \times \mathcal { S } \mapsto [ 0 , 1 ]$ maps each state action $\left( s _ { h } , a _ { h } \right)$ to a probability distribution $P _ { h } ( \cdot | s _ { h } , a _ { h } )$ and $P _ { h }$ can be different across the time. Besides, $r : S \times A \mapsto \mathbb { R }$ is the expected instantaneous reward function satisfying $0 \leq r \leq 1$ . $d _ { 1 }$ is the initial state distribution. $H$ is the horizon. A policy $\pi = ( \pi _ { 1 } , \ldots , \pi _ { H } )$ assigns each state $s _ { h } \in S$ a probability distribution over actions according to the map $s _ { h } \mapsto \pi _ { h } ( \cdot | s _ { h } ) \forall h \in [ H ]$ . An MDP together with a policy $\pi$ induce a random trajectory $s _ { 1 } , a _ { 1 } , r _ { 1 } , \ldots , s _ { H } , a _ { H } , r _ { H } , s _ { H + 1 }$ with $s _ { 1 } \sim d _ { 1 } , a _ { h } \sim \pi ( \cdot | s _ { h } ) , s _ { h + 1 } \sim P _ { h } ( \cdot | s _ { h } , a ) , \forall h \in [ H ]$ and $r _ { h }$ is a random realization given the observed $s _ { h } , a _ { h }$ .
+
+$Q$ -values, Bellman (optimality) equations. The value function $V _ { h } ^ { \pi } ( \cdot ) \in \mathbb { R } ^ { S }$ and $\mathbf { Q }$ -value function $Q _ { h } ^ { \pi } ( \cdot , \cdot ) \in \mathbb { R } ^ { S \times A }$ for any policy $\pi$ is defined as: $V _ { h } ^ { \pi } ( s ) ~ = ~ \mathbb { E } _ { \pi } [ \sum _ { t = h } ^ { H } r _ { t } | s _ { h } = s ]$ , $Q _ { h } ^ { \pi } ( s , a ) =$ $\begin{array} { r } { \mathbb { E } _ { \pi } [ \sum _ { t = h } ^ { H } r _ { t } | s _ { h } , a _ { h } \ = \ s , a ] } \end{array}$ , $\forall s , a \in \mathcal { S } , \mathcal { A } , h \in [ H ]$ . The performance is defined as $v ^ { \pi } : =$ $\begin{array} { r } { \mathbb { E } _ { d _ { 1 } } \left[ V _ { 1 } ^ { \pi } \right] = \mathbb { E } _ { \pi , d _ { 1 } } \left[ \sum _ { t = 1 } ^ { H } r _ { t } \right] } \end{array}$ , where we denote $V _ { h } ^ { \pi } , Q _ { h } ^ { \pi }$ as column vectors and $P _ { h } ~ \in ~ \mathbb { R } ^ { S A \times S }$ the transition matrix, then the vector form Bellman (optimality) equations follow $\forall h \in [ H ]$ : $\begin{array} { r } { Q _ { h } ^ { \pi } = r _ { h } + P _ { h } V _ { h + 1 } ^ { \pi } } \end{array}$ , $V _ { h } ^ { \pi } = \mathbb { E } _ { a \sim \pi _ { h } } [ Q _ { h } ^ { \pi } ]$ , $Q _ { h } ^ { \star } = r _ { h } + P _ { h } V _ { h + 1 } ^ { \star }$ , $V _ { h } ^ { \star } = \operatorname* { m a x } _ { a } Q _ { h } ^ { \star } ( \cdot , a )$ .In addition, we denote the per-step marginal state-action occupancy $d _ { h } ^ { \pi } ( s , a )$ as: $d _ { h } ^ { \pi } ( s , a ) : = \mathbb { P } [ s _ { h } =$ $s | s _ { 1 } \sim d _ { 1 } , \pi ] \cdot \pi _ { h } ( a | s )$ , which is the marginal state-action probability at time $h$ .
+
+Offline setting and the goal. The offline RL requires the agent to find a policy $\pi$ such that the performance $v ^ { \pi }$ is maximized, given only the episodic data $\mathcal { D } = \{ ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } , r _ { h } ^ { \tau } , s _ { h + 1 } ^ { \tau } ) \} _ { \tau \in [ n ] } ^ { h \in [ H ] }$ rolled out from some behavior policy $\mu$ . The offline nature requires we cannot change $\mu$ and in particular we do not assume the functional knowledge of $\mu$ . That is to say, given the batch data $\mathcal { D }$ and a targeted accuracy $\epsilon > 0$ , the offline RL seeks to find a policy $\pi _ { \mathrm { a l g } }$ such that $v ^ { \star } - v ^ { \pi _ { \mathrm { a l g } } } \leq \epsilon$ .
+
+# 2.1 Assumptions in offline RL
+
+We revise several types of assumptions proposed by existing studies that can yield provably efficient results. Recall $d _ { h } ^ { \mu } ( s _ { h } , a _ { h } )$ is the marginal state-action probability and $\mu$ is the behavior policy.
+
+Assumption 2.1 (Uniform data coverage [Yin et al., 2021a]). The behavior policy obeys that $d _ { m } : =$ $\begin{array} { r } { \operatorname* { m i n } _ { h , s _ { h } , a _ { h } } d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) > 0 . } \end{array}$ . Here the infimum is over all the states satisfying there exists certain policy so that this state can be reached by the current MDP with this policy.
+
+This is the strongest assumption in offline RL as it requires $\mu$ to explore each state-action pairs with positive probability. Under 2.1, it mostly holds $1 / d _ { m } \ge S A$ . This reveals offline learning is generically harder than the generative model setting [Agarwal et al., 2020] in the statistical sense. On the other hand, this is required for the uniform $O P E$ task in Yin et al. [2021a] as it seeks to simultaneously evaluate all the policies within the policy class and it is in general a harder task than offline learning itself.
+
+Assumption 2.2 (Uniform concentrability Szepesvári and Munos [2005], Chen and Jiang [2019]).   
+For all the policies, $\begin{array} { r } { C _ { \mu } : = \operatorname* { s u p } _ { \pi , h } | | d _ { h } ^ { \pi } ( \cdot , \cdot ) / d _ { h } ^ { \mu } ( \cdot , \cdot ) | | _ { \infty } < \infty . } \end{array}$ .
+
+This is a classical offline RL condition that is commonly assumed in the function approximation scheme (e.g. Fitted Q-Iteration). Qualitatively, this is a uniform data-coverage assumption that is similar to Assumption 2.1, but quantitatively, the coefficient $C _ { \mu }$ can be smaller than $1 / \bar { d _ { m } } \bar { }$ due the $d _ { h } ^ { \pi }$ term in the numerator.
+
+Assumption 2.3 (Liu et al. [2019]). There exists one optimal policy $\pi ^ { \star }$ , s.t. $\forall s _ { h } , a _ { h } \ \in \ S , A ,$ , $d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) \ > \ 0$ if $d _ { h } ^ { \pi ^ { \star } } ( s _ { h } , a _ { h } ) > 0$ . We further denote the trackable set as $\mathcal C _ { h } : = \{ ( s _ { h } , a _ { h } )$ : $d _ { h } ^ { \tilde { \mu } } ( s _ { h } , a _ { h } ) > 0 \}$ .
+
+Assumption 2.3 is (arguably) the weakest assumption needed for accurately learning the optimal value $v ^ { \star }$ and we will use 2.3 for most parts of this paper. It only requires $\mu$ to trace the state-action space of one optimal policy and can be agnostic at other locations. Rashidinejad et al. [2021], Xie et al. [2021b] considers this assumption and provide analysis is based on the single concentrability coefficient $\begin{array} { r } { C ^ { \star } : = \operatorname* { m a x } _ { s , a } { d ^ { \pi ^ { \star } } ( s , a ) } / { d ^ { \mu } ( s , a ) } } \end{array}$ . The dependence on $C ^ { \star }$ makes their result less adaptive since there can be lots of locations that have the ratio ${ d ^ { \pi } } ^ { \star } ( s , a ) / { d ^ { \mu } } ( s , a )$ much smaller than $C ^ { \star }$ . Furthermore, what could we end up with when 2.3 is not met? We will provide our answers in the subsequent sections.
+
+# 3 A warm-up case study: Vanilla Pessimistic Value Iteration
+
+As a step towards the optimal and strong adaptive offline RL bound, we analyze the vanilla pessimistic value iteration (VPVI), a tabular counterpart of pessimistic value iteration (PEVI initiated in Jin et al. [2020]), to understand what is missing for achieving the fully adaptivity. In particular, VPVI relies on the model-based construction.
+
+Model-based Components. Given data n $\begin{array} { r } {  { \mathcal { D } } \ = \ \left\{ \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } , r _ { h } ^ { \tau } , s _ { h + 1 } ^ { \tau } \right) \right\} _ { \tau \in [ n ] } ^ { h \in [ H ] } } \end{array}$ , we denote $n _ { s _ { h } , a _ { h } } : =$ $\begin{array} { r } { \sum _ { \tau = 1 } ^ { n } \mathbf { 1 } [ s _ { h } ^ { \tau } , { a } _ { h } ^ { \tau } \ = \ s _ { h } , { a } _ { h } ] } \end{array}$ be the total counts that vis construct the estimators for $\left( s _ { h } , a _ { h } \right)$ pair at time as: , then we use the $P _ { h }$ $r _ { h }$
+
+$$
+\widehat { P } _ { h } ( s ^ { \prime } | s _ { h } , a _ { h } ) = \frac { \sum _ { \tau = 1 } ^ { n } \mathbf { 1 } [ ( s _ { h + 1 } ^ { \tau } , a _ { h } ^ { \tau } , s _ { h } ^ { \tau } ) = ( s ^ { \prime } , s _ { h } , a _ { h } ) ] } { n _ { s _ { h } , a _ { h } } } , \widehat { r } _ { h } ( s _ { h } , a _ { h } ) = \frac { \sum _ { \tau = 1 } ^ { n } \mathbf { 1 } [ ( a _ { h } ^ { \tau } , s _ { h } ^ { \tau } ) = ( s _ { h } , a _ { h } ) ] \cdot r _ { h } ^ { \tau } } { n _ { s _ { h } , a _ { h } } } ,
+$$
+
+if $n _ { s _ { h } , a _ { h } } > 0$ and $\widehat { P } _ { h } ( s ^ { \prime } | s _ { h } , a _ { h } ) = 1 / S , \widehat { r } _ { h } ( s _ { h } , a _ { h } ) = 0$ if $n _ { s _ { h } , a _ { h } } = 0$ . In particular, we use the word b“vanilla” as it directly mirrors Jin et al. [2020] with a pessimistic penalty of order $O ( H / \sqrt { n _ { s _ { h } , a _ { h } } } )$ . 2 With $\widehat { P } _ { h } , \widehat { r } _ { h }$ in Algorithm 2 (which we defer to Appendix), VPVI guarantees the following:
+
+Theorem 3.1. Under the Assumption 2.3, denote $\begin{array} { r } { \bar { d } _ { m } : = \operatorname* { m i n } _ { h \in [ H ] } \{ d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) : d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) > 0 \} } \end{array}$ . For any $0 < \delta < 1$ , there exists absolute constants $c _ { 0 } , C ^ { \prime } > 0$ , such that when $n > c _ { 0 } \cdot 1 / \bar { d } _ { m } \cdot \iota$ $( \iota = \log ( H S A / \delta ) )$ , with probability $1 - \delta$ , the output policy $\widehat { \pi }$ of VPVI satisfies
+
+$$
+0 \leq v ^ { \star } - v ^ { \widehat { \pi } } \leq C ^ { \prime } H \sum _ { h = 1 } ^ { H } \sum _ { ( s _ { h } , a _ { h } ) \in \mathcal { C } _ { h } } d _ { h } ^ { \pi ^ { \star } } ( s _ { h } , a _ { h } ) \cdot \sqrt { \frac { \iota } { n \cdot d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) } } .
+$$
+
+The full proof can be found in Appendix C. Theorem 3.1 makes some improvements over the existing works. First, it is more adaptive than the results with uniform data-coverage Assumption 2.1 (Yin et al. [2021a], Ren et al. [2021]). In addition, by straightforward calculation (3) can be bounded by $\tilde { O } ( \sqrt { H ^ { 4 } S C ^ { \star } / n } )$ which improves VI-LCB [Rashidinejad et al., 2021] by a factor of $H$ .3 Besides, the analysis of VPVI also improves the direct reduction of PEVI [Jin et al., 2020] in the tabular case by a factor $_ { S A }$ since their $\beta = S A H$ when $d = S A$ .
+
+However, VPVI is not optimal as the dependence on horizon is $H ^ { 4 }$ which does not match the optimal worst case guarantee $H ^ { 3 }$ [Yin et al., 2021a] in the nonstationary setting. Also, the explicit dependence on $H$ in (3) possibly hides some key features of the specific offline RL instances. For example, no improvement can be made if the system has the deterministic transition.
+
+# Algorithm 1 Adaptive (assumption-free) Pessimistic Value Iteration or LCBVI-Bernstein
+
+1: Input: Offline dataset $\boldsymbol { \mathcal { D } } = \{ ( \boldsymbol { s } _ { h } ^ { \tau } , \boldsymbol { a } _ { h } ^ { \tau } , \boldsymbol { r } _ { h } ^ { \tau } , \boldsymbol { s } _ { h + 1 } ^ { \tau } ) \} _ { \tau , h = 1 } ^ { n , H }$ . Set $C _ { 1 } = 2 , C _ { 2 } = 1 4$ , failure probability $\delta$ .   
+2: Initialization: Set $\widehat { V } _ { H + 1 } ( \cdot )  0$ . Set $\iota = \log ( H S A / \delta )$ . (if assumption-free, set $M ^ { \dagger } , \widehat { M } ^ { \dagger }$ as in Section 5.)   
+3: for time $h = H , H - 1 , \ldots , 1 \bullet$ do   
+4: Set $\widehat { Q } _ { h } ( \cdot , \cdot ) \gets \widehat { r } _ { h } ( \cdot , \cdot ) + ( \widehat { P } _ { h } \cdot \widehat { V } _ { h + 1 } ) ( \cdot , \cdot ) \mathrm { ~ } ( \mathrm { u s e } \widehat { r } _ { h } ^ { \dag } + ( \widehat { P } _ { h } ^ { \dag } \cdot \widehat { V } _ { h + 1 } )$ if assumption-free)   
+5: $\forall s _ { h } , a _ { h }$ , set $\begin{array} { r } { \Gamma _ { h } ( s _ { h } , a _ { h } ) = C _ { 1 } \sqrt { \frac { \operatorname { V a r } _ { \widehat { P } _ { s _ { h } , a _ { h } } } ( \widehat { r } _ { h } + \widehat { V } _ { h + 1 } ) \cdot \iota } { { n _ { s _ { h } , a _ { h } } } } } + \frac { C _ { 2 } H \cdot \iota } { n _ { s _ { h } , a _ { h } } } } \end{array}$ if $n _ { s _ { h } , a _ { h } } \ge 1$ , o.w. set to $\frac { C H \iota } { 1 }$ .   
+6: (If assumption-free, use C1qVarPb†sh,ah ( $\begin{array} { r } { C _ { 1 } \sqrt { \mathrm { V a r } _ { \widehat { P } _ { s _ { h } , a _ { h } } ^ { \dagger } } ( \widehat { r } _ { h } ^ { \dagger } + \widehat { V } _ { h + 1 } ) \cdot \iota / n _ { s _ { h } , a _ { h } } } + \frac { C _ { 2 } H \cdot \iota } { n _ { s _ { h } , a _ { h } } } } \end{array}$ C2H·ιn if nsh,ah ≥ 1, o.w. use 0.)   
+7: Set $\widehat { Q } _ { h } ^ { p } ( \cdot , \cdot ) \gets \widehat { Q } _ { h } ( \cdot , \cdot ) - \dot { \Gamma _ { h } } ( \cdot , \cdot )$ . Set $\overline { { { Q } } } _ { h } ( \cdot , \cdot )  \operatorname* { m i n } \{ \widehat { Q } _ { h } ^ { p } ( \cdot , \cdot ) , H - h + 1 \} ^ { + }$ . // Pessmistic update   
+8: $\forall s _ { h }$ , Select $\begin{array} { r } { \widehat \pi _ { h } ( \cdot | s _ { h } ) \gets \operatorname * { a r g m a x } _ { \pi _ { h } } \langle \overline { Q } _ { h } ( s _ { h } , \cdot ) , \pi _ { h } ( \cdot | s _ { h } ) \rangle } \end{array}$ . Set $\widehat { V } _ { h } ( s _ { h } ) \gets \langle \overline { { Q } } _ { h } ( s _ { h } , \cdot ) , \widehat { \pi } _ { h } ( \cdot | s _ { h } ) \rangle$ .   
+9: end for   
+10: Output: $\left\{ \widehat { \pi } _ { h } \right\}$ .
+
+# 4 Intrinsic Offline Reinforcement Learning bound
+
+Now we go deeper to understand what is the more intrinsic characterization for offline reinforcement learning. From the study of VPVI, penalizing the Q-function by $\widetilde { O } ( H / \sqrt { n _ { s _ { h } , a _ { h } } } )$ is crude as it estimates the confidence width of ${ \widehat { Q } } _ { h }$ in Algorithm 2 too conservatively therefore loses the accuracy (the bound is suboptimal). This motivates us to use empirical standard deviation instead to create a more adaptive (and also less conservative) Bernstein-type confidence width as the pessimistic penalty:
+
+$$
+\Gamma _ { h } ( s _ { h } , a _ { h } ) = \widetilde { \mathcal { O } } \bigg [ \sqrt { \frac { \mathrm { V a r } _ { \widetilde { P } _ { s _ { h } , a _ { h } } } ( \widehat { r } _ { h } + \widehat { V } _ { h + 1 } ) } {  { n } _ { s _ { h } , a _ { h } } } } + \frac { H } {  { n } _ { s _ { h } , a _ { h } } } \bigg ] \ ( \mathrm { i f } \ n _ { s _ { h } , a _ { h } } > 0 ) ; \ = \widetilde { \mathcal { O } } ( H ) \ ( \mathrm { i f } \ n _ { s _ { h } , a _ { h } } = 0 ) .
+$$
+
+and update $\widehat { Q } _ { h } \gets \widehat { Q } _ { h } - \Gamma _ { h }$ . On one hand, $\sqrt { \mathrm { V a r } _ { \widehat { P } _ { s _ { h } , a _ { h } } } ( \widehat { r } _ { h } + \widehat { V } _ { h + 1 } ) / n _ { s _ { h } , a _ { h } } }$ is a “less pessimistic” penalty than VPVI due to $\sqrt { \mathrm { V a r } _ { \widehat { P } } ( \widehat { r } _ { h } + \widehat { V } _ { h + 1 } ) } \le H$ and critically this design is more data-adaptive since it holds negative view towards the locations with high uncertainties and recommends the locations that we are confident about, as opposed to the online RL (which encourages exploration in the uncertain locations). Such principles are not reflected by the isotropic design in VPVI. On the other hand, it carries the extremely negative view towards fully agnostic locations ${ \widetilde { O } } ( H )$ which in turn causes the agent unlikely to choose them. We summarized the this adaptive pessimistic value
+
+iteration (APVI) into the Algorithm 1, with $\widehat { P } _ { h } , \widehat { r } _ { h }$ defined in (2). APVI has the following guarantee.   
+bA sketch of the analysis is presented in Section B and Appendix F includes the full proof.
+
+Theorem 4.1 (Intrinsic offline RL bound). Under the Assumption 2.3, denote $\bar { d } _ { m } : =$ $\mathrm { m i n } _ { h \in [ H ] } \{ d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) \ : \ d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) \ > \ 0 \}$ . For any $0 ~ < ~ \delta ~ < ~ 1$ , there exists absolute constants $c _ { 0 } , C ^ { \prime } > 0$ , such that when $n > c _ { 0 } \cdot 1 / \bar { d } _ { m } \cdot \iota ( \iota = \log ( H S A / \delta ) )$ , with probability $1 - \delta$ , the output policy $\widehat { \pi }$ of APVI (Algorithm $I$ ) satisfies $\widetilde O$ hides log factor and higher order terms)
+
+$$
+0 \leq v ^ { \star } - v ^ { \widehat { \pi } } \leq C ^ { \prime } \sum _ { h = 1 } ^ { H } \sum _ { ( s _ { h } , a _ { h } ) \in \mathcal { C } _ { h } } d _ { h } ^ { \pi ^ { \star } } \left( s _ { h } , a _ { h } \right) \cdot \sqrt { \frac { \mathrm { V a r } _ { P _ { s _ { h } , a _ { h } } } \left( r _ { h } + V _ { h + 1 } ^ { \star } \right) \cdot \iota } { n \cdot d _ { h } ^ { \mu } \left( s _ { h } , a _ { h } \right) } } + \widetilde O \left( \frac { H ^ { 3 } } { n \cdot \bar { d } _ { m } } \right)
+$$
+
+Remark 4.2. APVI (Algorithm $I$ ) can also be called LCBVI-Bernstein as it creates the offline counterpart of UCBVI in Azar et al. [2017]. However, to highlight that the resulting bound fully adapts to the specific system structure, we use the word “adaptive” instead.
+
+APVI makes significant improvements in a lot of aspects. First and foremost, the dominate term is fully expressed by the system quantities that admits no explicit dependence on $H , S , A$ . To the best of our knowledge, this is the first offline RL bound that concretely depicts the interrelations within the problem when the problem instance is a tuple $( M , \pi ^ { \star } , \mu )$ : an MDP $M$ (coupled with the optimal policy $\pi ^ { \star }$ ) with the data rolling from an offline logging policy $\mu$ . As we will discuss later, this result indicates (nearly) all the optimal worst-case non-adaptive bounds (and clearly also the VPVI) under their respective regimes / assumptions. Thus, (5) is generic. More interestingly, Theorem 4.1 caters to the specific MDP structures and adaptively yields improved sample complexities (e.g. faster convergence in deterministic systems) that existing works cannot imply. Such features are crucial as it helps us to understand what type of problems are harder / easier than others, and even more, in a quantitative way. Last but not least, to illustrate this bound exhibits the intrinsic nature of offline RL, we prove a per-instance dependent information-theoretical lower bound that shares a similar formulation. The proof of Theorem 4.3 can be found in Appendix G.
+
+Theorem 4.3 (Instance-dependent information theoretical offline lower bound). Denote $\begin{array} { c c c c c c } { { \mathcal G } } & { { : = } } & { { \{ ( \mu , M ) } } & { { : } } & { { \exists \pi ^ { \star } } } & { { s } } \end{array}$ .t. $d _ { h } ^ { \mu } ( s , a ) \quad > \quad 0 \quad i f \quad d _ { h } ^ { \pi ^ { \star } } ( s , a ) \quad > \quad 0 \}$ . Fix an instance $\begin{array} { r l r l r } { { \mathcal { P } } } & { { } = } & { ( \mu , M ) } & { { } \in { } } & { { \mathcal { G } } } \end{array}$ . Let $\mathcal { D }$ consists of $n$ episodes and define $\begin{array} { r l } { \xi } & { { } = } \end{array}$ $\begin{array} { r } { \operatorname* { s u p } _ { h , s _ { h } , a _ { h } , s _ { h + 1 } , d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) \cdot \operatorname { V a r } _ { P _ { s _ { h } , a _ { h } } } ( V _ { h + 1 } ^ { \star } ) > 0 } \frac { P _ { h } ( s _ { h + 1 } | s _ { h } , a _ { h } ) \big ( V _ { h + 1 } ^ { \star } ( s _ { h + 1 } ) - \mathbb { E } _ { P _ { s _ { h } , a _ { h } } } ( V _ { h + 1 } ^ { \star } | ) \big ) } { \sqrt { 2 \cdot d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) \cdot \operatorname { V a r } _ { P _ { s _ { h } , a _ { h } } } ( V _ { h + 1 } ^ { \star } ) } } } \end{array}$ Ph(sh+1|sh,ah)(V ?h+1(sh+1)−EPsh,ah [V ?h+1]) . Let π to be the output of any algorithm. Define the local non-asymptotic minimax risk as
+
+$$
+\mathfrak { R } _ { n } ( \mathcal { P } ) : = \operatorname* { s u p } _ { \mathcal { P } ^ { \prime } \in \mathcal { G } } \operatorname* { i n f } _ { \widehat { \pi } } \operatorname* { m a x } _ { \mathcal { Q } \in \{ \mathcal { P } , \mathcal { P } ^ { \prime } \} } \sqrt { n } \cdot \mathbb { E } _ { \mathcal { Q } } \left[ v ^ { \star } ( \mathcal { Q } ) - v ^ { \widehat { \pi } } \right]
+$$
+
+where $v ^ { \star } ( \mathcal { Q } )$ denotes the optimal value under the instance $\mathcal { Q }$ . Then there exists universal constants $c _ { 0 } , p , C > 0$ , such that if $\begin{array} { r } { { \bf \ddot { \sigma } } n \geq c _ { 0 } H ^ { 6 } \xi ^ { 4 } / ( \sum _ { h = 1 } ^ { H } \sum _ { s _ { h } , a _ { h } } { d _ { h } ^ { \pi } } ^ { \star } ( s _ { h } , a _ { h } ) \sqrt { \frac { \mathrm { V a r } { P _ { s _ { h } , a _ { h } } } ( V _ { h + 1 } ^ { \star } ) } { \zeta \cdot d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) } } ) ^ { 2 } } \end{array}$ , with constant probability $p > 0$ , Then we have (here $\zeta = H / \bar { d } _ { m , }$ ):
+
+$$
+\mathfrak { R } _ { n } ( \mathcal { P } ) \geq C \cdot \sum _ { h = 1 } ^ { H } \sum _ { ( s _ { h } , a _ { h } ) \in \mathcal { C } _ { h } } d _ { h } ^ { \pi ^ { \star } } ( s _ { h } , a _ { h } ) \cdot \sqrt { \frac { \mathrm { V a r } _ { P _ { s _ { h } , a _ { h } } } ( r _ { h } + V _ { h + 1 } ^ { \star } ) } { \zeta \cdot d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) } } ,
+$$
+
+where $\mathcal { P } = ( \mu , M )$ and $M = \left( S , A , P , r , H , d _ { 1 } \right)$ .
+
+The interpretation of Theorem 4.3 is: for any instance $\mathcal { P }$ , learning requires (7) (divided by $1 / \sqrt { n } )$ for any algorithm. Note this notion is significantly stronger than the previous minimax offline lower bounds [Yin et al., 2021a, Rashidinejad et al., 2021, Xie et al., 2021b, Jin et al., 2020] (where they only select a particular family of hard problems), therefore, their lower bounds in general do not hold for individual instances.
+
+The quantity (1) nearly-matches the per-instance lower bound (7) (they deviate by a factor of $\zeta = \hat { H } / \hat { d } _ { m }$ due to the technical reason) and, in addition, we provide a matching minimax lower bound in Appendix H. These results certify Theorem 4.1 is not only adaptive but also near-optimal.
+
+Hence, we call the quantity $\begin{array} { r l } & { \sum _ { h = 1 } ^ { H } \sum _ { ( s _ { h } , a _ { h } ) \in \mathcal { C } _ { h } } { d _ { h } ^ { \pi ^ { \star } } ( s _ { h } , a _ { h } ) \cdot \sqrt { \frac { \operatorname { V a r } _ { P _ { s _ { h } , a _ { h } } } ( r _ { h } + V _ { h + 1 } ^ { \star } ) } { n \cdot d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) } } } } \end{array}$ intrinsic offline reinforcement learning bound. In the sequel, we provide thorough discussions to explain the intrinsic bound embraces the fundamental challenges in offline RL and the strong adaptivity. The detailed technical derivations that are missing in Section 4.1-4.4 are deferred to Appendix I.
+
+![](images/6b83607ca2951ec3cf5d8f65a3f909d38e7e57c936199b24d666e4d9988b9fb3.jpg)  
+Figure 1: A visualization on how intrinsic learning bound subsumes existing best-known results: uniform visitation, single concentrability (partial coverage) and adaptive domain.
+
+# 4.1 Optimality under Uniform data-coverage assumption
+
+Under the uniform exploration Assumption 2.1 with parameter $d _ { m } : = \operatorname* { m i n } _ { h , s _ { h , } a _ { h } } d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) > 0$ Yin et al. [2021a] analyzes the model-based plug-in approach and obtains the optimal sample complexity $\widetilde { O } ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } )$ and shows $\Omega ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } )$ is also the lower bound. Indeed, this rate can ebe directly implied by the intrinsic RL bound via Cauchy inequality and the Sum of Total Variance (Lemma J.6):4
+
+$$
+\begin{array} { r l } & { \displaystyle \sum _ { h = 1 } ^ { H } \langle d _ { h } ^ { \pi ^ { \star } } ( \cdot ) , \sqrt { \frac { \mathrm { V a r } _ { P ( \cdot ) } ( r _ { h } + V _ { h + 1 } ^ { \star } ) } { n \cdot d _ { h } ^ { \mu } ( \cdot ) } } \rangle = \displaystyle \sum _ { h = 1 } ^ { H } \langle \sqrt { d _ { h } ^ { \pi ^ { \star } } ( \cdot ) } , \sqrt { \frac { d _ { h } ^ { \pi ^ { \star } } ( \cdot ) \odot \mathrm { V a r } _ { P ( \cdot ) } ( r _ { h } + V _ { h + 1 } ^ { \star } ) } { n \cdot d _ { m } } } \rangle } \\ & { \displaystyle \leq \displaystyle \sum _ { h = 1 } ^ { H } \left\| \sqrt { d _ { h } ^ { \pi ^ { \star } } ( \cdot ) } \right\| _ { 2 } \left\| \sqrt { \frac { d _ { h } ^ { \pi ^ { \star } } ( \cdot ) \odot \mathrm { V a r } _ { P ( \cdot ) } ( r _ { h } + V _ { h + 1 } ^ { \star } ) } { n \cdot d _ { m } } } \right\| _ { 2 } \leq \sqrt { \frac { H \cdot \mathrm { V a r } _ { \pi ^ { \star } } ( \cdot \sum _ { h = 1 } ^ { H } r _ { h } ) } { n \cdot d _ { m } } } \leq \sqrt { \frac { H ^ { 3 } } { n \cdot d _ { m } } } } \end{array}
+$$
+
+which translates to $\widetilde { O } ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } )$ complexity. Our result maintains the optimal worst-case guarantee when $\mu$ has the uniform data-coverage:
+
+Proposition 4.4. Under Assumption 2.1 and apply Theorem 4.1, APVI achieves the sample complexity of minimax-rate $\widetilde { O } ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } )$ (Theorem 4.1 and Theorem G.2 in Yin et al. [2021a]).
+
+Remark 4.5. We believe if the MDP is time-invariant, then by a modified construction of $\widehat { P }$ , $\widehat { r }$ in (2) our result will imply the minimax-rate of $\widetilde { O } ( H ^ { 2 } / d _ { m } \epsilon ^ { 2 } )$ as achieved in Yin et al. [2021b]. We include this discussion in Appendix I.
+
+# 4.2 Bounded sum of total rewards and the Horizon-Free case
+
+$\begin{array} { r } { r _ { h } \geq 0 , \sum _ { h = 1 } ^ { H } r _ { h } \in [ 0 , 1 ] } \end{array}$ f studies that follow the bounded sum of total rewards assumption: i.e. [Krishnamurthy et al., 2016, Jiang et al., 2017, Zhang et al., 2021]. Such a Jiang and Agarwal [2018]. In offline RL, Ren et al. [2021] derives the nearly horizon-free worst case bound $\widetilde { O } ( \sqrt { 1 / n d _ { m } } )$ for the time-invariant MDPs, under the Assumption 2.1. As a comparison, our Theorem 4.1 achieves the following guarantee for the time-varying (non-stationary) MDPs.
+
+Proposition 4.6. Assume $r _ { h } \ge 0 ,$ , $\textstyle \sum _ { h = 1 } ^ { H } r _ { h } \leq 1$ . Then in the time-varying case AVPI (Theorem 4.1) outputs a policy $\widehat { \pi }$ such that the suboptimality gap $\boldsymbol { v } ^ { \star } - \boldsymbol { v } ^ { \widehat { \pi } }$ is bounded by $\widetilde { O } ( \sqrt { H / n d _ { m } } )$ with high bprobability under the Assumption 2.1.
+
+The derivation is straightforward by using $\begin{array} { r } { \operatorname { V a r } _ { \pi ^ { \star } } ( \sum _ { h = 1 } ^ { H } r _ { h } ) \le 1 } \end{array}$ in (8). This proposition is interesting since it indicates when the MDP is non-stationary, $\widetilde { O } ( H / d _ { m } \epsilon ^ { 2 } )$ is required in the worst case even under $\textstyle \sum _ { h = 1 } ^ { H } r _ { h } \leq 1$ .5 The extra $H$ factor resembles the challenge that we have $H$ transitions $( P _ { 1 } , \dots , P _ { H } )$ to learn, as opposed to the bandit-type $1 / d _ { m } \epsilon ^ { 2 }$ result due to there is only one $P$ throughout (time-invariant). This reveals that one hardness in solving the MDP is in proportion to the number of different transition kernels within the MDP. Such a finding could help researchers understand the special settings like low switching cost in transitions [Bai et al., 2019] or nonstationarity [Cheung et al., 2020].
+
+# 4.3 Optimality with Single Concentrability
+
+In the finite horizon discounted setting, Rashidinejad et al. [2021] proposes the single policy concentrability assumption which is defined as C? := maxh,s,a dh (s,a)dµ(s,a) in the current episodic nonstationary MDP setting. As discussed in Appendix D, their lower bound translates to $\Omega ( { \sqrt { \frac { H ^ { 3 } S C ^ { \star } } { n } } } )$ and their VI-LCB algorithm yields q H5SC?n ) suboptimality gap in H-horizon case. Since single policy concentrability is strictly weaker than its uniform version (Assumption 2.2), we only discuss this set up. In particular, we have the following implication from our Theorem 4.1 (whose derivation can be found in Appendix I):
+
+Proposition 4.7. Let π? be a deterministic policy such that C? := maxh,s,a dπ h (s,a)dµ(s,a) . Then by Theorem 4.1, with high probability the output policy of APVI satisfies the suboptimality gap $\widetilde { O } ( \sqrt { \frac { H ^ { 3 } S C ^ { \star } } { n } } )$ in the time-varying (non-stationary) MDPs.
+
+This can computed similar to (8) except we use $\begin{array} { r } { \frac { \boldsymbol { d } _ { h } ^ { \pi ^ { \star } } ( s , a ) } { \boldsymbol { d } _ { h } ^ { \mu } ( s , a ) } \leq C ^ { \star } } \end{array}$ . Our implication improves the VI-LCB by the factor $H ^ { 2 }$ (in terms of sample complexity) and is optimal (recover the concurrent Xie et al. [2021b]). Qualitatively, single concentrability is the same as Assumption 2.3, but the use of $C ^ { \star }$ makes the bound highly problem independent and limits the adaptivity. Problem dependent bound is a more interesting domain as it tailors to each MDP separately. We discuss it now.
+
+# 4.4 Problem dependent domain
+
+We define the pre-step environmental norm (the finite horizon counterpart of Maillard et al. [2014]) as: $\begin{array} { r } { \mathbb { Q } _ { h } ^ { \star } = \operatorname* { m a x } _ { s _ { h } , a _ { h } } { \operatorname { V a r } _ { P _ { s _ { h } , a _ { h } } } { \left( r _ { h } + V _ { h + 1 } ^ { \star } \right) } } } \end{array}$ for all $h \in [ H ]$ , and relax the total sum of rewards to be bounded by any arbitrary value $\boldsymbol { B }$ (i.e. $\textstyle \sum _ { h = 1 } ^ { H } r _ { h } \leq B )$ , then Theorem 4.1 implies:
+
+Proposition 4.8. Under Assumption 2.1, with high probability, subopmality of AVPI is bounded by
+
+$$
+\operatorname* { m i n } \bigg \{ \widetilde O \big ( \sum _ { h = 1 } ^ { H } \sqrt { \frac { \mathbb { Q } _ { h } ^ { \star } } { n \bar { d } _ { m } } } \big ) , \widetilde O \big ( \sqrt { \frac { H \cdot \mathcal { B } ^ { 2 } } { n \bar { d } _ { m } } } \big ) \bigg \} + \widetilde O ( \frac { H ^ { 3 } } { n \bar { d } _ { m } } ) .
+$$
+
+Such a result mirrors the online version of the tight problem-dependent bound Zanette and Brunskill [2019] but with a more general pre-step environmental norm for the non-stationary MDPs.6 For the problem instances with either small $\boldsymbol { B }$ or small $\mathbb { Q } _ { h } ^ { \star }$ , our result yields much better performances, as discussed in the following.
+
+Deterministic systems. For many practical applications of interest, the systems are equipped with low stochasticity, e.g. robotics, or even deterministic dynamics, e.g. the game of GO. In those scenarios, the agent needs less experience for each state-action therefore the learning procedure could be much faster. In particular, when the system is fully deterministic (in both transitions and rewards) then $\mathbb { Q } _ { h } ^ { \star } = 0$ for all $h$ . This enables a faster convergence rate of order $\frac { H ^ { 3 } } { n \bar { d } _ { m } }$ and significantly improves over the existing non-adaptive results that have order $\scriptstyle { \frac { 1 } { \sqrt { n } } }$ . The convergence rate $\textstyle { \frac { 1 } { n } }$ matches Wen and ) regret into the PAC bound.
+
+Partially deterministic systems. Practical worlds are complicated and we could sometimes have a mixture model which contains both deterministic and stochastic steps. In those scenarios, the main complexity is decided by the number of stochastic stages: suppose there are $t$ stochastic $P _ { h } , r _ { h }$ ’s and $H - t$ deterministic $P _ { h ^ { \prime } } , r _ { h ^ { \prime } }$ ’s, then completing the offline learning guarantees $t \cdot \sqrt { \operatorname* { m a x } Q _ { h } ^ { \star } / n \bar { d } _ { m } }$ suboptimality gap, which could be much smaller than $H \cdot \sqrt { \operatorname* { m a x } Q _ { h } ^ { \star } / n \bar { d } _ { m } }$ when $t \ll H$ .
+
+Fast mixing domains. Consider a class of highly mixing non-stationary MDPs (a variant of Zanette and Brunskill [2018]) that satisfies the transition $P _ { h } ( \cdot | s _ { h } , a _ { h } ) : = \nu _ { h } ( \cdot ) \overline { { } }$ depends on neither the state $s _ { h }$ nor the action $a _ { h }$ . Define $\bar { s } _ { t } : = \arg \operatorname* { m a x } V _ { t } ^ { \star } ( s )$ and $\underline { { s } } _ { t } : = \arg \operatorname* { m a x } V _ { t } ^ { \star } ( s$ . Also, denote $\mathrm { r n g } V _ { h } ^ { \star }$ to be the range of $V _ { h } ^ { \star }$ . In such cases, Bellman optimality equations have the form
+
+$$
+V _ { h } ^ { \star } \left( \bar { s } _ { h } \right) = \operatorname* { m a x } _ { a } \left( r _ { h } \left( \bar { s } _ { h } , a \right) + \nu _ { h } ^ { \top } V _ { h + 1 } ^ { \star } \right) , \ V _ { h } ^ { \star } \left( { { s } _ { h } } \right) = \operatorname* { m a x } _ { a } \left( r _ { h } \left( { { s } _ { h } } , a \right) + \nu _ { h } ^ { \top } V _ { h + 1 } ^ { \star } \right)
+$$
+
+which yield $\begin{array} { r } { \mathfrak { s } \mathrm { \ r n g } { V } _ { h } ^ { \star } = V _ { h } ^ { \star } \left( \bar { s } _ { h } \right) - V _ { h } ^ { \star } \left( \underline { { s } } _ { h } \right) = \operatorname* { m a x } _ { a } r _ { h } \left( \bar { s } _ { h } , a \right) - \operatorname* { m i n } _ { a } r _ { h } \left( \underline { { s } } _ { h } , a \right) \le 1 , } \end{array}$ , and this in turn gives $\mathbb { Q } _ { h } ^ { \star } \le 1 + ( \mathrm { r n g } V _ { h } ^ { \star } ) ^ { 2 } = 2$ . As a result, the suboptimality is bounded by $\widetilde { O } ( \sqrt { H ^ { 2 } / n d _ { m } } )$ in the worst case. This result reveals, although this is a family of stochastic non-stationary MDPs, but it is only as hard as the family of stationary MDPs in the minimax sense $\left( \Omega ( H ^ { 2 } / d _ { m } \epsilon ^ { 2 } ) \right)$ ).
+
+Tabular contextual bandits. Our result also implies $\begin{array} { r l } & { \widetilde { \cal O } ( \sum _ { x _ { 1 } , a _ { 1 } } { d _ { 1 } ^ { \pi ^ { \star } } ( x _ { 1 } , a _ { 1 } ) \sqrt { \frac { \mathrm { V a r } ( r _ { 1 } ) } { n \cdot d _ { 1 } ^ { \mu } ( x _ { 1 } , a _ { 1 } ) } } } ) } \end{array}$ q Var(r1)n·dµ1 (x1,a1) ) gap for the offline tabular contextual bandit problem and improves to $\widetilde { \cal O } ( 1 / n d _ { m } )$ when the reward is deterministic. In either cases, the result is optimal and this is due to: when $r _ { 1 }$ is deterministic, the agent only needs one sample at every location (see Bubeck and Cesa-Bianchi [2012] for a survey).
+
+# 5 Towards Assumption-Free Offline RL
+
+While assumption 2.3 is (arguably) the weakest assumption for correctly learning the optimal value, for the real-world applications even this might not be guaranteed. Can we still learn something meaningful? In this section, we consider this most general setting where the behavior policy $\mu$ can be arbitrary. In this case, $\mu$ might not cover any optimal policy $\pi ^ { \star }$ (i.e. there might be high reward location $( s , a )$ that $\mu$ can never visit, e.g. in the extreme case where a clumsy doctor only uses one treatment all the time), and, irrelevant to the number of episode $n$ , a constant suboptimality gap needs to be suffered. To tackle this problem, we create a fictitious augmented MDP $M ^ { \dagger }$ that can help characterize the discrepancy of the values between the original MDP $M$ and the estimated MDP $\widehat { M } ^ { \dag }$ . In particular, $M ^ { \dagger }$ is negative towards agnostic state-actions $s _ { h } , a _ { h }$ by setting $r _ { h } ^ { \dagger } = 0$ cand transitions to an absorbing state s†h+1.
+
+Pessimistic augmented MDP. $M ^ { \dagger }$ is defined with one extra state $s _ { h } ^ { \dagger }$ for all $h \in \{ 2 , \dots , H + 1 \}$ with the augmented state space $S ^ { \dagger } = S \cup \{ s _ { h } ^ { \dagger } \}$ . The transition and the reward are defined as follows:
+
+$$
+P _ { h } ^ { \dagger } ( \cdot \mid s _ { h } , a _ { h } ) = \left\{ \begin{array} { l l } { P _ { h } ( \cdot \mid s _ { h } , a _ { h } ) , n _ { s _ { h } , a _ { h } } > 0 , } \\ { \delta _ { s _ { h + 1 } ^ { \dagger } } , s _ { h } = s _ { h } ^ { \dagger } \mathrm { o r } n _ { s _ { h } , a _ { h } } = 0 . } \end{array} \right. \quad r ^ { \dagger } ( s _ { h } , a _ { h } ) = \left\{ \begin{array} { l l } { r ( s _ { h } , a _ { h } ) , n _ { s _ { h } , a _ { h } } > 0 , } \\ { 0 , s _ { h } = s _ { h } ^ { \dagger } \mathrm { o r } n _ { s _ { h } , a _ { h } } = 0 . } \end{array} \right.
+$$
+
+here $\delta _ { s }$ is the Dirac measure and we denote $V _ { h } ^ { \dag \pi }$ and $v ^ { \dag \pi }$ to be the values under $M ^ { \dagger }$ . $\widehat { M } ^ { \dagger }$ is the empirical counterpart of $M ^ { \dagger }$ with $\widehat { P }$ , $\widehat { r }$ (the same as (2)) replacing $P , r .$ By Algorithm 1, we have bTheorem 5.1 (Assumption-free offline reinforcement learning). Let us make no assumption for $\mu$ and still denote $\bar { d } _ { m } : = \operatorname * { m i n } _ { h \in [ H ] } \{ d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) : d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) > 0 \}$ . For any $0 < \delta < 1$ , there exists absolute constants $c _ { 0 } , C ^ { \prime } > 0$ , such that when $n > c _ { 0 } \cdot 1 / \bar { d } _ { m } \cdot \iota ( \iota = \log ( H S A / \delta ) )$ , with probability $1 - \delta$ , the output policy $\widehat { \pi }$ of APVI satisfies (recall $\mathcal { C } _ { h } : = \{ ( s _ { h } , a _ { h } ) : d _ { h } ^ { \mu } ( s _ { h } , a _ { h } ) > 0 \}$ )
+
+$$
+v ^ { \star } - v ^ { \widehat { \pi } } \leq \sum _ { h = 2 } ^ { H + 1 } d _ { h } ^ { \dagger \pi ^ { \star } } \left( s _ { h } ^ { \dagger } \right) + C ^ { \prime } \sum _ { h = 1 } ^ { H } \sum _ { ( s _ { h } , a _ { h } ) \in { \mathcal { C } } _ { h } } d _ { h } ^ { \dagger \pi ^ { \star } } \left( s _ { h } , a _ { h } \right) \cdot \sqrt { \frac { \operatorname { V a r } _ { P _ { \hat { s } _ { h } , a _ { h } } ^ { \dagger } } \left( r _ { h } ^ { \dagger } + V _ { h + 1 } ^ { \dagger \pi ^ { \star } } \right) \cdot \iota } { n \cdot d _ { h } ^ { \mu } \left( s _ { h } , a _ { h } \right) } } + \widetilde { O } \left( \frac { H ^ { 3 } } { n \bar { d } _ { m } } \right) ,
+$$
+
+where $d _ { h } ^ { \dagger \pi ^ { \star } } ( s _ { h } , a _ { h } ) \leq d _ { h } ^ { \pi ^ { \star } } ( s _ { h } , a _ { h } ) , V _ { h } ^ { \dagger \pi ^ { \star } } ( s _ { h } ) \leq V _ { h } ^ { \star } ( s _ { h } ) .$ for all $s _ { h } , a _ { h } \in S \times \mathcal { A } ,$ , and for all $h \in [ H ]$ , $\begin{array} { r } { d _ { h } ^ { \dagger \pi ^ { \star } } ( s _ { h } ^ { \dagger } ) = \sum _ { t = 1 } ^ { h - 1 } \sum _ { ( s _ { t } , a _ { t } ) \in \mathcal { S } \times \mathcal { A } \backslash \mathcal { C } _ { t } } d _ { t } ^ { \dagger \pi ^ { \star } } \big ( s _ { t } , a _ { t } \big ) } \end{array}$ . The proof is in Appendix $E$ .
+
+Take-aways of Theorem 5.1. In $M ^ { \dagger }$ , there is no agnostic location any more since the original unknown spaces now all have known deterministic transitions to $s ^ { \dagger }$ in $M ^ { \dagger }$ . At a price, the algorithm has to suffer the constant suboptimality PH+1h=2 $\textstyle \sum _ { h = 2 } ^ { H + 1 } d _ { h } ^ { \dagger \pi ^ { \star } } ( s _ { h } ^ { \dagger } )$ due to no data in the region. The quantity $\textstyle \sum _ { h = 2 } ^ { H + 1 } d _ { h } ^ { \dag \pi ^ { \star } } ( s _ { h } ^ { \dag } )$ helps characterize the hardness when nothing is assumed about $\mu$ : it is always less than $H$ (cannot suffer more than suboptimality); under Assumption 2.1, it is 0 since with high probability (by Chernoff bound) and this causes $\boldsymbol { \mathcal { S } } \times \boldsymbol { \mathcal { A } } \backslash \boldsymbol { \bar { \mathcal { C } } } _ { h } = \boldsymbol { \emptyset }$ ; under Assumption 2.3, it is also 0 and 5.1 reduces to Theorem 4.1 (see Appendix F).
+
+# 5.1 Assumption Free vs Without Great Coverage (Partial Coverage)
+
+Recently there is a surge of studies that aim at weakening the assumptions of provable offline $/$ batch RL. Those learning bounds are derived (mostly) under the insufficient data coverage assumptions. One type of works consider the assumption without great coverage (or partial coverage): Chang et al. [2021], Uehara and Sun [2021] assume $\begin{array} { r } { \operatorname* { m a x } _ { s , a } d ^ { \pi _ { e } } ( s , a ) / \mu ( s , a ) < \infty } \end{array}$ where $\pi _ { e }$ is either an expert policy or a policy of great quality and they further compete against with this policy $\pi _ { e }$ . Those assumptions are similar to 2.3 and therefore are stronger than the assumption-free RL we considered in 5.1.
+
+In addition, there are other studies that apply to the case where $\mu$ can be arbitrary: Liu et al. [2020] considers the behavior policy with insufficient coverage probability $\epsilon _ { \zeta }$ (see their Definition 1), and they end up with the constant suboptimality gap Vmaxζ1−γ (their Theorem 1), when the insufficient coverage probability $\epsilon _ { \zeta } > 0$ , this gap has order $( 1 - \gamma ) ^ { - 2 }$ , which is larger in order than the biggest possible suboptimality gap $( 1 - \gamma ) ^ { - 1 }$ therefore unable to characterize the essential statistical gap over the region that can never be visited by the behavior policy (and this happens similarly in Kidambi et al. [2020], see their Theorem 1); Jin et al. [2020] derive the nice assumption-free result via regularization and their bound can incur $O ( H ^ { 2 } )$ constant gap when there is at least one $\left( s _ { h } , a _ { h } \right)$ cannot be obtained by $\mu$ for all $h \in [ H ]$ (i.e. replacing $n d _ { h } ^ { \bar { \mu } } ( { \bar { s } } _ { h } , a _ { h } )$ by 1 in (3)). The concurrent work Xie et al. [2021a] provides a better characterization (and they call it off-support error) with roughly $\begin{array} { r } { \frac { 1 } { 1 - \gamma } \sum _ { \left( s , a \right) \in S \times A } \left( d _ { \pi } \backslash \nu \right) \left( s , a \right) \left[ \Delta f _ { \pi } ( s , a ) - ( \mathcal { T } ^ { \pi } \Delta f _ { \pi } ) \left( s , \bar { a } \right) \right] } \end{array}$ , however, in the worst case $\Delta f _ { \pi } ( s , a ) - ( T ^ { \pi } \Delta f _ { \pi } ) ( s , a )$ might be large (which depends on the quality (assumption) of the function approximation class).
+
+In contrast, our 1) describes the $\textstyle \sum _ { h = 2 } ^ { H + 1 } d _ { h } ^ { \dag \pi ^ { \star } } ( s _ { h } ^ { \dag } )$ quantity (with p in a more pr $\begin{array} { r } { d _ { h } ^ { \dagger \pi ^ { \star } } ( s _ { h } ^ { \dagger } ) = \sum _ { t = 1 } ^ { h - 1 } \sum _ { ( s _ { t } , a _ { t } ) \in \mathcal { S } \times \mathcal { A } \backslash \mathcal { C } _ { t } } d _ { t } ^ { \dagger \pi ^ { \star } } ( s _ { t } , a _ { t } ) \leq } \end{array}$ into $s ^ { \dagger }$ and it is always bounded between 0 and $H$ . It reduces to 0 when $\pi ^ { \star }$ is covered. The gap is always of order $H$ (as opposed to $O ( H ^ { 2 } ) _ { * }$ ).
+
+# 6 Discussion and Conclusion
+
+This work studies the offline reinforcement learning problem and contributes the intrinsic offline learning bound which is a near-optimal and strong adaptive bound that subsumes existing worst-case bounds under various assumptions. The adaptive characterization of the intrinsic bound abandons the explicit dependence on $H , S , A , C ^ { \star } , d _ { m }$ and helps reveal the fundamental hardness of each individual instances. In this sense, it draws a clearer picture of what offline reinforcement learning looks like and serves as a step towards instance optimality in offline RL.
+
+Nevertheless, it is still unclear whether (5) is optimal over all the instances. For example, for fully deterministic systems, our bound provides a faster convergence $H ^ { 3 } / n \bar { d } _ { m }$ , however, $\bar { H } ^ { 3 }$ might be very suboptimal comparing to algorithms that are designed specifically for deterministic MDPs, since the agent only need to experience each location $( s , a )$ once to fully acquire the dynamic $P ( \cdot | s , a )$ and $r ( s , a )$ . Recently, Xiao et al. [2021] goes beyond the minimax (worst case) optimality and studies the instance optimality behavior for the simplified batch bandit setting. One of their findings is: for “easy enough” tasks, different type of algorithms can be equally good, provably. This seems to suggest instance optimality only matters for problems that are hard to learn. How to formally define the instance optimality metric for different problems remains an open problem and how to design a single algorithm that can achieve optimality for all instances could be challenging (or even infeasible). We leave those as the future works.
+
+# Acknowledgment
+
+The research is partially supported by NSF Awards #2007117 and #2003257. MY would like to thank Chenjun Xiao for bringing up a related literature [Xiao et al., 2021] and Masatoshi Uehara, Yu Bai for helpful suggestions.
+
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parse/train/6OoCDvFV4m/6OoCDvFV4m.md

@@ -0,0 +1,321 @@
+# Deep Residual Learning in Spiking Neural Networks
+
+Wei Fang1,2, Zhaofei $\mathrm { { Y u ^ { 1 , 2 * } } }$ , Yanqi Chen1,2,
+
+Tiejun Huang1,2, Timothée Masquelier3, Yonghong Tian1,2∗
+
+1Department of Computer Science and Technology, Peking University 2Peng Cheng Laboratory, Shenzhen 518055, China 3Centre de Recherche Cerveau et Cognition, UMR5549 CNRS - Univ. Toulouse 3 , Toulouse, France
+
+# Abstract
+
+Deep Spiking Neural Networks (SNNs) present optimization difficulties for gradient-based approaches due to discrete binary activation and complex spatialtemporal dynamics. Considering the huge success of ResNet in deep learning, it would be natural to train deep SNNs with residual learning. Previous Spiking ResNet mimics the standard residual block in ANNs and simply replaces ReLU activation layers with spiking neurons, which suffers the degradation problem and can hardly implement residual learning. In this paper, we propose the spikeelement-wise (SEW) ResNet to realize residual learning in deep SNNs. We prove that the SEW ResNet can easily implement identity mapping and overcome the vanishing/exploding gradient problems of Spiking ResNet. We evaluate our SEW ResNet on ImageNet, DVS Gesture, and CIFAR10-DVS datasets, and show that SEW ResNet outperforms the state-of-the-art directly trained SNNs in both accuracy and time-steps. Moreover, SEW ResNet can achieve higher performance by simply adding more layers, providing a simple method to train deep SNNs. To our best knowledge, this is the first time that directly training deep SNNs with more than 100 layers becomes possible. Our codes are available at https: //github.com/fangwei123456/Spike-Element-Wise-ResNet.
+
+# 1 Introduction
+
+Artificial Neural Networks (ANNs) have achieved great success in many tasks, including image classification [28, 52, 55], object detection [9, 34, 44], machine translation [2], and gaming [37, 51]. One of the critical factors for ANNs’ success is deep learning [29], which uses multi-layers to learn representations of data with multiple levels of abstraction. It has been proved that deeper networks have advantages over shallower networks in computation cost and generalization ability [3]. The function represented by a deep network can require an exponential number of hidden units by a shallow network with one hidden layer [38]. In addition, the depth of the network is closely related to the network’s performance in practical tasks [52, 55, 27, 52]. Nevertheless, recent evidence [13, 53, 14] reveals that with the network depth increasing, the accuracy gets saturated and then degrades rapidly. To solve this degradation problem, residual learning is proposed [14, 15] and the residual structure is widely exploited in “very deep” networks that achieve the leading performance [22, 59, 18, 57].
+
+Spiking Neural Networks (SNNs) are regarded as a potential competitor of ANNs for their high biological plausibility, event-driven property, and low power consumption [45]. Recently, deep learning methods are introduced into SNNs, and deep SNNs have achieved close performance as ANNs in some simple classification datasets [56], but still worse than ANNs in complex tasks, e.g., classifying the ImageNet dataset [47]. To obtain higher performance SNNs, it would be natural to explore deeper network structures like ResNet. Spiking ResNet [25, 60, 21, 17, 49, 12, 30, 64, 48, 42, 43], as the spiking version of ResNet, is proposed by mimicking the residual block in ANNs and replacing ReLU activation layers with spiking neurons. Spiking ResNet converted from ANN achieves state-of-the-art accuracy on nearly all datasets, while the directly trained Spiking ResNet has not been validated to solve the degradation problem.
+
+In this paper, we show that Spiking ResNet is inapplicable to all neuron models to achieve identity mapping. Even if the identity mapping condition is met, Spiking ResNet suffers from the problems of vanishing/exploding gradient. Thus, we propose the Spike-Element-Wise (SEW) ResNet to realize residual learning in SNNs. We prove that the SEW ResNet can easily implement identity mapping and overcome the vanishing/exploding gradient problems at the same time. We evaluate Spiking ResNet and SEW ResNet on both the static ImageNet dataset and the neuromorphic DVS Gesture dataset [1], CIFAR10-DVS dataset [32]. The experiment results are consistent with our analysis, indicating that the deeper Spiking ResNet suffers from the degradation problem — the deeper network has higher training loss than the shallower network, while SEW ResNet can achieve higher performance by simply increasing the network’s depth. Moreover, we show that SEW ResNet outperforms the state-of-the-art directly trained SNNs in both accuracy and time-steps. To the best of our knowledge, this is the first time to explore the directly-trained deep SNNs with more than 100 layers.
+
+# 2 Related Work
+
+# 2.1 Learning Methods of Spiking Neural Networks
+
+ANN to SNN conversion (ANN2SNN) [20, 4, 46, 49, 12, 11, 6, 54, 33] and backpropagation with surrogate gradient [40] are the two main methods to get deep SNNs. The ANN2SNN method firstly trains an ANN with ReLU activation, then converts the ANN to an SNN by replacing ReLU with spiking neurons and adding scaling operations like weight normalization and threshold balancing. Some recent conversion methods have achieved near loss-less accuracy with VGG-16 and ResNet [12, 11, 6, 33]. However, the converted SNN needs a longer time to rival the original ANN in precision as the conversion is based on rate-coding [46], which increases the SNN’s latency and restricts the practical application. The backpropagation methods can be classified into two categories [26]. The method in the first category computes the gradient by unfolding the network over the simulation timesteps [31, 19, 58, 50, 30, 40], which is similar to the idea of backpropagation through time (BPTT). As the gradient with respect to the threshold-triggered firing is non-differentiable, the surrogate gradient is often used. The SNN trained by the surrogate method is not limited to rate-coding, and can also be applied on temporal tasks, e.g., classifying neuromorphic datasets [58, 8, 16]. The second method computes the gradients of the timings of existing spikes with respect to the membrane potential at the spike timing [5, 39, 24, 65, 63].
+
+# 2.2 Spiking Residual Structure
+
+Previous ANN2SNN methods noticed the distinction between plain feedforward ANNs and residual ANNs, and made specific normalization for conversion. Hu et al. [17] were the first to apply the residual structure in ANN2SNN with scaled shortcuts in SNN to match the activations of the original ANN. Sengupta et al. [49] proposed Spike-Norm to balance SNN’s threshold and verified their method by converting VGG and ResNet to SNNs. Existing backpropagation-based methods use nearly the same structure from ResNet. Lee et al. [30] evaluated their custom surrogate methods on shallow ResNets whose depths are no more than ResNet-11. Zheng et al. [64] proposed the threshold-dependent batch normalization (td-BN) to replace naive batch normalization (BN) [23] and successfully trained Spiking ResNet-34 and Spiking ResNet-50 directly with surrogate gradient by adding td-BN in shortcuts.
+
+# 3 Methods
+
+# 3.1 Spiking Neuron Model
+
+The spiking neuron is the fundamental computing unit of SNNs. Similar to Fang et al. [8], we use a unified model to describe the dynamics of all kinds of spiking neurons, which includes the following
+
+![](images/b5ec2614e6cf2120ea600d72dfbd318f14729a9a283e05c316dc83083eca6ae6.jpg)  
+Figure 1: Residual blocks in ResNet, Spiking ResNet and SEW ResNet.
+
+discrete-time equations:
+
+$$
+\begin{array} { l } { { H [ t ] = f ( V [ t - 1 ] , X [ t ] ) , } } \\ { { S [ t ] = \Theta ( H [ t ] - V _ { t h } ) , } } \\ { { V [ t ] = H [ t ] \ ( 1 - S [ t ] ) + V _ { r e s e t } \ S [ t ] , } } \end{array}
+$$
+
+where $X [ t ]$ is the input current at time-step $t , H [ t ]$ and $V [ t ]$ denote the membrane potential after neuronal dynamics and after the trigger of a spike at time-step $t$ , respectively. $V _ { t h }$ is the firing threshold, $\Theta ( x )$ is the Heaviside step function and is defined by $\Theta ( x ) = 1$ for $x \geq 0$ and $\Theta ( x ) = { \bar { 0 } }$ for $x < 0 , S [ t ]$ is the output spike at time-step $t$ , which equals 1 if there is a spike and 0 otherwise. $V _ { r e s e t }$ denotes the reset potential. The function $f ( \cdot )$ in Eq. (1) describes the neuronal dynamics and takes different forms for different spiking neuron models. For example, the function $f ( \cdot )$ for the Integrate-and-Fire (IF) model and Leaky Integrate-and-Fire (LIF) model can be described by Eq. (4) and Eq. (5), respectively.
+
+$$
+\begin{array} { l } { \displaystyle H [ t ] = V [ t - 1 ] + X [ t ] , } \\ { \displaystyle H [ t ] = V [ t - 1 ] + \frac { 1 } { \tau } ( X [ t ] - ( V [ t - 1 ] - V _ { r e s e t } ) ) , } \end{array}
+$$
+
+where $\tau$ represents the membrane time constant. Eq. (2) and Eq. (3) describe the spike generation and resetting processes, which are the same for all kinds of spiking neuron models. In this paper, the surrogate gradient method is used to define $\Theta ^ { \prime } ( x ) \triangleq \sigma ^ { \prime } ( x )$ during error back-propagation, with $\sigma ( x )$ denoting the surrogate function.
+
+# 3.2 Drawbacks of Spiking ResNet
+
+The residual block is the key component of ResNet. Fig. 1(a) shows the basic block in ResNet [14], where $X ^ { l } , Y ^ { l }$ are the input and output of the $l$ -th block in ResNet, Conv is the convolutional layer, BN denotes batch normalization, and ReLU denotes the rectified linear unit activation layer. The basic block of Spiking ResNet used in [64, 17, 30] simply mimics the block in ANNs by replacing ReLU activation layers with spiking neurons (SN), which is illustrated in Fig. 1(b). Here ${ \dot { S } } ^ { l } [ t { \bar { ] } } , O ^ { l } [ t ]$ are the input and output of the $l$ -th block in Spiking ResNet at time-step $t$ . Based on the above definition, we will analyze the drawbacks of Spiking ResNet below.
+
+Spiking ResNet is inapplicable to all neuron models to achieve identity mapping. One of the critical concepts in ResNet is identity mapping. He et al. [14] noted that if the added layers implement the identity mapping, a deeper model should have training error no greater than its shallower counterpart. However, it is unable to train the added layers to implement identity mapping in a feasible time, resulting in deeper models performing worse than shallower models (the degradation problem). To solve this problem, the residual learning is proposed by adding a shortcut connection (shown in Fig. 1(a)). If we use $\mathcal { F } ^ { l }$ to denote the residual mapping, e.g., a stack of two convolutional layers, of the $l$ -th residual block in ResNet and Spiking ResNet, then the residual block in Fig.1(a)
+
+and Fig.1(b) can be formulated as
+
+$$
+\begin{array} { r } { Y ^ { l } = \mathrm { R e L U } ( \mathcal { F } ^ { l } ( X ^ { l } ) + X ^ { l } ) , } \\ { O ^ { l } [ t ] = \mathrm { S N } ( \mathcal { F } ^ { l } ( S ^ { l } [ t ] ) + S ^ { l } [ t ] ) . } \end{array}
+$$
+
+The residual block of Eq. (6) make it easy to implement identity mapping in ANNs. To see this, when $\mathcal { F } ^ { l } ( X ^ { l } ) \equiv 0$ , $Y ^ { l } = \mathrm { R e L U } ( \mathbf { X } ^ { \ l } )$ . In most cases, $X ^ { l }$ is the activation of the previous ReLU layer and $X ^ { l } \ge 0$ . Thus, $Y ^ { l } = \mathrm { R e L U } ( X ^ { l } ) = X ^ { l }$ , which is identity mapping.
+
+Different from ResNet, the residual block in Spiking ResNet (Eq. (7)) restricts the models of spiking neuron to implement identity mapping. When $\dot { \mathcal { F } } ^ { l } ( S ^ { \tilde { l } } [ t ] ) \equiv 0$ , $O ^ { l } [ t ] = \operatorname { S N } ( S ^ { l } [ t ] ) \neq S ^ { l } [ t ]$ . To transmit $S ^ { l } [ t ]$ and make $\mathrm { S N } ( S ^ { l } [ t ] ) = S ^ { l } [ t ]$ , the last spiking neuron (SN) in the $l$ -th residual block needs to fire a spike after receiving a spike, and keep silent after receiving no spike at time-step $t$ . It works for IF neuron described by Eq. (4). Specifically, we can set $0 < V _ { t h } \le 1$ and $V [ t - 1 ] = 0$ to ensure that $X [ t ] = 1$ leads to $H [ t ] \geq V _ { t h }$ , and $X [ t ] \stackrel { \cdot } { = } 0$ leads to $H [ t ] < V _ { t h }$ . However, when considering some spiking neuron models with complex neuronal dynamics, it is hard to achieve $\mathrm { S N } ( S ^ { l } [ t ] ) = S ^ { l } [ t ]$ . For example, the LIF neuron used in [66, 8, 61] considers a learnable membrane time constant $\tau$ , the neuronal dynamics of which can be described with Eq. (5). When $X [ t ] = 1$ and $V [ t - 1 ] = 0$ , $\begin{array} { r } { H [ t ] = \frac { 1 } { \tau } } \end{array}$ . It is difficult to find a firing threshold that ensures $H [ t ] > V _ { t h }$ as $\tau$ is being changed in training by the optimizer.
+
+Spiking ResNet suffers from the problems of vanishing/exploding gradient. Consider a spiking ResNet with $k$ sequential blocks to transmit $S ^ { l } [ t ]$ , and the identity mapping condition is met, e.g., the spiking neurons are the IF neurons with $0 < V _ { t h } \le 1$ , then we have $S ^ { l } [ t ] = S ^ { l + 1 } [ t ] = \ldots =$ $S ^ { l + k - 1 } [ t ] = O ^ { l + k - 1 } [ t ]$ . Denote the $j$ -th element in $S ^ { l } [ t ]$ and $O ^ { l } [ t ]$ as $S _ { j } ^ { l } [ t ]$ and $O _ { j } ^ { l } [ t ]$ respectively, the gradient of the output of the $( l + k - 1 )$ -th residual block with respect to the input of the $l$ -th residual block can be calculated layer by layer:
+
+$$
+\frac { \partial O _ { j } ^ { l + k - 1 } [ t ] } { \partial S _ { j } ^ { l } [ t ] } = \prod _ { i = 0 } ^ { k - 1 } \frac { \partial O _ { j } ^ { l + i } [ t ] } { \partial S _ { j } ^ { l + i } [ t ] } = \prod _ { i = 0 } ^ { k - 1 } \Theta ^ { \prime } ( S _ { j } ^ { l + i } [ t ] - V _ { t h } )  \{ \begin{array} { l l } { 0 , \mathbf { i f } \Theta < \Theta ^ { \prime } ( S _ { j } ^ { l } [ t ] - V _ { t h } ) < 1 } \\ { 1 , \mathbf { i f } \Theta ^ { \prime } ( S _ { j } ^ { l } [ t ] - V _ { t h } ) = 1 } \\ { + \infty , \mathbf { i f } \Theta ^ { \prime } ( S _ { j } ^ { l } [ t ] - V _ { t h } ) > 1 } \end{array}  ,
+$$
+
+where $\Theta ( x )$ is the Heaviside step function and $\Theta ^ { \prime } ( x )$ is defined by the surrogate gradient. The second equality hold as $O _ { j } ^ { l + i } [ t ] = \mathrm { S N } ( S _ { j } ^ { l + i } [ t ] )$ . In view of the fact that $S _ { j } ^ { l } [ t ]$ can only take 0 or 1, $\Theta ^ { \prime } ( S _ { j } ^ { l } [ t ] - V _ { t h } ) = 1$ is not satisfied for commonly used surrogate functions mentioned in [40]. Thus, the vanishing/exploding gradient problems are prone to happen in deeper Spiking ResNet.
+
+Based on the above analysis, we believe that the previous Spiking ResNet ignores the highly nonlinear caused by spiking neurons, and can hardly implement residual learning. Nonetheless, the basic block in Fig. 1(b) is still decent for ANN2SNN with extra normalization [17, 49], as the SNN converted from ANN aims to use firing rates to match the origin ANN’s activations.
+
+# 3.3 Spike-Element-Wise ResNet
+
+Here we propose the Spike-Element-Wise (SEW) residual block to realize the residual learning in SNNs, which can easily implement identity mapping and overcome the vanishing/exploding gradient problems at the same time. As illustrated in Fig. 1(c), the SEW residual block can be formulated as:
+
+$$
+O ^ { l } [ t ] = g ( \mathrm { S N } ( \mathcal { F } ^ { l } ( S ^ { l } [ t ] ) ) , S ^ { l } [ t ] ) = g ( A ^ { l } [ t ] , S ^ { l } [ t ] ) ,
+$$
+
+where $g$ represents an element-wise function with two spikes tensor as inputs. Here we use $A ^ { l } [ t ]$ to denote the residual mapping to be learned as $A ^ { l } [ t ] = \mathrm { S N } ( \mathcal { F } ^ { l } ( S ^ { l } [ t ] ) )$ .
+
+SEW ResNet can easily implement identity mapping. By utilizing the binary property of spikes, we can find different element-wise functions $g$ that satisfy identity mapping (shown in Tab. 1). To be specific, when choosing $A D D$ and IAND as element-wise functions $g$ , identity mapping is achieved by setting $A ^ { l } [ t ] \equiv 0$ , which can be implemented simply by setting the weights and the bias of the last batch normalization layer (BN) in $\mathcal { F } ^ { l }$ to zero. Then we can get $O ^ { l } [ t ] \stackrel { - } { = } g ( A ^ { l } [ t ] , S ^ { l } [ t ] ) = g ( \mathrm { S N } ( 0 ) , S ^ { l } [ t ] ) = g ( 0 , S ^ { l } [ t ] ) \stackrel { - } { = } S ^ { l } [ t ]$ . This is applicable to all neuron models. When using AND as the element-wise function $g$ , we set $A ^ { l } [ t ] \equiv 1$ to get identity mapping. It can be implemented by setting the last BN’s weights to zero and the bias to a large enough constant to cause spikes, e.g., setting the bias as $V _ { t h }$ when the last SN is IF neurons. Then we have $O ^ { l } [ t ] = 1 \land S ^ { l } [ \dot { t } ] = S ^ { l } [ t ]$ . Note that using AND may suffer from the same problem as Spiking ResNet. It is hard to control some spiking neuron models with complex neuronal dynamics to generate spikes at a specified time-step.
+
+Table 1: List of element-wise functions $g$   
+
+<table><tr><td>Name</td><td>Expression of g(A[t],S[t])</td></tr><tr><td>ADD</td><td>A[t]+S[t]</td></tr><tr><td>AND</td><td>A[t]△s[]=A[]·S[]</td></tr><tr><td>IAND</td><td>(-Al[t])△S‘[t]=(1-A[t]）):S[t]</td></tr></table>
+
+![](images/4ac730fadf39ed303c4ebfcef523e701db046c069196c72a74a7658946d3934b.jpg)  
+Figure 2: Downsample blocks in Spiking ResNet and SEW ResNet.
+
+Formulation of downsample block. Remarkably, when the input and output of one block have different dimensions, the shortcut is set as convolutional layers with stride $> 1$ , rather than the identity connection, to perform downsampling. The ResNet and the Spiking ResNet utilize {ConvBN} without ReLU in shortcut (Fig. 2(a)). In contrast, we add a SN in shortcut (Fig. 2(b)).
+
+SEW ResNet can overcome vanishing/exploding gradient. The SEW block is similar to ReLU before addition (RBA) block [15] in ANNs, which can be formulated as
+
+$$
+Y ^ { l } = \mathrm { R e L U } ( \mathcal { F } ^ { l } ( X ^ { l } ) ) + X ^ { l } .
+$$
+
+The RBA block is criticized by He et al. [15] for $X ^ { l + 1 } = Y ^ { l } \geq X ^ { l }$ , which will cause infinite outputs in deep layers. The experiment results in [15] also showed that the performance of the RBA block is worse than the basic block (Fig.1(a)). To some extent, the SEW block is an extension of the RBA block. Note that using $A N D$ and IAND as $g$ will output spikes (i.e. binary tensors), which means that the infinite outputs problem in ANNs will never occur in SNNs with SEW blocks, since all spikes are less or equal than 1. When choosing $A D D$ as $g$ , the infinite outputs problem can be relieved as the output of $k$ sequential SEW blocks will be no larger than $k + 1$ . In addition, a downsample SEW block will regulate the output to be no larger than 2 when $g$ is $A D D$ .
+
+When the identity mapping is implemented, the gradient of the output of the $( l + k - 1 )$ -th SEW block with respect to the input of the $l$ -th SEW block can be calculated layer by layer:
+
+$$
+\frac { \partial O _ { j } ^ { l + k - 1 } [ t ] } { \partial S _ { j } ^ { l } [ t ] } = \prod _ { i = 0 } ^ { k - 1 } \frac { \partial g ( A _ { j } ^ { l + i } [ t ] , S _ { j } ^ { l + i } [ t ] ) } { \partial S _ { j } ^ { l + i } [ t ] } = \left\{ \prod _ { i = 0 } ^ { k - 1 } \frac { \partial ( ( \boldsymbol { 0 } + S _ { j } ^ { l + i } [ t ] ) } { \partial S _ { j } ^ { l + i } [ t ] } , \mathrm { i f ~ } g = A D D \right. \qquad = 1 .
+$$
+
+The second equality holds as identity mapping is achieved by setting $A ^ { l + i } [ t ] \equiv 1$ for $g = A N D$ , and $A ^ { l + i } [ t ] \equiv 0$ for $g = A D D / I A N D$ . Since the gradient in Eq. (11) is a constant, the SEW ResNet can overcome the vanishing/exploding gradient problems.
+
+# 4 Experiments
+
+# 4.1 ImageNet Classification
+
+As the test server of ImageNet 2012 is no longer available, we can not report the actual test accuracy.   
+Instead, we use the accuracy on the validation set as the test accuracy, which is the same as [17, 64].
+
+![](images/6b9086656ad4506861229ab27e5507189e49c21944b1855ee1e6014b42ff6689.jpg)  
+Figure 3: Comparison of the training loss, training accuracy and test accuracy on ImageNet.
+
+<table><tr><td rowspan="2">Network</td><td colspan="2">SEWResNet (ADD)</td><td colspan="2">Spiking ResNet</td></tr><tr><td>Acc@1(%)</td><td>Acc@5(%)</td><td>Acc@1(%)</td><td>Acc@5(%)</td></tr><tr><td>ResNet-18</td><td>63.18</td><td>84.53</td><td>62.32</td><td>84.05</td></tr><tr><td>ResNet-34</td><td>67.04</td><td>87.25</td><td>61.86</td><td>83.69</td></tr><tr><td>ResNet-50</td><td>67.78</td><td>87.52</td><td>57.66</td><td>80.43</td></tr><tr><td>ResNet-101</td><td>68.76</td><td>88.25</td><td>31.79</td><td>54.91</td></tr><tr><td>ResNet-152</td><td>69.26</td><td>88.57</td><td>10.03</td><td>23.57</td></tr></table>
+
+Table 2: Test accuracy on ImageNet.
+
+He et al. [14] evaluated the 18/34/50/101/152-layer ResNets on the ImageNet dataset. For comparison, we consider the SNNs with the same network architectures, except that the basic residual block (Fig.1(a)) is replaced by the spiking basic block (Fig.1(b)) and SEW block (Fig.1(c)) with $g$ as $A D D$ , respectively. We denote the SNN with the basic block as Spiking ResNet and the SNN with the SEW block as SEW ResNet. The IF neuron model is adopted for the static ImageNet dataset. During training on ImageNet, we find that the Spiking ResNet-50/101/152 can not converge unless we use the zero initialization [10], which sets all blocks to be an identity mapping at the start of training. Thus, the results of Spiking ResNet-18/34/50/101/152 reported in this paper are with zero initialization.
+
+Spiking ResNet vs. SEW ResNet. We first evaluate the performance of Spiking ResNet and SEW ResNet. Tab. 2 reports the test accuracy on ImageNet validation. The results show that the deeper 34-layer Spiking ResNet has lower test accuracy than the shallower 18-layer Spiking ResNet. As the layer increases, the test accuracy of Spiking ResNet decreases. To reveal the reason, we compare the training loss, training accuracy, and test accuracy of Spiking ResNet during the training procedure, which is shown in Fig. 3. We can find the degradation problem of the Spiking ResNet — the deeper network has higher training loss than the shallower network. In contrast, the deeper 34-layer SEW ResNet has higher test accuracy than the shallower 18-layer SEW ResNet (shown in Tab. 2). More importantly, it can be found from Fig. 3 that the training loss of our SEW ResNet decreases and the training/test accuracy increases with the increase of depth, which indicates that we can obtain higher performance by simply increasing the network’s depth. All these results imply that the degradation problem is well addressed by SEW ResNet.
+
+Comparisons with State-of-the-art Methods. In Tab. 3, we compare SEW ResNet with previous Spiking ResNets that achieve the best results on ImageNet. To our best knowledge, the SEW ResNet101 and the SEW ResNet-152 are the only SNNs with more than 100 layers to date, and there are no other networks with the same structure to compare. When the network structure is the same, our SEW ResNet outperforms the state-of-the-art accuracy of directly trained Spiking ResNet, even with fewer time-steps $T$ . The accuracy of SEW ResNet-34 is slightly lower than Spiking ResNet-34 (large) with td-BN $( 6 7 . 0 4 \%$ v.s. $6 7 . 0 5 \%$ ), which uses 1.5 times as many simulating time-steps $T$ (6 v.s. 4) and 4 times as many the number of parameters (85.5M v.s. 21.8M), compared with our SEW ResNet. The state-of-the-art ANN2SNN methods [33, 17] have better accuracy than our SEW ResNet, but they respectively use 64 and 87.5 times as many time-steps as ours.
+
+<table><tr><td>Network</td><td>Methods</td><td>Accuracy(%)</td><td>T</td></tr><tr><td>SEW ResNet-34</td><td>Spike-based BP</td><td>67.04</td><td>4</td></tr><tr><td>Spiking ResNet-34(large)† with td-BN [64]</td><td>Spike-based BP</td><td>67.05</td><td>6</td></tr><tr><td>Spiking ResNet-34 with td-BN [64]</td><td>Spike-based BP</td><td>63.72</td><td>6</td></tr><tr><td>Spiking ResNet-34 [12]</td><td>ANN2SNN</td><td>69.89</td><td>4096</td></tr><tr><td>Spiking ResNet-34 [49]</td><td>ANN2SNN</td><td>65.47</td><td>2000</td></tr><tr><td>Spiking ResNet-34 [33]</td><td>ANN2SNN</td><td>74.61</td><td>256</td></tr><tr><td>Spiking ResNet-34 [43]</td><td>ANN2SNN and Spike-based BP</td><td>61.48</td><td>250</td></tr><tr><td>SEW ResNet-50</td><td>Spike-basedBP</td><td>67.78</td><td>4</td></tr><tr><td>Spiking ResNet-50 with td-BN [64]</td><td>Spike-based BP</td><td>64.88</td><td>6</td></tr><tr><td>Spiking ResNet-50 [17]</td><td>ANN2SNN</td><td>72.75</td><td>350</td></tr><tr><td>SEWResNet-101</td><td>Spike-based BP</td><td>68.76</td><td>4</td></tr><tr><td>SEWResNet-152</td><td>Spike-basedBP</td><td>69.26</td><td>4</td></tr></table>
+
+Table 3: Comparison with previous Spiking ResNet on ImageNet. † has the same network structure as the standard Spiking ResNet-34, but uses four times as many the number of convolution kernels.
+
+![](images/083eb00ec81924defce6805a3223ddb459767168347f3b8473343acc55f30ac7.jpg)  
+Figure 4: Firing rates of $A ^ { l }$ in SEW blocks on ImageNet.
+
+Analysis of spiking response of SEW blocks. Fig. 4 shows the firing rates of $A ^ { l }$ in SEW ResNet18/34/50/101/152 on ImageNet. There are 7 blocks in SEW ResNet-18, 15 blocks in SEW ResNet-34 and SEW ResNet-50, 33 blocks in SEW ResNet-101, and 50 blocks in SEW ResNet-152. The downsample SEW blocks are marked by the triangle down symbol $\bigtriangledown$ . As we choose $A D D$ as elementwise functions $g$ , a lower firing rate means that the SEW block gets closer to implementing identity mapping, except for downsample blocks. Note that the shortcuts of downsample blocks are not identity mapping, which is illustrated in Fig. 2(b). Fig. 4 shows that all spiking neurons in SEW blocks have low firing rates, and the spiking neurons in the last two blocks even have firing rates of almost zero. As the time-steps $T$ is 4 and firing rates are no larger than 0.25, all neurons in SEW ResNet-18/34/50 fire on average no more than one spike during the whole simulation. Besides, all firing rates in SEW ResNet-101/152 are not larger than 0.5, indicating that all neurons fire on average not more than two spikes. In general, the firing rates of $A ^ { l }$ in SEW blocks are at a low level, verifying that most SEW blocks act as identity mapping.
+
+Gradients Check on ResNet-152 Structure. Eq. (8) and Eq. (11) analyze the gradients of multiple blocks with identity mapping. To verify that SEW ResNet can overcome vanishing/exploding gradient, we check the gradients of Spiking ResNet-152 and SEW ResNet-152, which are the deepest standard ResNet structure. We consider the same initialization parameters and with/without zero initialization.
+
+As the gradients of SNNs are significantly influenced by firing rates (see Sec.A.4), we analyze the firing rate firstly. Fig. 5(a) shows the initial firing rate of $l$ -th block’s output $O ^ { l }$ . The indexes of downsample blocks are marked by vertical dotted lines. The blocks between two adjacent dotted lines represent the identity mapping areas, and have inputs and outputs with the same shape. When using zero initialization, Spiking ResNet, SEW AND ResNet, SEW IAND ResNet, and SEW ADD ResNet have the same firing rates (green curve), which is the zero init curve. Without zero initialization, the silence problem happens in the SEW AND network (red curve), and is relieved by the SEW IAND network (purple curve). Fig. 5(b) shows the firing rate of $A ^ { l }$ , which represents the output of last SN in $l$ -th block. It can be found that although the firing rate of $O ^ { l }$ in SEW ADD ResNet increases linearly in the identity mapping areas, the last SN in each block still maintains a stable firing rate. Note that when $g$ is $A D D$ , the output of the SEW block is not binary, and the firing rate is actually the mean value. The SNs of SEW IAND ResNet maintain an adequate firing rate and decay slightly with depth (purple curve), while SNs in deep layers of SEW AND ResNet keep silent (orange curve). The silence problem can be explained as follows. When using $A N D$ , $O ^ { l } [ \dot { t } ] = \mathrm { S N } ( \mathcal { F } ^ { l } ( \tilde { O } ^ { l - 1 } [ t ] ) ) \wedge O ^ { l - 1 } [ t ] \leq \dot { O } ^ { l - 1 } [ t ] .$ . Since it is hard to keep $\mathrm { S N } ( \mathcal { F } ^ { l } ( O ^ { l - 1 } [ t ] ) ) \equiv 1$ at each time-step $t$ , the silence problem may frequently happen in SEW ResNet with AND as $g$ . Using IAND as a substitute of $A N D$ can relieve this problem because it is easy to keep $\mathrm { S N } ( \mathcal { F } ^ { l } ( O ^ { l - 1 } [ t ] ) ) \equiv \breve { 0 }$ at each time-step $t$ .
+
+![](images/31ff982586d6ff21bc23fef94a8beedfc34af8d2284a28b9925488a098afa5a2.jpg)  
+Figure 5: The initial firing rates of output $O ^ { l }$ and $A ^ { l }$ in $l$ -th block on 152-layer network.
+
+The surrogate gradient function we used in all experiments is $\begin{array} { r } { \sigma ( x ) = \frac { 1 } { \pi } \arctan ( \frac { \pi } { 2 } \alpha x ) + \frac { 1 } { 2 } } \end{array}$ , thus $\begin{array} { r } { \sigma ^ { \prime } ( x ) = \frac { \alpha } { 2 ( 1 + ( \frac { \pi } { 2 } \alpha x ) ^ { 2 } ) } . } \end{array}$ . When $V _ { t h } = 1 , \alpha = 2$ , the gradient amplitude $\Vert \frac { \partial L } { \partial S ^ { l } } \Vert$ of each block is shown in Fig. 6. Note that $\alpha = 2$ , $\sigma ^ { \prime } ( x ) \leq \sigma ^ { \prime } ( 0 ) = \sigma ^ { \prime } ( 1 - V _ { t h } ) = 1$ and $\sigma ^ { \prime } ( 0 - V _ { t h } ) = 0 . 0 9 2 < 1$ . It can be found that the gradients in Spiking ResNet-152 decay from deeper layers to shallower layers in the identity mapping areas without zero initialization, which is caused by $\begin{array} { r } { \dot { \sigma } ^ { \prime } ( x ) \leq 1 } \end{array}$ . It is worth noting that the decay also happens in Spiking ResNet-152 with zero initialization. The small convex $\Lambda$ near the dotted lines is caused by the vanishing gradients of those $S _ { j } ^ { l } [ t ] = 0$ . After these gradients decays to 0 completely, $\Vert \frac { \partial L } { \partial S ^ { l } } \Vert$ will be a constant because the rest gradients are calculated by $S _ { j } ^ { l } [ t ] = 1$ and $\sigma ^ { \prime } ( 1 - V _ { t h } ) = 1$ , which can also explain why the gradient-index curve is horizontal at some areas. When referring to SEW ResNet-152 with zero initialization, it can be found that all gradient-index curves are similar no matter what $g$ we choose. This is caused by that in the identity mapping areas, $S ^ { l }$ is constant for all index $l$ , and the gradient also becomes a constant as it will not flow through SNs. Without zero initialization, the vanishing gradient happens in the SEW AND ResNet-152, which is caused by the silence problem. The gradients of SEW ADD, IAND network increase slowly when propagating from deeper layers to shallower layers, due to the adequate firing rates shown in Fig. 5.
+
+When $V _ { t h } = 0 . 5 , \alpha = 2$ , $\sigma ^ { \prime } ( 0 - V _ { t h } ) = \sigma ^ { \prime } ( 1 - V _ { t h } ) = 0 . 2 8 8 < 1$ , indicating that transmitting spikes to SNs is prone to causing vanishing gradient, as shown in Fig. 7. With zero initialization, the decay in Spiking ResNet-152 is more serious because gradient from $\mathcal { F } ^ { l }$ can not contribute. The SEW ResNet-152 will not be affected no matter what $g$ we choose. When $V _ { t h } = 1 , \alpha = 3$ , $\sigma ^ { \prime } ( 1 - V _ { t h } ) = 1 . 5 > 1$ , indicating that transmitting spikes to SNs is prone to causing exploding gradient. Fig. 8 shows the gradient in this situation. Same with the reason in Fig. 6, the change of surrogate function will increase gradients of all networks without zero initialization, but not affect SEW ResNet-152 with zero initialization. The Spiking ResNet-152 meets exploding gradient, while this problem in SEW ADD, IAND ResNet-152 is not serious.
+
+# 4.2 DVS Gesture Classification
+
+The origin ResNet, which is designed for classifying the complex ImageNet dataset, is too large for the DVS Gesture dataset. Hence, we design a tiny network named 7B-Net, whose structure is c32k3s1-BN-PLIF-{SEW Block-MPk2s2}\*7-FC11. Here c32k3s1 means the convolutional layer with channels 32, kernel size 3, stride 1. MPk2s2 is the max pooling with kernel size 2, stride 2. The symbol $\{ \} ^ { * } $ denotes seven repeated structure, and PLIF denotes the Parametric Leaky-Integrate-andFire Spiking Neuron with a learnable membrane time constant, which is proposed in [8] and can be described by Eq. (5). See Sec.A.1 for AER data pre-processing details.
+
+![](images/8073af5219e66c4db06c8fadc705ab27b014379a3c1ffa798fba5f23925374a6.jpg)  
+Figure 7: Gradient amplitude $\Big | \Big | \frac { \partial L } { \partial S ^ { l } } \Big | \Big |$ of $l$ -th block when $V _ { t h } = 0 . 5 , \alpha = 2$
+
+Spiking ResNet vs. SEW ResNet. We first compare the performance of SEW ResNet with $A D D$ element-wise function (SEW ADD ResNet) and Spiking ResNet by replacing SEW blocks with basic blocks. As shown in Fig. 9 and Tab. 4, although the training loss of Spiking ResNet (blue curve) is lower than SEW ADD ResNet (orange curve), the test accuracy is lower than SEW ADD ResNet $( 9 0 . 9 7 \%$ v.s. $9 7 . 9 2 \%$ ), which implies that Spiking ResNet is easier to overfit than SEW ADD ResNet.
+
+Evaluation of different element-wise functions and plain block. As the training cost of SNNs on the DVS Gesture dataset is much lower than on ImageNet, we carry out more ablation experiments on the DVS Gesture dataset. We replace SEW blocks with the plain blocks (no shortcut connection) and test the performance. We also evaluate all kinds of element-wise functions $g$ in Tab. 1. Fig. 9 shows the training loss and training/test accuracy on DVS Gesture. The sharp fluctuation during early epochs is caused by the large learning rate (see Sec.A.1). We can find that the training loss is SEW IAND $<$ Spiking ResNe $<$ <SEW ADD $<$ Plain Net<SEW AND. Due to the overfitting problem, a lower loss does not guarantee a higher test accuracy.
+
+Tab. 4 shows the test accuracy of all networks. The SEW ADD ResNet gets the highest accuracy than others.
+
+Comparisons with State-of-the-art Methods. Tab. 5 compares our network with SOTA methods. It can be found that our SEW ResNet outperforms the SOTA works in accuracy, parameter numbers, and simulating time-steps.
+
+Table 4: Test accuracy on DVS Gesture. The networks’ order is ranked by accuracy.   
+
+<table><tr><td>Network</td><td>Element-Wise Function g</td><td>Accuracy(%)</td></tr><tr><td>SEWResNet</td><td>ADD</td><td>97.92</td></tr><tr><td>SEWResNet</td><td>IAND</td><td>95.49</td></tr><tr><td>Plain Net</td><td>1</td><td>91.67</td></tr><tr><td>Spiking ResNet</td><td>1</td><td>90.97</td></tr><tr><td>SEWResNet</td><td>AND</td><td>70.49</td></tr></table>
+
+<table><tr><td>Network</td><td>Accuracy(%)</td><td>Parameters</td><td>T</td></tr><tr><td>c32k3s1-BN-PLIF-{SEW Block (c32) -MPk2s2}*7-FC11 (7B-Net)</td><td>97.92</td><td>0.13M</td><td>16</td></tr><tr><td>{c128k3s1-BN-PLIF-MPk2s2}*5-DP-</td><td>97.57</td><td>1.70M</td><td>20</td></tr><tr><td>FC512-PLIF-DP-FC110-PLIF-APk10s10[8]</td><td></td><td>11.18M</td><td></td></tr><tr><td>SpikingResNet-17with td-BN[64] MPk4-c64k3-LIF-c128k3-LIF-APk2-c128k3-LIF-APk2-FC256-LIF-FC11[16]</td><td>96.87 93.40</td><td>23.23M</td><td>40 60</td></tr></table>
+
+Table 5: Comparison with the state-of-the-art (SOTA) methods on DVS Gesture dataset.
+
+# 4.3 CIFAR10-DVS Classification
+
+We also report SEW ResNet on the CIFAR10-DVS dataset, which is obtained by recording the moving images of the CIFAR-10 dataset on a LCD monitor by a DVS camera. As CIFAR10-DVS is more complicated than DVS Gesture, we use the network structure named Wide-7B-Net, which is similar to 7B-Net but with more channels. The structure of Wide-7B-Net is c64k3s1-BN-PLIF-{SEW Block (c64)-MPk2s2}\*4-c128k3s1-BN-PLIF-{SEW Block (c128)-MPk2s2}\*3-FC10.
+
+![](images/adf33effbd45f2f127239f214ada7e8b50cddf2be808e866b1d288cd44499759.jpg)  
+Figure 8: Gradient amplitude $\Big | \Big | \frac { \partial L } { \partial S ^ { l } } \Big | \Big |$ of $l$ -th block when $V _ { t h } = 1 , \alpha = 3$
+
+![](images/2f787e59c9a353014adc1b3c571413dd5228c89ea36b3ec6a0cecc30ee07713a.jpg)  
+Figure 9: Comparison of the training loss, training accuracy and test accuracy on DVS Gesture dataset.
+
+<table><tr><td>Network</td><td>Accuracy(%)</td><td>Parameters</td><td>T</td></tr><tr><td>c64k3s1-BN-PLIF-{SEWBlock (c64)-MPk2s2}*4-c128k3s1- BN-PLIF-{SEWBlock (c128)-MPk2s2}*3-FCi0 (Wide-7B-Net)</td><td>64.8, 70.2, 74.4</td><td>1.19M</td><td>4,8,16</td></tr><tr><td>{c128k3s1-BN-PLIF-MPk2s2}*4-DP-FC512-PLIF-DP- FC100-PLIF-APk10s10[8]</td><td>74.8</td><td>17.4M</td><td>20</td></tr><tr><td>Spiking ResNet-19 with td-BN [64]</td><td>67.8</td><td>11.18M</td><td>10</td></tr></table>
+
+Table 6: Comparison with the state-of-the-art (SOTA) methods on CIFAR10-DVS dataset.
+
+In Tab.6, we compare SEW ResNet with the previous Spiking ResNet. One can find that our method achieves better performance $7 0 . 2 \%$ v.s. $6 7 . 8 \%$ and fewer time-steps (8 v.s. 10) than the Spiking ResNet [64]. We also compare our method with the state-of-the-art (SOTA) supervised learning methods on CIFAR10-DVS. The accuracy of our Wide-7B-Net is slightly lower than the current SOTA method [8] $7 4 . 4 \%$ v.s. $7 4 . 8 \%$ ), which uses 1.25 times as many simulation time-steps $T$ (20 v.s. 16) and 14.6 times as many the number of parameters (17.4M v.s. 1.19M). Moreover, when reducing $T$ shapely to $T = 4$ , our Wide-7B-Net can still get the accuracy of $6 4 . 8 \%$ .
+
+# 5 Conclusion
+
+In this paper, we analyze the previous Spiking ResNet whose residual block mimics the standard block of ResNet, and find that it can hardly implement identity mapping and suffers from the problems of vanishing/exploding gradient. To solve these problems, we propose the SEW residual block and prove that it can implement the residual learning. The experiment results on ImageNet, DVS Gesture, and CIFAR10-DVS datasets show that our SEW residual block solves the degradation problem, and SEW ResNet can achieve higher accuracy by simply increasing the network’s depth. Our work may shed light on the learning of “very deep” SNNs.
+
+# 6 Acknowledgment
+
+This work is supported by grants from the National Natural Science Foundation of China under contracts No.62027804, No.61825101, and No.62088102.
+
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+    {
+        "type": "text",
+        "text": "Deep Residual Learning in Spiking Neural Networks ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Wei Fang1,2, Zhaofei $\\mathrm { { Y u ^ { 1 , 2 * } } }$ , Yanqi Chen1,2, ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Tiejun Huang1,2, Timothée Masquelier3, Yonghong Tian1,2∗ ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1Department of Computer Science and Technology, Peking University 2Peng Cheng Laboratory, Shenzhen 518055, China 3Centre de Recherche Cerveau et Cognition, UMR5549 CNRS - Univ. Toulouse 3 , Toulouse, France ",
+        "bbox": [
+            178,
+            252,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            329,
+            535,
+            344
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep Spiking Neural Networks (SNNs) present optimization difficulties for gradient-based approaches due to discrete binary activation and complex spatialtemporal dynamics. Considering the huge success of ResNet in deep learning, it would be natural to train deep SNNs with residual learning. Previous Spiking ResNet mimics the standard residual block in ANNs and simply replaces ReLU activation layers with spiking neurons, which suffers the degradation problem and can hardly implement residual learning. In this paper, we propose the spikeelement-wise (SEW) ResNet to realize residual learning in deep SNNs. We prove that the SEW ResNet can easily implement identity mapping and overcome the vanishing/exploding gradient problems of Spiking ResNet. We evaluate our SEW ResNet on ImageNet, DVS Gesture, and CIFAR10-DVS datasets, and show that SEW ResNet outperforms the state-of-the-art directly trained SNNs in both accuracy and time-steps. Moreover, SEW ResNet can achieve higher performance by simply adding more layers, providing a simple method to train deep SNNs. To our best knowledge, this is the first time that directly training deep SNNs with more than 100 layers becomes possible. Our codes are available at https: //github.com/fangwei123456/Spike-Element-Wise-ResNet. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+            618,
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+            636
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Artificial Neural Networks (ANNs) have achieved great success in many tasks, including image classification [28, 52, 55], object detection [9, 34, 44], machine translation [2], and gaming [37, 51]. One of the critical factors for ANNs’ success is deep learning [29], which uses multi-layers to learn representations of data with multiple levels of abstraction. It has been proved that deeper networks have advantages over shallower networks in computation cost and generalization ability [3]. The function represented by a deep network can require an exponential number of hidden units by a shallow network with one hidden layer [38]. In addition, the depth of the network is closely related to the network’s performance in practical tasks [52, 55, 27, 52]. Nevertheless, recent evidence [13, 53, 14] reveals that with the network depth increasing, the accuracy gets saturated and then degrades rapidly. To solve this degradation problem, residual learning is proposed [14, 15] and the residual structure is widely exploited in “very deep” networks that achieve the leading performance [22, 59, 18, 57]. ",
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+        "text": "Spiking Neural Networks (SNNs) are regarded as a potential competitor of ANNs for their high biological plausibility, event-driven property, and low power consumption [45]. Recently, deep learning methods are introduced into SNNs, and deep SNNs have achieved close performance as ANNs in some simple classification datasets [56], but still worse than ANNs in complex tasks, e.g., classifying the ImageNet dataset [47]. To obtain higher performance SNNs, it would be natural to explore deeper network structures like ResNet. Spiking ResNet [25, 60, 21, 17, 49, 12, 30, 64, 48, 42, 43], as the spiking version of ResNet, is proposed by mimicking the residual block in ANNs and replacing ReLU activation layers with spiking neurons. Spiking ResNet converted from ANN achieves state-of-the-art accuracy on nearly all datasets, while the directly trained Spiking ResNet has not been validated to solve the degradation problem. ",
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+    {
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+        "text": "In this paper, we show that Spiking ResNet is inapplicable to all neuron models to achieve identity mapping. Even if the identity mapping condition is met, Spiking ResNet suffers from the problems of vanishing/exploding gradient. Thus, we propose the Spike-Element-Wise (SEW) ResNet to realize residual learning in SNNs. We prove that the SEW ResNet can easily implement identity mapping and overcome the vanishing/exploding gradient problems at the same time. We evaluate Spiking ResNet and SEW ResNet on both the static ImageNet dataset and the neuromorphic DVS Gesture dataset [1], CIFAR10-DVS dataset [32]. The experiment results are consistent with our analysis, indicating that the deeper Spiking ResNet suffers from the degradation problem — the deeper network has higher training loss than the shallower network, while SEW ResNet can achieve higher performance by simply increasing the network’s depth. Moreover, we show that SEW ResNet outperforms the state-of-the-art directly trained SNNs in both accuracy and time-steps. To the best of our knowledge, this is the first time to explore the directly-trained deep SNNs with more than 100 layers. ",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 Learning Methods of Spiking Neural Networks ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "ANN to SNN conversion (ANN2SNN) [20, 4, 46, 49, 12, 11, 6, 54, 33] and backpropagation with surrogate gradient [40] are the two main methods to get deep SNNs. The ANN2SNN method firstly trains an ANN with ReLU activation, then converts the ANN to an SNN by replacing ReLU with spiking neurons and adding scaling operations like weight normalization and threshold balancing. Some recent conversion methods have achieved near loss-less accuracy with VGG-16 and ResNet [12, 11, 6, 33]. However, the converted SNN needs a longer time to rival the original ANN in precision as the conversion is based on rate-coding [46], which increases the SNN’s latency and restricts the practical application. The backpropagation methods can be classified into two categories [26]. The method in the first category computes the gradient by unfolding the network over the simulation timesteps [31, 19, 58, 50, 30, 40], which is similar to the idea of backpropagation through time (BPTT). As the gradient with respect to the threshold-triggered firing is non-differentiable, the surrogate gradient is often used. The SNN trained by the surrogate method is not limited to rate-coding, and can also be applied on temporal tasks, e.g., classifying neuromorphic datasets [58, 8, 16]. The second method computes the gradients of the timings of existing spikes with respect to the membrane potential at the spike timing [5, 39, 24, 65, 63]. ",
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+    {
+        "type": "text",
+        "text": "2.2 Spiking Residual Structure ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Previous ANN2SNN methods noticed the distinction between plain feedforward ANNs and residual ANNs, and made specific normalization for conversion. Hu et al. [17] were the first to apply the residual structure in ANN2SNN with scaled shortcuts in SNN to match the activations of the original ANN. Sengupta et al. [49] proposed Spike-Norm to balance SNN’s threshold and verified their method by converting VGG and ResNet to SNNs. Existing backpropagation-based methods use nearly the same structure from ResNet. Lee et al. [30] evaluated their custom surrogate methods on shallow ResNets whose depths are no more than ResNet-11. Zheng et al. [64] proposed the threshold-dependent batch normalization (td-BN) to replace naive batch normalization (BN) [23] and successfully trained Spiking ResNet-34 and Spiking ResNet-50 directly with surrogate gradient by adding td-BN in shortcuts. ",
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+    {
+        "type": "text",
+        "text": "3 Methods ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 Spiking Neuron Model ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The spiking neuron is the fundamental computing unit of SNNs. Similar to Fang et al. [8], we use a unified model to describe the dynamics of all kinds of spiking neurons, which includes the following ",
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+    {
+        "type": "image",
+        "img_path": "images/b5ec2614e6cf2120ea600d72dfbd318f14729a9a283e05c316dc83083eca6ae6.jpg",
+        "image_caption": [
+            "Figure 1: Residual blocks in ResNet, Spiking ResNet and SEW ResNet. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "discrete-time equations: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/565dedc93f1803e6b581c90e560a7e39b651e59a146d1676bff639584eacd68e.jpg",
+        "text": "$$\n\\begin{array} { l } { { H [ t ] = f ( V [ t - 1 ] , X [ t ] ) , } } \\\\ { { S [ t ] = \\Theta ( H [ t ] - V _ { t h } ) , } } \\\\ { { V [ t ] = H [ t ] \\ ( 1 - S [ t ] ) + V _ { r e s e t } \\ S [ t ] , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $X [ t ]$ is the input current at time-step $t , H [ t ]$ and $V [ t ]$ denote the membrane potential after neuronal dynamics and after the trigger of a spike at time-step $t$ , respectively. $V _ { t h }$ is the firing threshold, $\\Theta ( x )$ is the Heaviside step function and is defined by $\\Theta ( x ) = 1$ for $x \\geq 0$ and $\\Theta ( x ) = { \\bar { 0 } }$ for $x < 0 , S [ t ]$ is the output spike at time-step $t$ , which equals 1 if there is a spike and 0 otherwise. $V _ { r e s e t }$ denotes the reset potential. The function $f ( \\cdot )$ in Eq. (1) describes the neuronal dynamics and takes different forms for different spiking neuron models. For example, the function $f ( \\cdot )$ for the Integrate-and-Fire (IF) model and Leaky Integrate-and-Fire (LIF) model can be described by Eq. (4) and Eq. (5), respectively. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6b68cef925c0ee0d39ee2a6ec70c7ecb5c3bbab8ec9d5b8abc23fb228a5d8e9d.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle H [ t ] = V [ t - 1 ] + X [ t ] , } \\\\ { \\displaystyle H [ t ] = V [ t - 1 ] + \\frac { 1 } { \\tau } ( X [ t ] - ( V [ t - 1 ] - V _ { r e s e t } ) ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\tau$ represents the membrane time constant. Eq. (2) and Eq. (3) describe the spike generation and resetting processes, which are the same for all kinds of spiking neuron models. In this paper, the surrogate gradient method is used to define $\\Theta ^ { \\prime } ( x ) \\triangleq \\sigma ^ { \\prime } ( x )$ during error back-propagation, with $\\sigma ( x )$ denoting the surrogate function. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.2 Drawbacks of Spiking ResNet ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The residual block is the key component of ResNet. Fig. 1(a) shows the basic block in ResNet [14], where $X ^ { l } , Y ^ { l }$ are the input and output of the $l$ -th block in ResNet, Conv is the convolutional layer, BN denotes batch normalization, and ReLU denotes the rectified linear unit activation layer. The basic block of Spiking ResNet used in [64, 17, 30] simply mimics the block in ANNs by replacing ReLU activation layers with spiking neurons (SN), which is illustrated in Fig. 1(b). Here ${ \\dot { S } } ^ { l } [ t { \\bar { ] } } , O ^ { l } [ t ]$ are the input and output of the $l$ -th block in Spiking ResNet at time-step $t$ . Based on the above definition, we will analyze the drawbacks of Spiking ResNet below. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Spiking ResNet is inapplicable to all neuron models to achieve identity mapping. One of the critical concepts in ResNet is identity mapping. He et al. [14] noted that if the added layers implement the identity mapping, a deeper model should have training error no greater than its shallower counterpart. However, it is unable to train the added layers to implement identity mapping in a feasible time, resulting in deeper models performing worse than shallower models (the degradation problem). To solve this problem, the residual learning is proposed by adding a shortcut connection (shown in Fig. 1(a)). If we use $\\mathcal { F } ^ { l }$ to denote the residual mapping, e.g., a stack of two convolutional layers, of the $l$ -th residual block in ResNet and Spiking ResNet, then the residual block in Fig.1(a) ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "and Fig.1(b) can be formulated as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/02155f34d080660276ea3e1f1bf5fdc73c98bd966a3985a3e6d01c85214d2c35.jpg",
+        "text": "$$\n\\begin{array} { r } { Y ^ { l } = \\mathrm { R e L U } ( \\mathcal { F } ^ { l } ( X ^ { l } ) + X ^ { l } ) , } \\\\ { O ^ { l } [ t ] = \\mathrm { S N } ( \\mathcal { F } ^ { l } ( S ^ { l } [ t ] ) + S ^ { l } [ t ] ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The residual block of Eq. (6) make it easy to implement identity mapping in ANNs. To see this, when $\\mathcal { F } ^ { l } ( X ^ { l } ) \\equiv 0$ , $Y ^ { l } = \\mathrm { R e L U } ( \\mathbf { X } ^ { \\ l } )$ . In most cases, $X ^ { l }$ is the activation of the previous ReLU layer and $X ^ { l } \\ge 0$ . Thus, $Y ^ { l } = \\mathrm { R e L U } ( X ^ { l } ) = X ^ { l }$ , which is identity mapping. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Different from ResNet, the residual block in Spiking ResNet (Eq. (7)) restricts the models of spiking neuron to implement identity mapping. When $\\dot { \\mathcal { F } } ^ { l } ( S ^ { \\tilde { l } } [ t ] ) \\equiv 0$ , $O ^ { l } [ t ] = \\operatorname { S N } ( S ^ { l } [ t ] ) \\neq S ^ { l } [ t ]$ . To transmit $S ^ { l } [ t ]$ and make $\\mathrm { S N } ( S ^ { l } [ t ] ) = S ^ { l } [ t ]$ , the last spiking neuron (SN) in the $l$ -th residual block needs to fire a spike after receiving a spike, and keep silent after receiving no spike at time-step $t$ . It works for IF neuron described by Eq. (4). Specifically, we can set $0 < V _ { t h } \\le 1$ and $V [ t - 1 ] = 0$ to ensure that $X [ t ] = 1$ leads to $H [ t ] \\geq V _ { t h }$ , and $X [ t ] \\stackrel { \\cdot } { = } 0$ leads to $H [ t ] < V _ { t h }$ . However, when considering some spiking neuron models with complex neuronal dynamics, it is hard to achieve $\\mathrm { S N } ( S ^ { l } [ t ] ) = S ^ { l } [ t ]$ . For example, the LIF neuron used in [66, 8, 61] considers a learnable membrane time constant $\\tau$ , the neuronal dynamics of which can be described with Eq. (5). When $X [ t ] = 1$ and $V [ t - 1 ] = 0$ , $\\begin{array} { r } { H [ t ] = \\frac { 1 } { \\tau } } \\end{array}$ . It is difficult to find a firing threshold that ensures $H [ t ] > V _ { t h }$ as $\\tau$ is being changed in training by the optimizer. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Spiking ResNet suffers from the problems of vanishing/exploding gradient. Consider a spiking ResNet with $k$ sequential blocks to transmit $S ^ { l } [ t ]$ , and the identity mapping condition is met, e.g., the spiking neurons are the IF neurons with $0 < V _ { t h } \\le 1$ , then we have $S ^ { l } [ t ] = S ^ { l + 1 } [ t ] = \\ldots =$ $S ^ { l + k - 1 } [ t ] = O ^ { l + k - 1 } [ t ]$ . Denote the $j$ -th element in $S ^ { l } [ t ]$ and $O ^ { l } [ t ]$ as $S _ { j } ^ { l } [ t ]$ and $O _ { j } ^ { l } [ t ]$ respectively, the gradient of the output of the $( l + k - 1 )$ -th residual block with respect to the input of the $l$ -th residual block can be calculated layer by layer: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4bbc968c9c55af91d63a5eb4af0b3b4eab4781d22b8e29c8a0992642d796bd7d.jpg",
+        "text": "$$\n\\frac { \\partial O _ { j } ^ { l + k - 1 } [ t ] } { \\partial S _ { j } ^ { l } [ t ] } = \\prod _ { i = 0 } ^ { k - 1 } \\frac { \\partial O _ { j } ^ { l + i } [ t ] } { \\partial S _ { j } ^ { l + i } [ t ] } = \\prod _ { i = 0 } ^ { k - 1 } \\Theta ^ { \\prime } ( S _ { j } ^ { l + i } [ t ] - V _ { t h } )  \\{ \\begin{array} { l l } { 0 , \\mathbf { i f } \\Theta < \\Theta ^ { \\prime } ( S _ { j } ^ { l } [ t ] - V _ { t h } ) < 1 } \\\\ { 1 , \\mathbf { i f } \\Theta ^ { \\prime } ( S _ { j } ^ { l } [ t ] - V _ { t h } ) = 1 } \\\\ { + \\infty , \\mathbf { i f } \\Theta ^ { \\prime } ( S _ { j } ^ { l } [ t ] - V _ { t h } ) > 1 } \\end{array}  ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\Theta ( x )$ is the Heaviside step function and $\\Theta ^ { \\prime } ( x )$ is defined by the surrogate gradient. The second equality hold as $O _ { j } ^ { l + i } [ t ] = \\mathrm { S N } ( S _ { j } ^ { l + i } [ t ] )$ . In view of the fact that $S _ { j } ^ { l } [ t ]$ can only take 0 or 1, $\\Theta ^ { \\prime } ( S _ { j } ^ { l } [ t ] - V _ { t h } ) = 1$ is not satisfied for commonly used surrogate functions mentioned in [40]. Thus, the vanishing/exploding gradient problems are prone to happen in deeper Spiking ResNet. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Based on the above analysis, we believe that the previous Spiking ResNet ignores the highly nonlinear caused by spiking neurons, and can hardly implement residual learning. Nonetheless, the basic block in Fig. 1(b) is still decent for ANN2SNN with extra normalization [17, 49], as the SNN converted from ANN aims to use firing rates to match the origin ANN’s activations. ",
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+    {
+        "type": "text",
+        "text": "3.3 Spike-Element-Wise ResNet ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Here we propose the Spike-Element-Wise (SEW) residual block to realize the residual learning in SNNs, which can easily implement identity mapping and overcome the vanishing/exploding gradient problems at the same time. As illustrated in Fig. 1(c), the SEW residual block can be formulated as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1f07c471955c609b578b1c1520c9c76f4bec43129ceeeeb2060a5cc2255e1e47.jpg",
+        "text": "$$\nO ^ { l } [ t ] = g ( \\mathrm { S N } ( \\mathcal { F } ^ { l } ( S ^ { l } [ t ] ) ) , S ^ { l } [ t ] ) = g ( A ^ { l } [ t ] , S ^ { l } [ t ] ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $g$ represents an element-wise function with two spikes tensor as inputs. Here we use $A ^ { l } [ t ]$ to denote the residual mapping to be learned as $A ^ { l } [ t ] = \\mathrm { S N } ( \\mathcal { F } ^ { l } ( S ^ { l } [ t ] ) )$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "SEW ResNet can easily implement identity mapping. By utilizing the binary property of spikes, we can find different element-wise functions $g$ that satisfy identity mapping (shown in Tab. 1). To be specific, when choosing $A D D$ and IAND as element-wise functions $g$ , identity mapping is achieved by setting $A ^ { l } [ t ] \\equiv 0$ , which can be implemented simply by setting the weights and the bias of the last batch normalization layer (BN) in $\\mathcal { F } ^ { l }$ to zero. Then we can get $O ^ { l } [ t ] \\stackrel { - } { = } g ( A ^ { l } [ t ] , S ^ { l } [ t ] ) = g ( \\mathrm { S N } ( 0 ) , S ^ { l } [ t ] ) = g ( 0 , S ^ { l } [ t ] ) \\stackrel { - } { = } S ^ { l } [ t ]$ . This is applicable to all neuron models. When using AND as the element-wise function $g$ , we set $A ^ { l } [ t ] \\equiv 1$ to get identity mapping. It can be implemented by setting the last BN’s weights to zero and the bias to a large enough constant to cause spikes, e.g., setting the bias as $V _ { t h }$ when the last SN is IF neurons. Then we have $O ^ { l } [ t ] = 1 \\land S ^ { l } [ \\dot { t } ] = S ^ { l } [ t ]$ . Note that using AND may suffer from the same problem as Spiking ResNet. It is hard to control some spiking neuron models with complex neuronal dynamics to generate spikes at a specified time-step. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/de7f1a69c0ae6816b2bd0bc7d648d7cd32c58824e51c826ae9be24398bf66ba1.jpg",
+        "table_caption": [
+            "Table 1: List of element-wise functions $g$ "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Name</td><td>Expression of g(A[t],S[t])</td></tr><tr><td>ADD</td><td>A[t]+S[t]</td></tr><tr><td>AND</td><td>A[t]△s[]=A[]·S[]</td></tr><tr><td>IAND</td><td>(-Al[t])△S‘[t]=(1-A[t]）):S[t]</td></tr></table>",
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/4ac730fadf39ed303c4ebfcef523e701db046c069196c72a74a7658946d3934b.jpg",
+        "image_caption": [
+            "Figure 2: Downsample blocks in Spiking ResNet and SEW ResNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Formulation of downsample block. Remarkably, when the input and output of one block have different dimensions, the shortcut is set as convolutional layers with stride $> 1$ , rather than the identity connection, to perform downsampling. The ResNet and the Spiking ResNet utilize {ConvBN} without ReLU in shortcut (Fig. 2(a)). In contrast, we add a SN in shortcut (Fig. 2(b)). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "SEW ResNet can overcome vanishing/exploding gradient. The SEW block is similar to ReLU before addition (RBA) block [15] in ANNs, which can be formulated as ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/01b7af130788bfb2f927ecead13361a66ad7ac34499b0418e0d114a11e8cb8ea.jpg",
+        "text": "$$\nY ^ { l } = \\mathrm { R e L U } ( \\mathcal { F } ^ { l } ( X ^ { l } ) ) + X ^ { l } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The RBA block is criticized by He et al. [15] for $X ^ { l + 1 } = Y ^ { l } \\geq X ^ { l }$ , which will cause infinite outputs in deep layers. The experiment results in [15] also showed that the performance of the RBA block is worse than the basic block (Fig.1(a)). To some extent, the SEW block is an extension of the RBA block. Note that using $A N D$ and IAND as $g$ will output spikes (i.e. binary tensors), which means that the infinite outputs problem in ANNs will never occur in SNNs with SEW blocks, since all spikes are less or equal than 1. When choosing $A D D$ as $g$ , the infinite outputs problem can be relieved as the output of $k$ sequential SEW blocks will be no larger than $k + 1$ . In addition, a downsample SEW block will regulate the output to be no larger than 2 when $g$ is $A D D$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "When the identity mapping is implemented, the gradient of the output of the $( l + k - 1 )$ -th SEW block with respect to the input of the $l$ -th SEW block can be calculated layer by layer: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/95b89b3ab00baa6e3a12ad6ffb12cd36228ed1cc68eacdb0a71016383bb9dc73.jpg",
+        "text": "$$\n\\frac { \\partial O _ { j } ^ { l + k - 1 } [ t ] } { \\partial S _ { j } ^ { l } [ t ] } = \\prod _ { i = 0 } ^ { k - 1 } \\frac { \\partial g ( A _ { j } ^ { l + i } [ t ] , S _ { j } ^ { l + i } [ t ] ) } { \\partial S _ { j } ^ { l + i } [ t ] } = \\left\\{ \\prod _ { i = 0 } ^ { k - 1 } \\frac { \\partial ( ( \\boldsymbol { 0 } + S _ { j } ^ { l + i } [ t ] ) } { \\partial S _ { j } ^ { l + i } [ t ] } , \\mathrm { i f ~ } g = A D D \\right. \\qquad = 1 .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "The second equality holds as identity mapping is achieved by setting $A ^ { l + i } [ t ] \\equiv 1$ for $g = A N D$ , and $A ^ { l + i } [ t ] \\equiv 0$ for $g = A D D / I A N D$ . Since the gradient in Eq. (11) is a constant, the SEW ResNet can overcome the vanishing/exploding gradient problems. ",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "4.1 ImageNet Classification ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "As the test server of ImageNet 2012 is no longer available, we can not report the actual test accuracy.   \nInstead, we use the accuracy on the validation set as the test accuracy, which is the same as [17, 64]. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/6b9086656ad4506861229ab27e5507189e49c21944b1855ee1e6014b42ff6689.jpg",
+        "image_caption": [
+            "Figure 3: Comparison of the training loss, training accuracy and test accuracy on ImageNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/2d5bf166dc121d1a5cb2ae567f68069f3f72a75191027e46922bb60ccd247e6c.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: Test accuracy on ImageNet. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Network</td><td colspan=\"2\">SEWResNet (ADD)</td><td colspan=\"2\">Spiking ResNet</td></tr><tr><td>Acc@1(%)</td><td>Acc@5(%)</td><td>Acc@1(%)</td><td>Acc@5(%)</td></tr><tr><td>ResNet-18</td><td>63.18</td><td>84.53</td><td>62.32</td><td>84.05</td></tr><tr><td>ResNet-34</td><td>67.04</td><td>87.25</td><td>61.86</td><td>83.69</td></tr><tr><td>ResNet-50</td><td>67.78</td><td>87.52</td><td>57.66</td><td>80.43</td></tr><tr><td>ResNet-101</td><td>68.76</td><td>88.25</td><td>31.79</td><td>54.91</td></tr><tr><td>ResNet-152</td><td>69.26</td><td>88.57</td><td>10.03</td><td>23.57</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "He et al. [14] evaluated the 18/34/50/101/152-layer ResNets on the ImageNet dataset. For comparison, we consider the SNNs with the same network architectures, except that the basic residual block (Fig.1(a)) is replaced by the spiking basic block (Fig.1(b)) and SEW block (Fig.1(c)) with $g$ as $A D D$ , respectively. We denote the SNN with the basic block as Spiking ResNet and the SNN with the SEW block as SEW ResNet. The IF neuron model is adopted for the static ImageNet dataset. During training on ImageNet, we find that the Spiking ResNet-50/101/152 can not converge unless we use the zero initialization [10], which sets all blocks to be an identity mapping at the start of training. Thus, the results of Spiking ResNet-18/34/50/101/152 reported in this paper are with zero initialization. ",
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+    {
+        "type": "text",
+        "text": "Spiking ResNet vs. SEW ResNet. We first evaluate the performance of Spiking ResNet and SEW ResNet. Tab. 2 reports the test accuracy on ImageNet validation. The results show that the deeper 34-layer Spiking ResNet has lower test accuracy than the shallower 18-layer Spiking ResNet. As the layer increases, the test accuracy of Spiking ResNet decreases. To reveal the reason, we compare the training loss, training accuracy, and test accuracy of Spiking ResNet during the training procedure, which is shown in Fig. 3. We can find the degradation problem of the Spiking ResNet — the deeper network has higher training loss than the shallower network. In contrast, the deeper 34-layer SEW ResNet has higher test accuracy than the shallower 18-layer SEW ResNet (shown in Tab. 2). More importantly, it can be found from Fig. 3 that the training loss of our SEW ResNet decreases and the training/test accuracy increases with the increase of depth, which indicates that we can obtain higher performance by simply increasing the network’s depth. All these results imply that the degradation problem is well addressed by SEW ResNet. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Comparisons with State-of-the-art Methods. In Tab. 3, we compare SEW ResNet with previous Spiking ResNets that achieve the best results on ImageNet. To our best knowledge, the SEW ResNet101 and the SEW ResNet-152 are the only SNNs with more than 100 layers to date, and there are no other networks with the same structure to compare. When the network structure is the same, our SEW ResNet outperforms the state-of-the-art accuracy of directly trained Spiking ResNet, even with fewer time-steps $T$ . The accuracy of SEW ResNet-34 is slightly lower than Spiking ResNet-34 (large) with td-BN $( 6 7 . 0 4 \\%$ v.s. $6 7 . 0 5 \\%$ ), which uses 1.5 times as many simulating time-steps $T$ (6 v.s. 4) and 4 times as many the number of parameters (85.5M v.s. 21.8M), compared with our SEW ResNet. The state-of-the-art ANN2SNN methods [33, 17] have better accuracy than our SEW ResNet, but they respectively use 64 and 87.5 times as many time-steps as ours. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/5d23e3a5b8276d1a9f2049d4c1b45f454e8a11c85505d6c3bd5759e22d56e2e4.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 3: Comparison with previous Spiking ResNet on ImageNet. † has the same network structure as the standard Spiking ResNet-34, but uses four times as many the number of convolution kernels. "
+        ],
+        "table_body": "<table><tr><td>Network</td><td>Methods</td><td>Accuracy(%)</td><td>T</td></tr><tr><td>SEW ResNet-34</td><td>Spike-based BP</td><td>67.04</td><td>4</td></tr><tr><td>Spiking ResNet-34(large)† with td-BN [64]</td><td>Spike-based BP</td><td>67.05</td><td>6</td></tr><tr><td>Spiking ResNet-34 with td-BN [64]</td><td>Spike-based BP</td><td>63.72</td><td>6</td></tr><tr><td>Spiking ResNet-34 [12]</td><td>ANN2SNN</td><td>69.89</td><td>4096</td></tr><tr><td>Spiking ResNet-34 [49]</td><td>ANN2SNN</td><td>65.47</td><td>2000</td></tr><tr><td>Spiking ResNet-34 [33]</td><td>ANN2SNN</td><td>74.61</td><td>256</td></tr><tr><td>Spiking ResNet-34 [43]</td><td>ANN2SNN and Spike-based BP</td><td>61.48</td><td>250</td></tr><tr><td>SEW ResNet-50</td><td>Spike-basedBP</td><td>67.78</td><td>4</td></tr><tr><td>Spiking ResNet-50 with td-BN [64]</td><td>Spike-based BP</td><td>64.88</td><td>6</td></tr><tr><td>Spiking ResNet-50 [17]</td><td>ANN2SNN</td><td>72.75</td><td>350</td></tr><tr><td>SEWResNet-101</td><td>Spike-based BP</td><td>68.76</td><td>4</td></tr><tr><td>SEWResNet-152</td><td>Spike-basedBP</td><td>69.26</td><td>4</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/083eb00ec81924defce6805a3223ddb459767168347f3b8473343acc55f30ac7.jpg",
+        "image_caption": [
+            "Figure 4: Firing rates of $A ^ { l }$ in SEW blocks on ImageNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Analysis of spiking response of SEW blocks. Fig. 4 shows the firing rates of $A ^ { l }$ in SEW ResNet18/34/50/101/152 on ImageNet. There are 7 blocks in SEW ResNet-18, 15 blocks in SEW ResNet-34 and SEW ResNet-50, 33 blocks in SEW ResNet-101, and 50 blocks in SEW ResNet-152. The downsample SEW blocks are marked by the triangle down symbol $\\bigtriangledown$ . As we choose $A D D$ as elementwise functions $g$ , a lower firing rate means that the SEW block gets closer to implementing identity mapping, except for downsample blocks. Note that the shortcuts of downsample blocks are not identity mapping, which is illustrated in Fig. 2(b). Fig. 4 shows that all spiking neurons in SEW blocks have low firing rates, and the spiking neurons in the last two blocks even have firing rates of almost zero. As the time-steps $T$ is 4 and firing rates are no larger than 0.25, all neurons in SEW ResNet-18/34/50 fire on average no more than one spike during the whole simulation. Besides, all firing rates in SEW ResNet-101/152 are not larger than 0.5, indicating that all neurons fire on average not more than two spikes. In general, the firing rates of $A ^ { l }$ in SEW blocks are at a low level, verifying that most SEW blocks act as identity mapping. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Gradients Check on ResNet-152 Structure. Eq. (8) and Eq. (11) analyze the gradients of multiple blocks with identity mapping. To verify that SEW ResNet can overcome vanishing/exploding gradient, we check the gradients of Spiking ResNet-152 and SEW ResNet-152, which are the deepest standard ResNet structure. We consider the same initialization parameters and with/without zero initialization. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As the gradients of SNNs are significantly influenced by firing rates (see Sec.A.4), we analyze the firing rate firstly. Fig. 5(a) shows the initial firing rate of $l$ -th block’s output $O ^ { l }$ . The indexes of downsample blocks are marked by vertical dotted lines. The blocks between two adjacent dotted lines represent the identity mapping areas, and have inputs and outputs with the same shape. When using zero initialization, Spiking ResNet, SEW AND ResNet, SEW IAND ResNet, and SEW ADD ResNet have the same firing rates (green curve), which is the zero init curve. Without zero initialization, the silence problem happens in the SEW AND network (red curve), and is relieved by the SEW IAND network (purple curve). Fig. 5(b) shows the firing rate of $A ^ { l }$ , which represents the output of last SN in $l$ -th block. It can be found that although the firing rate of $O ^ { l }$ in SEW ADD ResNet increases linearly in the identity mapping areas, the last SN in each block still maintains a stable firing rate. Note that when $g$ is $A D D$ , the output of the SEW block is not binary, and the firing rate is actually the mean value. The SNs of SEW IAND ResNet maintain an adequate firing rate and decay slightly with depth (purple curve), while SNs in deep layers of SEW AND ResNet keep silent (orange curve). The silence problem can be explained as follows. When using $A N D$ , $O ^ { l } [ \\dot { t } ] = \\mathrm { S N } ( \\mathcal { F } ^ { l } ( \\tilde { O } ^ { l - 1 } [ t ] ) ) \\wedge O ^ { l - 1 } [ t ] \\leq \\dot { O } ^ { l - 1 } [ t ] .$ . Since it is hard to keep $\\mathrm { S N } ( \\mathcal { F } ^ { l } ( O ^ { l - 1 } [ t ] ) ) \\equiv 1$ at each time-step $t$ , the silence problem may frequently happen in SEW ResNet with AND as $g$ . Using IAND as a substitute of $A N D$ can relieve this problem because it is easy to keep $\\mathrm { S N } ( \\mathcal { F } ^ { l } ( O ^ { l - 1 } [ t ] ) ) \\equiv \\breve { 0 }$ at each time-step $t$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/31ff982586d6ff21bc23fef94a8beedfc34af8d2284a28b9925488a098afa5a2.jpg",
+        "image_caption": [
+            "Figure 5: The initial firing rates of output $O ^ { l }$ and $A ^ { l }$ in $l$ -th block on 152-layer network. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The surrogate gradient function we used in all experiments is $\\begin{array} { r } { \\sigma ( x ) = \\frac { 1 } { \\pi } \\arctan ( \\frac { \\pi } { 2 } \\alpha x ) + \\frac { 1 } { 2 } } \\end{array}$ , thus $\\begin{array} { r } { \\sigma ^ { \\prime } ( x ) = \\frac { \\alpha } { 2 ( 1 + ( \\frac { \\pi } { 2 } \\alpha x ) ^ { 2 } ) } . } \\end{array}$ . When $V _ { t h } = 1 , \\alpha = 2$ , the gradient amplitude $\\Vert \\frac { \\partial L } { \\partial S ^ { l } } \\Vert$ of each block is shown in Fig. 6. Note that $\\alpha = 2$ , $\\sigma ^ { \\prime } ( x ) \\leq \\sigma ^ { \\prime } ( 0 ) = \\sigma ^ { \\prime } ( 1 - V _ { t h } ) = 1$ and $\\sigma ^ { \\prime } ( 0 - V _ { t h } ) = 0 . 0 9 2 < 1$ . It can be found that the gradients in Spiking ResNet-152 decay from deeper layers to shallower layers in the identity mapping areas without zero initialization, which is caused by $\\begin{array} { r } { \\dot { \\sigma } ^ { \\prime } ( x ) \\leq 1 } \\end{array}$ . It is worth noting that the decay also happens in Spiking ResNet-152 with zero initialization. The small convex $\\Lambda$ near the dotted lines is caused by the vanishing gradients of those $S _ { j } ^ { l } [ t ] = 0$ . After these gradients decays to 0 completely, $\\Vert \\frac { \\partial L } { \\partial S ^ { l } } \\Vert$ will be a constant because the rest gradients are calculated by $S _ { j } ^ { l } [ t ] = 1$ and $\\sigma ^ { \\prime } ( 1 - V _ { t h } ) = 1$ , which can also explain why the gradient-index curve is horizontal at some areas. When referring to SEW ResNet-152 with zero initialization, it can be found that all gradient-index curves are similar no matter what $g$ we choose. This is caused by that in the identity mapping areas, $S ^ { l }$ is constant for all index $l$ , and the gradient also becomes a constant as it will not flow through SNs. Without zero initialization, the vanishing gradient happens in the SEW AND ResNet-152, which is caused by the silence problem. The gradients of SEW ADD, IAND network increase slowly when propagating from deeper layers to shallower layers, due to the adequate firing rates shown in Fig. 5. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "When $V _ { t h } = 0 . 5 , \\alpha = 2$ , $\\sigma ^ { \\prime } ( 0 - V _ { t h } ) = \\sigma ^ { \\prime } ( 1 - V _ { t h } ) = 0 . 2 8 8 < 1$ , indicating that transmitting spikes to SNs is prone to causing vanishing gradient, as shown in Fig. 7. With zero initialization, the decay in Spiking ResNet-152 is more serious because gradient from $\\mathcal { F } ^ { l }$ can not contribute. The SEW ResNet-152 will not be affected no matter what $g$ we choose. When $V _ { t h } = 1 , \\alpha = 3$ , $\\sigma ^ { \\prime } ( 1 - V _ { t h } ) = 1 . 5 > 1$ , indicating that transmitting spikes to SNs is prone to causing exploding gradient. Fig. 8 shows the gradient in this situation. Same with the reason in Fig. 6, the change of surrogate function will increase gradients of all networks without zero initialization, but not affect SEW ResNet-152 with zero initialization. The Spiking ResNet-152 meets exploding gradient, while this problem in SEW ADD, IAND ResNet-152 is not serious. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.2 DVS Gesture Classification ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The origin ResNet, which is designed for classifying the complex ImageNet dataset, is too large for the DVS Gesture dataset. Hence, we design a tiny network named 7B-Net, whose structure is c32k3s1-BN-PLIF-{SEW Block-MPk2s2}\\*7-FC11. Here c32k3s1 means the convolutional layer with channels 32, kernel size 3, stride 1. MPk2s2 is the max pooling with kernel size 2, stride 2. The symbol $\\{ \\} ^ { * } $ denotes seven repeated structure, and PLIF denotes the Parametric Leaky-Integrate-andFire Spiking Neuron with a learnable membrane time constant, which is proposed in [8] and can be described by Eq. (5). See Sec.A.1 for AER data pre-processing details. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/8073af5219e66c4db06c8fadc705ab27b014379a3c1ffa798fba5f23925374a6.jpg",
+        "image_caption": [
+            "Figure 7: Gradient amplitude $\\Big | \\Big | \\frac { \\partial L } { \\partial S ^ { l } } \\Big | \\Big |$ of $l$ -th block when $V _ { t h } = 0 . 5 , \\alpha = 2$ "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Spiking ResNet vs. SEW ResNet. We first compare the performance of SEW ResNet with $A D D$ element-wise function (SEW ADD ResNet) and Spiking ResNet by replacing SEW blocks with basic blocks. As shown in Fig. 9 and Tab. 4, although the training loss of Spiking ResNet (blue curve) is lower than SEW ADD ResNet (orange curve), the test accuracy is lower than SEW ADD ResNet $( 9 0 . 9 7 \\%$ v.s. $9 7 . 9 2 \\%$ ), which implies that Spiking ResNet is easier to overfit than SEW ADD ResNet. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Evaluation of different element-wise functions and plain block. As the training cost of SNNs on the DVS Gesture dataset is much lower than on ImageNet, we carry out more ablation experiments on the DVS Gesture dataset. We replace SEW blocks with the plain blocks (no shortcut connection) and test the performance. We also evaluate all kinds of element-wise functions $g$ in Tab. 1. Fig. 9 shows the training loss and training/test accuracy on DVS Gesture. The sharp fluctuation during early epochs is caused by the large learning rate (see Sec.A.1). We can find that the training loss is SEW IAND $<$ Spiking ResNe $<$ <SEW ADD $<$ Plain Net<SEW AND. Due to the overfitting problem, a lower loss does not guarantee a higher test accuracy. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Tab. 4 shows the test accuracy of all networks. The SEW ADD ResNet gets the highest accuracy than others. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Comparisons with State-of-the-art Methods. Tab. 5 compares our network with SOTA methods. It can be found that our SEW ResNet outperforms the SOTA works in accuracy, parameter numbers, and simulating time-steps. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/522c669d26d07a3caca5e8f8818a5be47f54161cd9bb501d3dc28f6227872d40.jpg",
+        "table_caption": [
+            "Table 4: Test accuracy on DVS Gesture. The networks’ order is ranked by accuracy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Network</td><td>Element-Wise Function g</td><td>Accuracy(%)</td></tr><tr><td>SEWResNet</td><td>ADD</td><td>97.92</td></tr><tr><td>SEWResNet</td><td>IAND</td><td>95.49</td></tr><tr><td>Plain Net</td><td>1</td><td>91.67</td></tr><tr><td>Spiking ResNet</td><td>1</td><td>90.97</td></tr><tr><td>SEWResNet</td><td>AND</td><td>70.49</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/87f97b8c6c4f4d3ca56f0ed9846af33b37f9edac715d30d69555aede06cd280b.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 5: Comparison with the state-of-the-art (SOTA) methods on DVS Gesture dataset. "
+        ],
+        "table_body": "<table><tr><td>Network</td><td>Accuracy(%)</td><td>Parameters</td><td>T</td></tr><tr><td>c32k3s1-BN-PLIF-{SEW Block (c32) -MPk2s2}*7-FC11 (7B-Net)</td><td>97.92</td><td>0.13M</td><td>16</td></tr><tr><td>{c128k3s1-BN-PLIF-MPk2s2}*5-DP-</td><td>97.57</td><td>1.70M</td><td>20</td></tr><tr><td>FC512-PLIF-DP-FC110-PLIF-APk10s10[8]</td><td></td><td>11.18M</td><td></td></tr><tr><td>SpikingResNet-17with td-BN[64] MPk4-c64k3-LIF-c128k3-LIF-APk2-c128k3-LIF-APk2-FC256-LIF-FC11[16]</td><td>96.87 93.40</td><td>23.23M</td><td>40 60</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.3 CIFAR10-DVS Classification ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We also report SEW ResNet on the CIFAR10-DVS dataset, which is obtained by recording the moving images of the CIFAR-10 dataset on a LCD monitor by a DVS camera. As CIFAR10-DVS is more complicated than DVS Gesture, we use the network structure named Wide-7B-Net, which is similar to 7B-Net but with more channels. The structure of Wide-7B-Net is c64k3s1-BN-PLIF-{SEW Block (c64)-MPk2s2}\\*4-c128k3s1-BN-PLIF-{SEW Block (c128)-MPk2s2}\\*3-FC10. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/adf33effbd45f2f127239f214ada7e8b50cddf2be808e866b1d288cd44499759.jpg",
+        "image_caption": [
+            "Figure 8: Gradient amplitude $\\Big | \\Big | \\frac { \\partial L } { \\partial S ^ { l } } \\Big | \\Big |$ of $l$ -th block when $V _ { t h } = 1 , \\alpha = 3$ "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "image",
+        "img_path": "images/2f787e59c9a353014adc1b3c571413dd5228c89ea36b3ec6a0cecc30ee07713a.jpg",
+        "image_caption": [
+            "Figure 9: Comparison of the training loss, training accuracy and test accuracy on DVS Gesture dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/4a2bf250a61343fe14828ee5184b156e1551d92b64407315677402ffac7cd288.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 6: Comparison with the state-of-the-art (SOTA) methods on CIFAR10-DVS dataset. "
+        ],
+        "table_body": "<table><tr><td>Network</td><td>Accuracy(%)</td><td>Parameters</td><td>T</td></tr><tr><td>c64k3s1-BN-PLIF-{SEWBlock (c64)-MPk2s2}*4-c128k3s1- BN-PLIF-{SEWBlock (c128)-MPk2s2}*3-FCi0 (Wide-7B-Net)</td><td>64.8, 70.2, 74.4</td><td>1.19M</td><td>4,8,16</td></tr><tr><td>{c128k3s1-BN-PLIF-MPk2s2}*4-DP-FC512-PLIF-DP- FC100-PLIF-APk10s10[8]</td><td>74.8</td><td>17.4M</td><td>20</td></tr><tr><td>Spiking ResNet-19 with td-BN [64]</td><td>67.8</td><td>11.18M</td><td>10</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "In Tab.6, we compare SEW ResNet with the previous Spiking ResNet. One can find that our method achieves better performance $7 0 . 2 \\%$ v.s. $6 7 . 8 \\%$ and fewer time-steps (8 v.s. 10) than the Spiking ResNet [64]. We also compare our method with the state-of-the-art (SOTA) supervised learning methods on CIFAR10-DVS. The accuracy of our Wide-7B-Net is slightly lower than the current SOTA method [8] $7 4 . 4 \\%$ v.s. $7 4 . 8 \\%$ ), which uses 1.25 times as many simulation time-steps $T$ (20 v.s. 16) and 14.6 times as many the number of parameters (17.4M v.s. 1.19M). Moreover, when reducing $T$ shapely to $T = 4$ , our Wide-7B-Net can still get the accuracy of $6 4 . 8 \\%$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, we analyze the previous Spiking ResNet whose residual block mimics the standard block of ResNet, and find that it can hardly implement identity mapping and suffers from the problems of vanishing/exploding gradient. To solve these problems, we propose the SEW residual block and prove that it can implement the residual learning. The experiment results on ImageNet, DVS Gesture, and CIFAR10-DVS datasets show that our SEW residual block solves the degradation problem, and SEW ResNet can achieve higher accuracy by simply increasing the network’s depth. Our work may shed light on the learning of “very deep” SNNs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6 Acknowledgment ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This work is supported by grants from the National Natural Science Foundation of China under contracts No.62027804, No.61825101, and No.62088102. ",
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parse/train/DNl5s5BXeBn/DNl5s5BXeBn.md

@@ -0,0 +1,446 @@
+# FAIR MIXUP: FAIRNESS VIA INTERPOLATION
+
+Ching-Yao Chuang∗ CSAIL, MIT cychuang@mit.edu
+
+Youssef Mroueh IBM Research AI mroueh@us.ibm.com
+
+# ABSTRACT
+
+Training classifiers under fairness constraints such as group fairness, regularizes the disparities of predictions between the groups. Nevertheless, even though the constraints are satisfied during training, they might not generalize at evaluation time. To improve the generalizability of fair classifiers, we propose fair mixup, a new data augmentation strategy for imposing the fairness constraint. In particular, we show that fairness can be achieved by regularizing the models on paths of interpolated samples between the groups. We use mixup, a powerful data augmentation strategy to generate these interpolates. We analyze fair mixup and empirically show that it ensures a better generalization for both accuracy and fairness measurement in tabular, vision, and language benchmarks. The code is available at https://github.com/chingyaoc/fair-mixup.
+
+# 1 INTRODUCTION
+
+Fairness has increasingly received attention in machine learning, with the aim of mitigating unjustified bias in learned models. Various statistical metrics were proposed to measure the disparities of model outputs and performance when conditioned on sensitive attributes such as gender or race. Equipped with these metrics, one can formulate constrained optimization problems to impose fairness as a constraint. Nevertheless, these constraints do not necessarily generalize since they are data-dependent, i.e they are estimated from finite samples. In particular, models that minimize the disparities on training sets do not necessarily achieve the same fairness metric on testing sets (Cotter et al., 2019). Conventionally, regularization is required to improve the generalization ability of a model (Zhang et al., 2016). On one hand, explicit regularization such as weight decay and dropout constrain the model capacity. On the other hand, implicit regularization such as data augmentation enlarge the support of the training distribution via prior knowledge (Hernandez-Garc ´ ´ıa & Konig ¨ , 2018).
+
+In this work, we propose a data augmentation strategy for optimizing group fairness constraints such as demographic parity (DP) and equalized odds (EO) (Barocas et al., 2019). Given two sensitive groups such as male and female, instead of directly restricting the disparity, we propose to regularize the model on interpolated distributions between them. Those augmented distributions form a path connecting the two sensitive groups. Figure 1 provides an illustrative example of the idea. The path simulates how the distribution transitions from one group to another via interpolation. Ideally, if the model is invariant to the sensitive attribute, the expected prediction of the model along the path should have a smooth behavior. Therefore, we propose a regularization that favors smooth transitions along the path, which provides a stronger prior on the model class.
+
+We adopt mixup (Zhang et al., 2018b), a powerful data augmentation strategy, to construct the interpolated samples. Owing to mixup’s simple form, the smoothness regularization we introduce has a closed form expression that can be easily optimized. One disadvantage of mixup is that the interpolated samples might not lie on the natural data manifold. Verma et al. (2019) propose Manifold Mixup, which generate the mixup samples in a latent space. Previous works (Bojanowski et al., 2018; Berthelot et al., 2018) have shown that interpolations between a pair of latent features correspond to semantically meaningful, smooth interpolation in the input space. By constructing the path in the latent space, we can better capture the semantic changes while traveling between the sensitive groups and hence result in a better fairness regularizer that we coin fair mixup. Empirically, fair mixup improves the generalizability for both DP and EO on tabular, computer- vision, and natural language benchmarks. Theoretically, we prove for a particular case that fair mixup corresponds to a Mahalanobis metric in the feature space in which we perform the classification. This metric ensures group fairness of the model, and involves the Jacobian of the feature map as we travel along the path.
+
+![](images/62ad9b5f7df98a9699e950e834bab96c4809883b1212474b889265c1d52dbd12.jpg)  
+Figure 1: (a) Visualization of the path constructed via mixup interpolations between groups that have distribution $P _ { 0 }$ and $P _ { 1 }$ , respectively. (b) Fair mixup penalizes the changes in model’s expected prediction with respect to the interpolated distributions. The regularized model (blue curve) has smaller slopes comparing to the unregularized one (orange curve) along the path from $P _ { 0 }$ to $P _ { 1 }$ , which eventually leads to smaller demographic parity $\Delta \mathbf { D P }$ .
+
+In short, this work makes the following contributions:
+
+• We develop fair mixup, a data augmentation strategy that improves the generalization of group fairness metrics; • We provide a theoretical analysis to deepen our understanding of the proposed method; • We evaluate our approach via experiments on tabular, vision, and language benchmarks;
+
+# 2 RELATED WORK
+
+Machine Learning Fairness To mitigate unjustified bias in machine learning systems, various fairness definitions have been proposed. The definitions can usually be classified into individual fairness or group fairness. A system that is individually fair will treat similar users similarly, where the similarity between individuals can be obtained via prior knowledge or metric learning (Dwork et al., 2012; Yurochkin et al., 2019). Group fairness metrics measure the statistical parity between subgroups defined by the sensitive attributes such as gender or race (Zemel et al., 2013; Louizos et al., 2015; Hardt et al., 2016). While fairness can be achieved via pre- or post-processing, optimizing fair metrics at training time can lead to the highest utility (Barocas et al., 2019). For instance, Woodworth et al. (2017) impose independence via regularizing the covariance between predictions and sensitive attributes. Zafar et al. (2017) regularize decision boundaries of convex margin-based classifier to minimize the disparaty between groups. Zhang et al. (2018a) mitigate the bias via minimizing an adversary’s ability to predict sensitive attributes from predictions.
+
+Nevertheless, these constraints are data-dependent, even though the constraints are satisfied during training, the model may behave differently at evaluation time. Agarwal et al. (2018) analyze the generalization error of fair classifiers obtained via two-player games. To improve the generalizability, Cotter et al. (2019) inherit the two-player setting while training each player on two separated datasets. In spite of the analytical solutions and theoretical guarantees, game-theoretic approaches could be hard to scale for complex model classes. In contrast, our proposed fair mixup, is a general data augmentation strategy for optimizing the fairness constraints, which is easily compatible with any dataset modality or model class.
+
+Data Augmentation and Regularization Data augmentation expands the training data with examples generated via prior knowledge, which can be seen as an implicit regularization (Zhang et al., 2016; Hernandez-Garc ´ ´ıa & Konig ¨ , 2018) where the prior is specified as virtual examples. Zhang et al. (2018b) proposes mixup, which generate augmented samples via convex combinations of pairs of examples. In particular, given two examples $\overline { { z } } _ { i } , z _ { j } \in \mathbb { R } ^ { d }$ where $z$ could include both input and label, mixup constructs virtual samples as $t z _ { i } + ( 1 - t ) z _ { j }$ for $t \in [ 0 , 1 ]$ . State-of-the-art results are obtained via training on mixup samples in different modalities. Verma et al. (2019) introduces manifold mixup and shows that performing mixup in a latent space further improves the generalization. While previous works focus on general learning scenarios, we show that regularizing models on mixup samples can lead to group fairness and improve generalization.
+
+# 3 GROUP FAIRNESS
+
+Without loss of generality, we consider the standard fair binary classification setup where we obtain inputs $\boldsymbol { X } \in \mathcal { X } \bar { \subset } \mathbb { R } ^ { d }$ , labels $Y \in \mathcal { Y } = \{ 0 , 1 \}$ , sensitive attribute $A \in \{ 0 , 1 \}$ , and prediction score $\hat { Y } \in [ 0 , 1 ]$ from model $f : \mathbb { R } ^ { d }  [ 0 , 1 ]$ . We will focus on demographic parity (DP) and equalized odds (EO) in this work, while our approach also encompasses other fairness metrics (detailed discussion in section 5). DP requires the predictions $\hat { Y }$ to be independent of the sensitive attribute $A$ , that is, $P ( \hat { Y } | A = 0 ) = P ( \hat { Y } | A = 1 )$ . However, DP ignores the possible correlations between $Y$ and $A$ and could rule out the perfect predictor if $Y$ 6⊥⊥ $A$ . EO overcomes the limit of DP by conditioning on the label $Y$ . In particular, EO requires $\hat { Y }$ and $A$ to be conditionally independent with respect to $Y$ , that is, $P ( \hat { Y } | A = 1 , Y = y ) = P ( \hat { Y } | A = 0 , Y = y )$ for $y \in \{ 0 , 1 \}$ . Given the difficulty of optimizing the independency constraints, Madras et al. (2018) propose the following relaxed metrics:
+
+$$
+\Delta \mathrm { D P } ( f ) = | \mathbb { E } _ { x \sim P _ { 0 } } f ( x ) - \mathbb { E } _ { x \sim P _ { 1 } } f ( x ) | \Delta \mathrm { E O } ( f ) = \sum _ { y \in \{ 0 , 1 \} } \left| \mathbb { E } _ { x \sim P _ { 0 } ^ { y } } f ( x ) - \mathbb { E } _ { x \sim P _ { 1 } ^ { y } } f ( x ) \right|
+$$
+
+where we define $P _ { a } = P ( \cdot | A = a )$ and $P _ { a } ^ { y } = P ( \cdot | A = a , Y = y ) , a , $ $a , y \in \{ 0 , 1 \}$ . We denote the joint distribution of $X$ and $Y$ by $P$ . Similar metrics have also been used in Agarwal et al. (2018), Wei et al. (2019), and Taskesen et al. (2020). One can formulate a penalized optimization problem to regularize the fairness measurement, for instance,
+
+$$
+( \mathrm { G a p \ R e g u l a r i z a t i o n } ) ; \quad \operatorname* { m i n } _ { f } \ \mathbb { E } _ { ( x , y ) \sim P } [ \ell ( f ( x ) , y ) ] + \lambda \Delta \mathrm { D P } ( f ) ,
+$$
+
+where $\ell$ is the classification loss. In spite of its simplicity, our experiments show that small training values of $\Delta \mathrm { D P } ( f )$ do not necessarily generalize well at evaluation time (See section 6). To improve the generalizability, we introduce a data augmentation strategy via a dynamic form of group fairness metrics.
+
+# 4 DYNAMIC FORMULATION OF FAIRNESS: PATHS BETWEEN GROUPS
+
+For simplicity, we will first consider $\Delta \mathbf { D P }$ as the fairness metric, and extend our development to $\Delta \mathrm { E O }$ in section 5. $\Delta \mathbf { D P }$ provides a static measurement by quantifying the expected difference at $P _ { 0 }$ and $P _ { 1 }$ . In contrast, one can consider a dynamic metric that measures the change of $\hat { Y }$ while transitioning gradually from $P _ { 0 }$ to $P _ { 1 }$ . To convert from the static to the dynamic formulations, we start with a simple Lemma that bridges two groups with an interpolator $T ( x _ { 0 } , x _ { 1 } , t )$ , which generates interpolated samples between $x _ { 0 }$ and $x _ { 1 }$ based on step $t$ .
+
+Lemma 1. Let $T : \mathcal { X } ^ { 2 } \times [ 0 , 1 ] \to \mathcal { X }$ be a function continuously differentiable w.r.t. t such that $T ( x _ { 0 } , x _ { 1 } , 0 ) = x _ { 0 }$ and $T ( x _ { 0 } , x _ { 1 } , 1 ) = x _ { 1 }$ . For any differentiable function $f$ , we have
+
+$$
+\Delta { \bf D } { \bf P } ( f ) = \left| \int _ { 0 } ^ { 1 } \frac { d } { d t } \int f ( \underbrace { T ( x _ { 0 } , x _ { 1 } , t ) } _ { \mathrm { i n t e r p o l a t i o n } } ) d P _ { 0 } ( x _ { 0 } ) d P _ { 1 } ( x _ { 1 } ) d t \right| = : \left| \int _ { 0 } ^ { 1 } \frac { d } { d t } \mu _ { f } ( t ) d t \right| ,
+$$
+
+where we define $\mu _ { f } ( t ) = \mathbb { E } _ { x _ { 0 } \sim P _ { 0 } , x _ { 1 } \sim P _ { 1 } } { f ( T ( x _ { 0 } , x _ { 1 } , t ) ) }$ , the expected output of $f$ with respect to $T ( x _ { 0 } , x _ { 1 } , t )$ .
+
+Figure 2 provides an illustrative example of the idea. Lemma 1 relaxes the binary sensitive attribute into a continuous variable $t \in [ 0 , 1 ]$ , where $\mu _ { f }$ captures the behavior of $f$ while traveling from group 0 to group 1 along the path constructed with the interpolator $T$ . In particular, given two examples $x _ { 0 }$ and $x _ { 1 }$ drawn from each group, $T$ generates interpolated samples that change smoothly with respect to $t$ .
+
+For instance, given two racial backgrounds in the dataset, $\mu _ { f }$ simulates how the prediction of $f$ changes while the data of one group smoothly transforms to another. We can then detect whether there are “unfair” changes in $\mu _ { f }$ along the path. The dynamic formulation allows us to measure the sensitivity of $f$ with respect to a relaxed continuous sensitive attribute $t$ via the derivative $\begin{array} { r } { \frac { d } { d t } \mu _ { f } ( t ) } \end{array}$ . Ideally, if $f$ is invariant to the sensitive attribute, $\begin{array} { r } { \frac { d } { d t } \mu _ { f } ( t ) } \end{array}$ should be small along the path from $t = 0$ to 1. Importantly, a small $\Delta \mathbf { D P }$ does not imply $\begin{array} { r } { | \frac { d } { d t } \mu _ { f } ( t ) | } \end{array}$ is small for $t \in [ 0 , 1 ]$ since the derivative could fluctuate as it can be seen in Figure 2.
+
+![](images/07eb0534a896a5eb81656f66e6cd0b9ea2531281232e9ba51b637dd628ff676a.jpg)  
+Figure 2: The expected output $\mu _ { f } ( t )$ gradually changes as $t \to 1$ . Even when $\Delta \mathbf { D P }$ is small, $\begin{array} { r } { | \frac { d } { d t } \mu _ { f } ( t ) | } \end{array}$ could still be large along the path.
+
+# 4.1 SMOOTHNESS REGULARIZATION
+
+To make $f$ invariant to $t$ , we propose to regularize the derivative along the path:
+
+$$
+\mathrm { ( S m o o t h n e s s ~ R e g u l a r i z e r ) } ; \quad R _ { T } ( f ) = \int _ { 0 } ^ { 1 } \left| \frac { d } { d t } \mu _ { f } ( t ) \right| d t .
+$$
+
+Interestingly, $R _ { T } ( f )$ is the arc length of the curve defined by $\mu _ { f } ( t )$ for $t \in [ 0 , 1 ]$ . Now, we can interpret the problem from a geometric point of view. The interpolator $T$ defines a curve $\mu _ { f } ( t ) :$ $[ 0 , 1 ] \to \mathbb { R } .$ , and $\Delta \mathrm { D P } ( f ) = | \bar { \mu _ { f } } ( 0 ) - \bar { \mu _ { f } ( 1 ) } |$ is the Euclidean distance between points $t = 0$ and 1. $\Delta \mathrm { D P } ( f )$ fails to capture the behavior of $f$ while transitioning from $P _ { 0 }$ to $P _ { 1 }$ . In contrast, regularizing the arc length $R _ { T } ( f )$ favors a smooth transition from $t = 0$ to 1, which constrains the fluctuation of the function as the sensitive attributes change. By Jensen’s inequality, $\Delta \mathrm { D P } ( f ) \le R _ { T } ( f )$ for any $f$ , which further justifies the validity of regularizing $\Delta \mathrm { D P } ( f )$ with $R _ { T } ( f )$ .
+
+# 5 FAIR MIXUP: REGULARIZING MIXUP PATHS
+
+It remains to determine the interpolator $T$ . A good interpolater shall (1) generate meaningful interpolations, and (2) the derivative of $\mu _ { f } ( . )$ with respect to $t$ should be easy to compute. In this section, we show that mixup (Zhang et al., 2018b), a powerful data augmentation strategy, is itself a valid interpolator that satisfies both criterions.
+
+Input Mixup We first adopt the standard mixup (Zhang et al., 2018b) by setting the interpolator as the linear interpolation in input space: $T ( x _ { 0 } , \dot { x } _ { 1 } , t ) \stackrel { - } { = } t x _ { 0 } + ( 1 - t ) x _ { 1 }$ . It can be verified that $T _ { \mathrm { m i x u p } }$ satisfies the interpolator criterion defined in Lemma 1. The resulting smoothness regularizer has the following closed form expression1:
+
+$$
+R _ { \mathrm { m i x u p } } ^ { \mathrm { { D P } } } ( f ) = \int _ { 0 } ^ { 1 } \left| \int \left. \nabla _ { x } f ( t x _ { 0 } + ( 1 - t ) x _ { 1 } ) , x _ { 0 } - x _ { 1 } \right. d P _ { 0 } ( x _ { 0 } ) d P _ { 1 } ( x _ { 1 } ) \right| d t .
+$$
+
+The regularizer can be easily optimized by computing the Jacobian of $f$ on mixup samples. Jacobian regularization is a common approach to regularize neural networks (Drucker $\&$ LeCun, 1992). For instance, regularizing the norm of the Jacobian can improve adversarial robustness (Chan et al., 2019; Hoffman et al., 2019). Here, we regularize the expected inner product between the Jacobian on mixup samples and the difference $x _ { 0 } - x _ { 1 }$ .
+
+Manifold Mixup One disadvantage of input mixup is that the curve is defined with mixup samples, which might not lie on the natural data manifold. Verma et al. (2019) propose Manifold Mixup, which generate the mixup samples in the latent space $\mathcal { Z }$ . In particular, manifold mixup assumes a compositional hypothesis $f \circ g$ where $g : \mathcal { X }  \mathcal { Z }$ is the feature encoder and the predictor $f : \mathcal { Z } \to \mathcal { V }$ takes the encoded feature to perform prediction. Similarly, we can establish the equivalence between
+
+$\Delta \mathbf { D P }$ and manifold mixup:
+
+$$
+\Delta \mathrm { D P } ( f \circ g ) = \left. \int _ { 0 } ^ { 1 } \frac { d } { d t } \int f ( \underbrace { t g ( x _ { 0 } ) + ( 1 - t ) g ( x _ { 1 } ) } _ { \mathrm { M a n i f o l d M i x u p } } ) d P _ { 0 } ( x _ { 0 } ) d P _ { 1 } ( x _ { 1 } ) d t \right. ,
+$$
+
+which results in the following smoothness regularizer:
+
+$$
+R _ { \mathrm { m - m i x u p } } ^ { \mathrm { D P } } ( f \circ g ) = \int _ { 0 } ^ { 1 } \left| \int \left. \nabla _ { z } f ( t g ( x _ { 0 } ) + ( 1 - t ) g ( x _ { 1 } ) ) , g ( x _ { 0 } ) - g ( x _ { 1 } ) \right. d P _ { 0 } ( x _ { 0 } ) d P _ { 1 } ( x _ { 1 } ) \right| d t .
+$$
+
+Previous works (Bojanowski et al., 2018; Berthelot et al., 2018) have showed that interpolations between a pair of latent features correspond to semantically meaningful, smooth interpolations in input space. By constructing a curve in the latent space, we can better capture the semantic changes while traveling from $P _ { 0 }$ to $P _ { 1 }$ .
+
+Extensions and Implementation The derivations presented so far, can be easily extended to Equalized Odds (EO). In particular, Lemma 1 can be extended to $\Delta \mathrm { E O }$ by interpolating $P _ { 0 } ^ { y }$ and $P _ { 1 } ^ { \bar { y } }$ for $y \in \{ 0 , 1 \}$ :
+
+$$
+\Delta \mathrm { E O } ( f ) = \sum _ { y \in \{ 0 , 1 \} } \left| \int _ { 0 } ^ { 1 } \frac { d } { d t } \int f ( T ( x _ { 0 } , x _ { 1 } , t ) ) d P _ { 0 } ^ { y } ( x _ { 0 } ) d P _ { 1 } ^ { y } ( x _ { 1 } ) d t \right| .
+$$
+
+The corresponding mixup regularizers can be obtained similarly by substituting $P _ { 0 }$ and $P _ { 1 }$ in $R _ { \mathrm { m i x u p } }$ and $R _ { \mathrm { m - m i x u p } }$ with $\dot { \Gamma } _ { 0 } ^ { y }$ and $P _ { 1 } ^ { \bar { y } }$ :
+
+$$
+R _ { \mathrm { m i x u p } } ^ { \mathrm { E O } } ( f ) = \sum _ { y \in \{ 0 , 1 \} } \int _ { 0 } ^ { 1 } \Big | \int \langle \nabla _ { x } f ( t x _ { 0 } + ( 1 - t ) x _ { 1 } ) , x _ { 0 } - x _ { 1 } \rangle d P _ { 0 } ^ { y } ( x _ { 0 } ) d P _ { 1 } ^ { y } ( x _ { 1 } ) \Big | d t .
+$$
+
+Our formulation also encompasses other fairness metrics that quantify the expected difference between groups. This includes group fairness metrics such as accuracy equality which compares the mistreatment rate between groups (Berk et al., 2018). Similar to equation (1), we formulate a penalized optimization problem to enforce fairness via fair mixup:
+
+$$
+( \mathrm { F a i r } \mathrm { M i x u p } ) { \boldsymbol { : } } \quad \operatorname* { m i n } _ { f } \mathbb { E } _ { ( { \boldsymbol { x } } , { \boldsymbol { y } } ) \sim P } [ \ell ( f ( { \boldsymbol { x } } ) , { \boldsymbol { y } } ) ] + \lambda R _ { \mathrm { m i x u p } } ( f ) .
+$$
+
+Implementation-wise, we follow Zhang et al. (2018b) where only one $t$ is sampled per batch to perform mixup. This strategy works well in practice and reduce the computational requirements.
+
+# 5.1 THEORETICAL ANALYSIS
+
+To gain deeper insight, we analyze the optimal solution of fair mixup in a simple case. In particular, we consider the classification loss $\ell ( f ( \bar { x ) } , y ) = - y f ( x )$ and the following hypothesis class:
+
+$$
+\mathcal { H } = \{ f | f ( x ) = \left. v , \Phi ( x ) \right. , v \in \mathbb { R } ^ { m } , \Phi : \mathcal { X } \to \mathbb { R } ^ { m } \} .
+$$
+
+Define $m _ { \pm } = \mathbb { E } _ { x \sim \mathbb { P } _ { \pm } } \Phi ( x )$ , the label conditional mean embeddings, and $m _ { 0 } = \mathbb { E } _ { x \sim \mathbb { P } _ { 0 } } \Phi ( x )$ and $m _ { 1 } = \mathbb { E } _ { x \sim \mathbb { P } _ { 1 } } \Phi ( x )$ , the group mean embeddings. We then define the expected difference $\delta _ { \pm } = $ $m _ { + } - m _ { - }$ and $\delta _ { 0 , 1 } = m _ { 0 } - m _ { 1 }$ . To derive an interpretable solution, we will consider the L2 variants of the penalized optimization problem. The following proposition gives the analytical solution when we regularize the model with $\Delta \mathbf { D P }$ .
+
+Proposition 2. (Gap Regularization) Consider the following minimization problem
+
+$$
+\operatorname* { m i n } _ { f \in \mathcal { H } } \mathbb { E } _ { ( x , y ) \sim P } [ \ell ( f ( x ) , y ) ] + \frac { \lambda _ { 1 } } { 2 } \Delta \mathrm { D } \mathbf { P } ( f ) ^ { 2 } + \frac { \lambda _ { 2 } } { 2 } | | f | | _ { \mathcal { H } } ^ { 2 } .
+$$
+
+For a fixed embedding $\Phi$ , the optimal solution $f ^ { * }$ corresponds to $v ^ { * }$ given by the following closed form:
+
+$$
+v ^ { \ast } = \frac { 1 } { \lambda _ { 2 } } \left( \delta _ { \pm } - p r o j _ { \delta _ { 0 , 1 } } ^ { \frac { \lambda _ { 2 } } { \lambda _ { 1 } } } ( \delta _ { \pm } ) \right) ,
+$$
+
+where proj is the soft projection defined as p $\begin{array} { r } { r o j _ { u } ^ { \beta } ( x ) = \frac { u \otimes u } { | | u | | ^ { 2 } + \beta } x } \end{array}$
+
+The solution $v ^ { * }$ can be interpreted as the projection of the label discriminating direction $\delta _ { \pm }$ on the subspace that is orthogonal to the group discriminating direction $\delta _ { 0 , 1 }$ . By projecting to this orthogonal subspace, we can prevent the model from using group specific directions, that are unfair directions when performing prediction. Interestingly, the projection trick has been used in Zhang et al. (2018a), where they subtract the gradient of the model parameters in each update step with its projection on unfair directions. We then prove the optimal solution of fair mixup with the same setup as above. Similarly, we introduce an L2 variant of the fair mixup regularizer defined as follows:
+
+$$
+R _ { \mathrm { m i x u p } } ^ { \mathrm { D P - 2 } } ( f ) = \int _ { 0 } ^ { 1 } \bigg | \int \left. \nabla _ { x } f ( t x _ { 0 } + ( 1 - t ) x _ { 1 } ) , x _ { 0 } - x _ { 1 } \right. d P _ { 0 } ( x _ { 0 } ) d P _ { 1 } ( x _ { 1 } ) \bigg | ^ { 2 } d t ,
+$$
+
+where we consider the squared absolute value of the derivative within the integral, in order to get a closed form solution.
+
+Proposition 3. (Fair Mixup) Consider the following minimization problem
+
+$$
+\operatorname* { m i n } _ { f \in \mathcal { H } } \mathbb { E } _ { ( x , y ) \sim P } [ \ell ( f ( x ) , y ) ] + \frac { \lambda _ { 1 } } { 2 } R _ { \mathrm { m i x u p } } ^ { \mathrm { D P - 2 } } ( f ) + \frac { \lambda _ { 2 } } { 2 } | | f | | _ { \mathcal { H } } ^ { 2 } .
+$$
+
+Let $m _ { t } = \mathbb { E } _ { x _ { 0 } \sim P _ { 0 } , x _ { 1 } \sim P _ { 1 } } [ \Phi ( t x _ { 0 } + ( 1 - t ) x _ { 1 } ) ]$ be the $t$ dependent mean embedding, and $\dot { m } _ { t }$ its derivative with respect to $t$ . Let $D$ be a positive-semi definite matrix defined as follows: $D =$ $\begin{array} { r } { \int _ { 0 } ^ { 1 } { \dot { m _ { t } } } \otimes { \dot { m } } _ { t } d t } \end{array}$ . Given an embedding $\Phi$ , the optimal solution $v ^ { * }$ has the following form:
+
+$$
+v ^ { * } = ( \lambda _ { 1 } D + \lambda _ { 2 } I _ { m } ) ^ { - 1 } \delta _ { \pm } .
+$$
+
+Hence the optimal fair mixup classifier can be finally written as :
+
+$$
+f ( x ) = \left. \delta _ { \pm } , ( \lambda _ { 1 } D + \lambda _ { 2 } I _ { m } ) ^ { - 1 } \Phi ( x ) \right. ,
+$$
+
+which means that fair mixup changes the geometry of the decision boundary via a new dot product in the feature space that ensures group fairness, instead of simply projecting on the subspace orthogonal to a single direction as in gap regularization. This dot product leads to a Mahalanobis distance in the feature space that is defined via the covariance of time derivatives of mean embeddings of intermediate densities between the groups. To understand this better, given two points $x _ { 0 }$ in group 0 and $x _ { 1 }$ in group 1, by the mean value theorem, there exists $x _ { c }$ such that:
+
+$$
+f ( x _ { 0 } ) = f ( x _ { 1 } ) + \langle \nabla f ( x _ { c } ) , x _ { 0 } - x _ { 1 } \rangle = f ( x _ { 1 } ) + \langle \delta _ { \pm } , ( \lambda _ { 1 } D + \lambda _ { 2 } I ) ^ { - 1 } J \Phi ( x _ { c } ) ( x _ { 0 } - x _ { 1 } ) \rangle
+$$
+
+Note that $D$ provides the correct average conditioning for $J \Phi ( x _ { c } ) ( x _ { 0 } - x _ { 1 } )$ , this can be seen from the expression of $\dot { m } _ { t }$ $\mathrm { D }$ is a covariance of $J \Phi ( x _ { c } ) ( x _ { 0 } - x _ { 1 } ) )$ . This conditioned Jacobian ensures that the function does not fluctuate a lot between the groups, which matches our motivation.
+
+# 6 EXPERIMENTS
+
+We now examine fair mixup with binary classification tasks on tabular benchmarks (Adult), visual recognition (CelebA), and language dataset (Toxicity). For evaluation, we show the trade-offs between average precision (AP) and fairness metrics (∆DP/∆EO) by varying the hyper-parameter $\lambda$ in the objective. We evaluate both AP and fairness metrics on a testing set to assess the generalizability of learned models. For a fair comparison, we will compare fair mixup with baselines that optimize the fairness constraint at training time. In particular, we compare our method with (a) empirical risk minimization (ERM) that trains the model without regularization, (b) gap regularization, which directly regularizes the model as given in Equation (1), and (c) adversarial debiasing (Zhang et al., 2018a) introduced in section 2. Details about the baselines and experimental setups for each dataset can be found in appendix.
+
+# 6.1 ADULT
+
+UCI Adult dataset (Dua & Graff, 2017) contains information about over 40,000 individuals from the 1994 US Census. The task is to predict whether the income of a person is greater than $\$ 50 k$ given attributes about the person. We consider gender as the sensitive attribute to measure the fairness of the algorithms. The models are two-layer ReLU networks with hidden size 200. We only evaluate input mixup for Adult dataset as the network is not deep enough to produce meaningful latent representations. We retrain each model 10 times and report the mean accuracy and fairness measurement. In each trial, the dataset is randomly randomly split into a training, validation, and testing set with partition $6 0 \%$ , $2 0 \%$ , and $2 0 \%$ , respectively. The models are then selected via the performance on the validation set.
+
+![](images/65119c01cf273da95a9c7b11d1c847be39533ab6522f39237b1ebb31640cc3e3.jpg)  
+Figure 3: Adult Dataset. (a,b) The tradeoff between AP and $\Delta \mathrm { D P } / \Delta \mathrm { E O }$ . (c) Visualization of the mixup path for models that regularize $\Delta \mathbf { D P }$ with different algorithms. We plot the calibrated curve $\mu _ { f } ^ { \prime } ( t ) : = \mu _ { f } ( t ) - \mu _ { f } ( 0 )$ for a better visualization. In this case, $\mu _ { f } ^ { \prime } ( 0 ) = 0$ and $| \mu _ { f } ^ { \prime } ( 1 ) | = \Delta \mathrm { D P }$ for all the calibrated curves $\mu _ { f } ^ { \prime }$ . Therefore, we can compare the $\Delta \mathbf { D P }$ of each method with the absolute value of the last points $\dot { \boldsymbol { t } } = 1$ ). The flatness of the path is highly correlated with the $\Delta \mathbf { D P }$ .
+
+![](images/51df03dd15d33607d8544c1de791d40c1c14596e4980ebf195ec0f7dab5ce072.jpg)  
+Figure 4: CelebA Dataset. The tradeoff between AP and $\Delta \mathrm { D P } / \Delta \mathrm { E O }$ are shown in the first/second row for each task. Manifold mixup consistently outperforms the baseline across tasks.
+
+Figures 3 (a) shows the tradeoff between AP and $\Delta \mathbf { D P }$ . We can see that fair mixup consistently achieves a better tradeoff compared to the baselines. We then show the tradeoff between AP and $\Delta \mathrm { E O }$ in figure 3 (b). For this metric, fair mixup performs slightly better than directly regularizing the EO gap. Interestingly, fair mixup even achieves a better AP compared to ERM, indicating that mixup regularization not only improves the generalization of fairness constraints but also overall accuracy. To understand the effect of fair mixup, we visualize the expected output $\mu _ { f }$ along the path for each method (i.e $\mu _ { f }$ as function of $t$ ). For a fair comparison, we select the models that have similar AP for the visualization. As we can see in figure 3 (c), the flatness of the path is highly correlated to $\Delta \mathbf { D P }$ . Traininig without any regularization leads to the largest derivative along the path, which eventually leads to large $\Delta \mathbf { D P }$ . All the fairness-aware algorithms regularize the slope to some extent, nevertheless, fair mixup achieves the shortest arc length and hence leads to the smallest $\Delta \mathbf { D P }$ .
+
+![](images/4a08b04e57db355f94da875c45227322763730847d175188284e9fc94c21a7a0.jpg)  
+Figure 5: Visualization of calibrated paths on attractive classification task for $\Delta \mathbf { D P }$ regularized models. The flatness of both input and latent path plays an important role in regularizing $\Delta \mathbf { D P }$ .
+
+# 6.2 CELEBA
+
+Next, we show that fair mixup generalizes well to high-dimensional tasks with the CelebA face attributes dataset (Liu et al.). CelebA contains over 200,000 images of celebrity faces, where each image is associated with 40 human-labeled binary attributes including gender. Among the attributes, we select attractive, smile, and wavy hair and use them to form three binary classification tasks while treating gender as the sensitive attribute2. The reason we choose these three attributes is that there exists in all these tasks, a sensitive group that has more positive samples than the other one. For each task, we train a ResNet-18 (He et al., 2016) along with two hidden layers for final prediction. To implement manifold fair mixup, we interpolate the representations before the average pooling layer.
+
+The first row in figure 4 shows the tradeoff between AP and $\Delta \mathbf { D P }$ for each task. Again, fair mixup consistently outperforms the baselines by a large margin. We also observe that manifold mixup further boosts the performance for all the tasks. The tradeoffs between AP and $\Delta \mathrm { E O }$ are shown in the second row of figure 4. Again, both input mixup and manifold mixup yields well generalizing classifiers. To gain further insights, we plot the path in both input space and latent space in figure 5 (a) and (b) for the “attractive” attribute classification task. Fair mixup leads to a smoother path in both cases. Without mixup augmentation, gap regularization and adversarial debiasing present similar paths and both have larger $\Delta \mathbf { D P }$ . We also observe that the expected output $\mu _ { f }$ in the latent path is almost linear with respect to the continuous sensitive attribute $t$ , manifold mixup being the curve with the smallest slope and hence smallest $\Delta \mathbf { D P }$ .
+
+# 6.3 TOXICITY CLASSIFICATION
+
+Lastly, we consider comment toxicity classification with Jigsaw toxic comment dataset (Jigsaw, 2018). The data was initially released by Civil Comments platform, which was then extended to a public Kaggle challenge. The task is to predict whether a comment is toxic or not while being fair across groups. A subset of comments have been labeled with identity attributes, including gender and race. It has been shown that some of the identities (e.g., black) are correlated with the toxicity label. In this work, we consider race as the sensitive attribute and select the subset of comments that contain identities black or asian, as these two groups have the largest gap in terms of probability of being associated with a toxic comment. We use pretrained BERT embeddings (Devlin et al., 2019) to encode each comment into a vector of size 768. A three layer ReLU network is then trained to perform the prediction with the encoded feature. We directly adopt manifold mixup since input mixup is equivalent to manifold mixup by simply setting the encoder $g$ to BERT. Similarly, we retrain each model 10 times using randomly split training, validation, and testing sets, and report mean accuracy and fairness measurement.
+
+Figures 6 (a) and (b) show the tradeoff between AP and $\Delta \mathrm { D P } / \Delta \mathrm { E O }$ , respectively. Again, fair mixup consistently achieves a better tradeoff for both $\Delta \mathbf { D P }$ and $\Delta \mathrm { E O }$ . We then show the visualization of calibrated paths for $\Delta \mathbf { D P }$ -regularized models in Figure 6 (c). We can see that even with the powerful BERT embedding, all the baselines present fluctuated paths with similar patterns. In contrast, fair mixup introduces a nearly linear curve with a small slope, which eventually leads to the smallest $\Delta \mathbf { D P }$ .
+
+![](images/e2af1e74704552b1d41bd209c55ee361e444a6f4773390df8b1bfebb5ea6f099.jpg)  
+Figure 6: Toxic Classification (a,b) The tradeoff between AP and $\Delta \mathrm { D P } / \Delta \mathrm { E O }$ . (c) Visualization of the calibrated paths for models that regularize $\Delta \mathbf { D P }$ with different algorithms. Interestingly, fair mixup presents a nearly linear curve with small slope, while the baselines introduce “inverted-U” shaped curves.
+
+# 7 CONCLUSION
+
+In this work, we propose fair mixup, a data augmentation strategy to optimize fairness constraints. By bridging sensitive groups with interpolated samples, fair mixup consistently improves the generalizability of fairness constraints across benchmarks with different modalities. Interesting future directions include (1) generating interpolated samples that lie on the natural data manifold with generative models or via dynamic optimal transport paths between the groups (Benamou & Brenier, 2000), (2) extending fair mixup to other group fairness metrics such as accuracy equality, and (3) estimating the generalization of fairness constraints (Chuang et al., 2020).
+
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+Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. In International Conference on Learning Representations, 2018b.
+
+# A PROOFS
+
+A.1 PROOF OF LEMMA 1
+
+Lemma 1. Let $T : \mathcal { X } ^ { 2 } \times [ 0 , 1 ] \to \mathcal { X }$ be a function continuously differentiable w.r.t. t such that $T ( x _ { 0 } , x _ { 1 } , 0 ) = x _ { 0 }$ and $T ( x _ { 0 } , x _ { 1 } , 1 ) = x _ { 1 }$ . For any differentiable function $f$ , we have
+
+$$
+\Delta \mathrm { D P } ( f ) = \left| \int _ { 0 } ^ { 1 } \frac { d } { d t } \int f ( \underbrace { T ( x _ { 0 } , x _ { 1 } , t ) } _ { \mathrm { i n t e r p o l a t i o n } } ) d P _ { 0 } ( x _ { 0 } ) d P _ { 1 } ( x _ { 1 } ) d t \right| .
+$$
+
+Proof. The result follows from the fundamental theorem of calculus. In particular, given an interpolator $T$ , we first rewrite the $\Delta \mathbf { D P }$ with the $T$ :
+
+$$
+\begin{array} { r l } & { \Delta \mathrm { D P } ( f ) = | \mathbb { E } _ { x \sim P _ { 0 } } f ( x ) - \mathbb { E } _ { x \sim P _ { 1 } } f ( x ) | } \\ & { \qquad = | \mathbb { E } _ { x _ { 0 } \sim P _ { 0 } , x _ { 1 } \sim P _ { 1 } } f ( x _ { 0 } ) - f ( x _ { 1 } ) | } \\ & { \qquad = | \mathbb { E } _ { x _ { 0 } \sim P _ { 0 } , x _ { 1 } \sim P _ { 1 } } f ( T ( x _ { 0 } , x _ { 1 } , 0 ) ) - \mathbb { E } _ { x _ { 0 } \sim P _ { 0 } , x _ { 1 } \sim P _ { 1 } } f ( T ( x _ { 0 } , x _ { 1 } , 1 ) ) | . } \end{array}
+$$
+
+Not that $\mathbb { E } _ { x _ { 0 } \sim P _ { 0 } , x _ { 1 } \sim P _ { 1 } } f ( T ( x _ { 0 } , x _ { 1 } , t ) )$ is a real-valued continuous function on $t \in [ 0 , 1 ]$ . Therefore, we have the following equivalence via the fundamental theorem of calculus:
+
+$$
+\begin{array} { l } { \displaystyle \Delta \mathrm { D P } ( f ) = \left| \mathbb { E } _ { x _ { 0 } \sim P _ { 0 } , x _ { 1 } \sim P _ { 1 } } f ( T ( x _ { 0 } , x _ { 1 } , 0 ) ) - \mathbb { E } _ { x _ { 0 } \sim P _ { 0 } , x _ { 1 } \sim P _ { 1 } } f ( T ( x _ { 0 } , x _ { 1 } , 1 ) ) \right| . } \\ { \displaystyle \qquad = \left| \int _ { 0 } ^ { 1 } \frac { d } { d t } \mathbb { E } _ { x _ { 0 } \sim P _ { 0 } , x _ { 1 } \sim P _ { 1 } } f ( T ( x _ { 0 } , x _ { 1 } , t ) ) \right| . } \end{array}
+$$
+
+# A.2 PROOF OF PROPOSITION 2
+
+Proposition 2. (Gap Regularization) Consider the following minimization problem
+
+$$
+\operatorname* { m i n } _ { f \in \mathcal { H } } \mathbb { E } _ { ( x , y ) \sim P } [ \ell ( f ( x ) , y ) ] + \frac { \lambda _ { 1 } } { 2 } \Delta \mathrm { D } \mathbf { P } ( f ) ^ { 2 } + \frac { \lambda _ { 2 } } { 2 } | | f | | _ { \mathcal { H } } ^ { 2 } .
+$$
+
+For a fixed embedding $\Phi$ , the optimal solution $f ^ { * }$ corresponds to $v ^ { * }$ given by following closed form:
+
+$$
+v ^ { \ast } = \frac { 1 } { \lambda _ { 2 } } \left( \delta _ { \pm } - p r o j _ { \delta _ { 0 , 1 } } ^ { \frac { \lambda _ { 2 } } { \lambda _ { 1 } } } ( \delta _ { \pm } ) \right) ,
+$$
+
+where proj is the soft projection defined as pro $\begin{array} { r } { j _ { u } ^ { \beta } ( x ) = \frac { u \otimes u } { | | u | | ^ { 2 } + \beta } x } \end{array}$
+
+Proof. The problem above can be written as follows:
+
+$$
+\operatorname* { m i n } _ { v \in \mathbb { R } ^ { m } } \mathcal { L } ( v ) : = - ( \langle v , \delta _ { \pm } \rangle ) + \frac { \lambda _ { 1 } } { 2 } | \langle v , \delta _ { 0 , 1 } \rangle | ^ { 2 } + \frac { \lambda _ { 2 } } { 2 } | | v | | _ { 2 } ^ { 2 }
+$$
+
+Setting first order condition to zero
+
+$$
+\begin{array} { r } { \nabla _ { v } \mathcal { L } ( v ) = - \delta _ { \pm } + \lambda _ { 1 } \delta _ { 0 , 1 } \otimes \delta _ { 0 , 1 } v + \lambda _ { 2 } v = 0 , } \end{array}
+$$
+
+we obtain
+
+$$
+( \lambda _ { 1 } \delta _ { 0 , 1 } \otimes \delta _ { 0 , 1 } + \lambda _ { 2 } I _ { m } ) v ^ { \ast } = \delta _ { \pm } .
+$$
+
+By inverting and applying the Sherman-Morrison Lemma, we have
+
+$$
+\begin{array} { l l l } { { v ^ { * } } } & { { = } } & { { \displaystyle ( \lambda _ { 1 } \delta _ { 0 , 1 } \otimes \delta _ { 0 , 1 } + \lambda _ { 2 } I _ { m } ) ^ { - 1 } \delta _ { \pm } } } \\ { { } } & { { = } } & { { \displaystyle \lambda _ { 1 } ^ { - 1 } \left( \frac { \lambda _ { 2 } } { \lambda _ { 1 } } I _ { m } + \delta _ { 0 , 1 } \otimes \delta _ { 0 , 1 } \right) ^ { - 1 } \delta _ { \pm } } } \\ { { } } & { { = } } & { { \displaystyle \lambda _ { 1 } ^ { - 1 } \left( I _ { m } - \frac { ( \lambda _ { 1 } ) ^ { 2 } \delta _ { 0 , 1 } \otimes \delta _ { 0 , 1 } } { 1 + | | \delta _ { 0 , 1 } | | ^ { 2 } \frac { \lambda _ { 1 } } { \lambda _ { 2 } } } \right) \delta _ { \pm } } } \\ { { } } & { { = } } & { { \displaystyle \frac { 1 } { \lambda _ { 2 } } \left( I _ { m } - \frac { \delta _ { 0 , 1 } \otimes \delta _ { 0 , 1 } } { \lambda _ { 2 } } \right) \delta _ { \pm } . } } \end{array}
+$$
+
+Note the soft projection on a vector $u$ is defined as follows:
+
+$$
+p r o j _ { u } ^ { \beta } ( x ) = \frac { u \otimes u } { | | u | | ^ { 2 } + \beta } x .
+$$
+
+It follows that
+
+$$
+v ^ { \ast } = \frac { 1 } { \lambda _ { 2 } } \left( \delta _ { \pm } - p r o j _ { \delta _ { 0 , 1 } } ^ { \frac { \lambda _ { 2 } } { \lambda _ { 1 } } } ( \delta _ { \pm } ) \right) ,
+$$
+
+which can be interpreted as the projection of the label discriminating direction $\delta _ { \pm }$ on the subspace that is orthogonal to the group discriminating direction $\delta _ { 0 , 1 }$ . □
+
+# A.3 PROOF OF PROPOSITION 3
+
+Proposition 3. (Fair Mixup) Consider the following minimization problem
+
+$$
+\operatorname* { m i n } _ { f \in \mathcal { H } } \mathbb { E } _ { ( x , y ) \sim P } [ \ell ( f ( x ) , y ) ] + \frac { \lambda _ { 1 } } { 2 } R _ { \mathrm { m i x u p } } ^ { \mathrm { D P - 2 } } ( f ) + \frac { \lambda _ { 2 } } { 2 } | | f | | _ { \mathcal { H } } ^ { 2 } .
+$$
+
+Define $m _ { t } = \mathbb { E } _ { x _ { 0 } \sim P _ { 0 } , x _ { 1 } \sim P _ { 1 } } [ \Phi ( t x _ { 0 } + ( 1 - t ) x _ { 1 } ) ]$ be the $t$ dependent mean embedding, and $\dot { m } _ { t }$ its derivative with respect to $t$ . Let $D$ be a positive-semi definite matrix defined as follows: $D =$ $\begin{array} { r } { \int _ { 0 } ^ { 1 } { \dot { m _ { t } } } \otimes { \dot { m } } _ { t } d t } \end{array}$ . Given an embedding $\Phi$ , the optimal solution $v ^ { * }$ has the following form:
+
+$$
+v ^ { * } = ( \lambda _ { 1 } D + \lambda _ { 2 } I _ { m } ) ^ { - 1 } \delta _ { \pm } .
+$$
+
+Proof. For $t \in [ 0 , 1 ]$ , we note by $\rho _ { t }$ the distribution of $T ( x _ { 0 } , x _ { 1 } , t )$ , for $x _ { 0 } \sim P _ { 0 } , x _ { 1 } \sim P _ { 1 }$ , and note $m _ { t } = \mathbb { E } _ { x \sim \rho _ { t } } \Phi ( x )$ , and $\dot { m } _ { t }$ its time derivative. We consider the $\ell _ { 2 }$ variant of $R _ { T } ( f )$ in the analysis:
+
+$$
+\begin{array} { r c l } { { R _ { T } ^ { 2 } ( f ) } } & { { = } } & { { \displaystyle \int _ { 0 } ^ { 1 } \left| \frac { d } { d t } \mu _ { f } ( t ) \right| ^ { 2 } d t = \displaystyle \int _ { 0 } ^ { 1 } \left| \frac { d } { d t } \left. v , m _ { t } \right. \right| ^ { 2 } d t } } \\ { { } } & { { = } } & { { \displaystyle \int _ { 0 } ^ { 1 } | \langle v , \dot { m } _ { t } \rangle | ^ { 2 } d t = \left. v , \left( \displaystyle \int _ { 0 } ^ { 1 } \dot { m } _ { t } \otimes \dot { m } _ { t } d t \right) v \right. , } } \end{array}
+$$
+
+We then expand $\dot { m } _ { t }$ when $T$ is mixup:
+
+$$
+\dot { m } _ { t } = \frac { d } { d t } \mathbb { E } _ { x _ { 0 } \sim \mathbb { P } _ { 0 } , x _ { 1 } \sim \mathbb { P } _ { 1 } } \Phi ( t x _ { 0 } + ( 1 - t ) x _ { 1 } ) = \mathbb { E } _ { x _ { 0 } \sim \mathbb { P } _ { 0 } , x _ { 1 } \sim \mathbb { P } _ { 1 } } J \Phi ( t x _ { 0 } + ( 1 - t ) x _ { 1 } ) ( x _ { 0 } - x _ { 1 } )
+$$
+
+where $J$ denotes the Jacobian. Note that $\begin{array} { r } { D = \int _ { 0 } ^ { 1 } \dot { m _ { t } } \otimes \dot { m } _ { t } d t } \end{array}$ , hence for the classification with fair mixup regularizer, the problem is equivalent to:
+
+$$
+\operatorname* { m i n } _ { v \in \mathbb { R } ^ { m } } \mathcal { L } ( v ) : = - ( \langle v , \delta _ { \pm } \rangle ) + \frac { \lambda _ { 1 } } { 2 } \langle v , D v \rangle + \frac { \lambda _ { 2 } } { 2 } | | v | | _ { 2 } ^ { 2 }
+$$
+
+Setting first order condition we obtain $( \lambda _ { 1 } D + \lambda _ { 2 } I _ { m } ) v ^ { \ast } = \delta _ { \pm }$ , which gives the optimal solution
+
+$$
+v ^ { * } = ( \lambda _ { 1 } D + \lambda _ { 2 } I _ { m } ) ^ { - 1 } \delta _ { \pm } .
+$$
+
+The corresponding optimal fair mixup classifier can be finally written as
+
+$$
+f ( x ) = \left. \delta _ { \pm } , ( \lambda _ { 1 } D + \lambda _ { 2 } I _ { m } ) ^ { - 1 } \Phi ( x ) \right. .
+$$
+
+# B EXPERIMENT DETAILS
+
+Adult We follow the preprocessing procedure of Yurochkin et al. (2019) by removing some features in the dataset3. We then encode the discrete and quantized continuous attributes with one-hot encoding. We retrain each model 10 times with batch size 1000 and report the mean accuracy and fairness measurement. The models are selected via the performance on validation set. In each trial, the dataset is randomly split into training and testing set with partition $8 0 \%$ and $2 0 \%$ , respectively. The models are optimized with Adam optimizer (Kingma & Ba, 2014) with learning rate $\mathrm { \bar { 1 } } \times e ^ { - \mathrm { \bar { 3 } } }$ For DP, we sample 500 datapoints for each $A \in \{ 0 , 1 \}$ to form a batch. Similarly, for EO, we sample 250 datapoints for each $( A , Y )$ pair where $A , Y \in \{ 0 , 1 \}$ .
+
+CelebA Model-wise, we extract the feature of size 512 after the average pooling layer of ResNet18. A two-layer ReLU network with hidden size 512 is then trained to perform prediction. Percentage of positive-labeled datapoints for attractive, wavy hair, and smiling that is male are $2 2 . 7 \%$ , $1 8 . 3 6 \%$ , and $3 4 . 6 \%$ , respectively. We use the original validation set of CelebA to perform model selection and report the accuracy and fairness metrics on the testing set. The visualization paths are also plotted with respect to the testing data. To implement manifold mixup, we interpolate the spatial features before the average pooling layer. Similarly, all the models are optimized with Adam optimizer with learning rate $1 \times e ^ { - 3 }$ .
+
+Toxicity Classification We download the Jigsaw toxic comment dataset from Kaggle website4. Percentage of positive-labeled datapoints for black and asian are $1 8 . 8 \%$ and $6 . 4 \%$ , respectively, which together results in a dataset of size 22835. We retrain each model 10 times with batch size 200 and report the mean accuracy and fairness measurement. The models are selected via the performance on validation set. The batch-sampling and data splitting procedure is the same as the one for Adult dataset. The models are again optimized with Adam optimizer with learning rate $1 \times e ^ { - 3 }$ .
+
+# C ADDITIONAL EXPERIMENTS
+
+# C.1 TRAINING PERFORMANCE
+
+In Figure 7, we show the training performance for Adult dataset. As expected, GapReg outperforms Fair mixup on the training set since it directly optimizes the fairness metric. The results also support our motivation: the constraints that are satisfied during training might not generalize at evaluation time.
+
+![](images/38d30429260f23fa3c4cb2c80e1c0feaa90205a73b7d9a5233050439ab5c215b.jpg)  
+Figure 7: Training Performance on Adult Dataset. The tradeoff between AP and $\Delta \mathrm { D P } / \Delta \mathrm { E O }$ on training set.
+
+# C.2 EVALUATION METRIC
+
+The relaxed evaluation metric could overestimate the performance when the predicted confidence is significantly different between groups. For instance, a classifier $f$ can be completely unfair while satisfying this condition: $f ( x ) { \overset { } { = } } 1$ w.p. $60 \%$ , 0 w.p. $40 \%$ on $P _ { 0 }$ , and $f ( x ) \stackrel { - } { = } 0 . 6$ w.p. $100 \%$ on $P _ { 1 }$ . This satisfies this expectation-based condition. However, it is highly unfair if we binarize the prediction by setting the threshold $= 0 . 5$ .
+
+To overcome this issue, let $f _ { t }$ be the binarized predictor $f _ { t } ( x ) = \mathbb { 1 } ( f ( x ) \geq t )$ , we evaluate the model with average $\Delta \mathrm { D P } ( \overline { { \Delta \mathrm { D P } } } )$ defined as follows:
+
+$$
+\begin{array} { r l } & { \overline { { \Delta \mathrm { D P } } } ( f ) = \displaystyle \frac { 1 } { | T | } \sum _ { t \in \mathcal { T } } \left| \mathbb { E } _ { \boldsymbol { x } \sim P _ { 0 } } f _ { t } ( \boldsymbol { x } ) - \mathbb { E } _ { \boldsymbol { x } \sim P _ { 1 } } f _ { t } ( \boldsymbol { x } ) \right| ; } \\ & { \overline { { \Delta \mathrm { E O } } } ( f ) = \displaystyle \frac { 1 } { | T | } \sum _ { t \in \mathcal { T } } \sum _ { y \in \{ 0 , 1 \} } \left| \mathbb { E } _ { \boldsymbol { x } \sim P _ { 0 } ^ { y } } f _ { t } ( \boldsymbol { x } ) - \mathbb { E } _ { \boldsymbol { x } \sim P _ { 1 } ^ { y } } f _ { t } ( \boldsymbol { x } ) \right| , } \end{array}
+$$
+
+where $\tau$ is a set of threshold values. $\overline { { \Delta \mathsf { D P } } }$ averages the $\Delta \mathbf { D P }$ with binarized predictions derived via different thresholds. For instance, by averaging the $\Delta \mathbf { D P }$ with thersholds $\mathcal { T } = [ 0 . 1 , 0 . 2 , \cdot \cdot \cdot , 0 . 9 ] ,$ , $\overline { { \Delta \mathrm { D P } } } = 0 . 5$ instead of 0 for the example above, which captures the unfairness between groups. We report $\overline { { \Delta \mathsf { D P } } }$ for each methods with thersholds $\mathcal { T } = [ 0 . 1 , \bar { 0 } . 2 , \cdot \cdot \cdot . , 0 . 9 ]$ in Figure 8. Similarly, fair mixup exhibits the best tradeoff comparing to the baselines for demographic parity. For equalized odds, the performances of fair mixup and GapReg are similar, where fair mixup achieves a better tradeoff when $\overline { { \Delta \mathrm { E O } } }$ is small.
+
+![](images/5ee5e26b6b027b544d99a89cf0f68d7011e47bc42ed3d946f586dec59c345bb7.jpg)  
+Figure 8: Average ∆DP and ∆EO on Adult Dataset.
+
+# C.3 SMALLER MODEL SIZE
+
+To examine the effect of model size, we reduce the hidden size from 200 to 50 and show the result in Figure 9. Overall, the performance does not vary significantly after reducing the model size. We can again observe that fair mixup outperform the baselines for $\Delta \mathbf { D P }$ . Similar to the results in section C.2, the performances of fair mixup and GapReg are similar, where fair mixup achieves a better tradeoff when $\Delta \mathrm { E O }$ is small.
+
+![](images/472cd71dd38bb056ebae1b91a67bd5645ae14173cedc6b390a003a86962f3536.jpg)  
+Figure 9: Reducing the Model Size on Adult Dataset.

parse/train/DNl5s5BXeBn/DNl5s5BXeBn_content_list.json

@@ -0,0 +1,2170 @@
+[
+    {
+        "type": "text",
+        "text": "FAIR MIXUP: FAIRNESS VIA INTERPOLATION ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            98,
+            715,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ching-Yao Chuang∗ CSAIL, MIT cychuang@mit.edu ",
+        "bbox": [
+            183,
+            145,
+            343,
+            186
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Youssef Mroueh IBM Research AI mroueh@us.ibm.com ",
+        "bbox": [
+            501,
+            145,
+            671,
+            185
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            223,
+            544,
+            238
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Training classifiers under fairness constraints such as group fairness, regularizes the disparities of predictions between the groups. Nevertheless, even though the constraints are satisfied during training, they might not generalize at evaluation time. To improve the generalizability of fair classifiers, we propose fair mixup, a new data augmentation strategy for imposing the fairness constraint. In particular, we show that fairness can be achieved by regularizing the models on paths of interpolated samples between the groups. We use mixup, a powerful data augmentation strategy to generate these interpolates. We analyze fair mixup and empirically show that it ensures a better generalization for both accuracy and fairness measurement in tabular, vision, and language benchmarks. The code is available at https://github.com/chingyaoc/fair-mixup. ",
+        "bbox": [
+            233,
+            256,
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+            409
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            436,
+            336,
+            453
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Fairness has increasingly received attention in machine learning, with the aim of mitigating unjustified bias in learned models. Various statistical metrics were proposed to measure the disparities of model outputs and performance when conditioned on sensitive attributes such as gender or race. Equipped with these metrics, one can formulate constrained optimization problems to impose fairness as a constraint. Nevertheless, these constraints do not necessarily generalize since they are data-dependent, i.e they are estimated from finite samples. In particular, models that minimize the disparities on training sets do not necessarily achieve the same fairness metric on testing sets (Cotter et al., 2019). Conventionally, regularization is required to improve the generalization ability of a model (Zhang et al., 2016). On one hand, explicit regularization such as weight decay and dropout constrain the model capacity. On the other hand, implicit regularization such as data augmentation enlarge the support of the training distribution via prior knowledge (Hernandez-Garc ´ ´ıa & Konig ¨ , 2018). ",
+        "bbox": [
+            173,
+            468,
+            825,
+            635
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose a data augmentation strategy for optimizing group fairness constraints such as demographic parity (DP) and equalized odds (EO) (Barocas et al., 2019). Given two sensitive groups such as male and female, instead of directly restricting the disparity, we propose to regularize the model on interpolated distributions between them. Those augmented distributions form a path connecting the two sensitive groups. Figure 1 provides an illustrative example of the idea. The path simulates how the distribution transitions from one group to another via interpolation. Ideally, if the model is invariant to the sensitive attribute, the expected prediction of the model along the path should have a smooth behavior. Therefore, we propose a regularization that favors smooth transitions along the path, which provides a stronger prior on the model class. ",
+        "bbox": [
+            173,
+            642,
+            825,
+            767
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We adopt mixup (Zhang et al., 2018b), a powerful data augmentation strategy, to construct the interpolated samples. Owing to mixup’s simple form, the smoothness regularization we introduce has a closed form expression that can be easily optimized. One disadvantage of mixup is that the interpolated samples might not lie on the natural data manifold. Verma et al. (2019) propose Manifold Mixup, which generate the mixup samples in a latent space. Previous works (Bojanowski et al., 2018; Berthelot et al., 2018) have shown that interpolations between a pair of latent features correspond to semantically meaningful, smooth interpolation in the input space. By constructing the path in the latent space, we can better capture the semantic changes while traveling between the sensitive groups and hence result in a better fairness regularizer that we coin fair mixup. Empirically, fair mixup improves the generalizability for both DP and EO on tabular, computer- vision, and natural language benchmarks. Theoretically, we prove for a particular case that fair mixup corresponds to a Mahalanobis metric in the feature space in which we perform the classification. This metric ensures group fairness of the model, and involves the Jacobian of the feature map as we travel along the path. ",
+        "bbox": [
+            174,
+            775,
+            825,
+            900
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/62ad9b5f7df98a9699e950e834bab96c4809883b1212474b889265c1d52dbd12.jpg",
+        "image_caption": [
+            "Figure 1: (a) Visualization of the path constructed via mixup interpolations between groups that have distribution $P _ { 0 }$ and $P _ { 1 }$ , respectively. (b) Fair mixup penalizes the changes in model’s expected prediction with respect to the interpolated distributions. The regularized model (blue curve) has smaller slopes comparing to the unregularized one (orange curve) along the path from $P _ { 0 }$ to $P _ { 1 }$ , which eventually leads to smaller demographic parity $\\Delta \\mathbf { D P }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            209,
+            103,
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+            231
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            330,
+            825,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In short, this work makes the following contributions: ",
+        "bbox": [
+            174,
+            406,
+            526,
+            421
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We develop fair mixup, a data augmentation strategy that improves the generalization of group fairness metrics; • We provide a theoretical analysis to deepen our understanding of the proposed method; • We evaluate our approach via experiments on tabular, vision, and language benchmarks; ",
+        "bbox": [
+            181,
+            434,
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+            494
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            512,
+            344,
+            529
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Machine Learning Fairness To mitigate unjustified bias in machine learning systems, various fairness definitions have been proposed. The definitions can usually be classified into individual fairness or group fairness. A system that is individually fair will treat similar users similarly, where the similarity between individuals can be obtained via prior knowledge or metric learning (Dwork et al., 2012; Yurochkin et al., 2019). Group fairness metrics measure the statistical parity between subgroups defined by the sensitive attributes such as gender or race (Zemel et al., 2013; Louizos et al., 2015; Hardt et al., 2016). While fairness can be achieved via pre- or post-processing, optimizing fair metrics at training time can lead to the highest utility (Barocas et al., 2019). For instance, Woodworth et al. (2017) impose independence via regularizing the covariance between predictions and sensitive attributes. Zafar et al. (2017) regularize decision boundaries of convex margin-based classifier to minimize the disparaty between groups. Zhang et al. (2018a) mitigate the bias via minimizing an adversary’s ability to predict sensitive attributes from predictions. ",
+        "bbox": [
+            173,
+            537,
+            825,
+            705
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Nevertheless, these constraints are data-dependent, even though the constraints are satisfied during training, the model may behave differently at evaluation time. Agarwal et al. (2018) analyze the generalization error of fair classifiers obtained via two-player games. To improve the generalizability, Cotter et al. (2019) inherit the two-player setting while training each player on two separated datasets. In spite of the analytical solutions and theoretical guarantees, game-theoretic approaches could be hard to scale for complex model classes. In contrast, our proposed fair mixup, is a general data augmentation strategy for optimizing the fairness constraints, which is easily compatible with any dataset modality or model class. ",
+        "bbox": [
+            173,
+            712,
+            825,
+            824
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Data Augmentation and Regularization Data augmentation expands the training data with examples generated via prior knowledge, which can be seen as an implicit regularization (Zhang et al., 2016; Hernandez-Garc ´ ´ıa & Konig ¨ , 2018) where the prior is specified as virtual examples. Zhang et al. (2018b) proposes mixup, which generate augmented samples via convex combinations of pairs of examples. In particular, given two examples $\\overline { { z } } _ { i } , z _ { j } \\in \\mathbb { R } ^ { d }$ where $z$ could include both input and label, mixup constructs virtual samples as $t z _ { i } + ( 1 - t ) z _ { j }$ for $t \\in [ 0 , 1 ]$ . State-of-the-art results are obtained via training on mixup samples in different modalities. Verma et al. (2019) introduces manifold mixup and shows that performing mixup in a latent space further improves the generalization. While previous works focus on general learning scenarios, we show that regularizing models on mixup samples can lead to group fairness and improve generalization. ",
+        "bbox": [
+            174,
+            840,
+            823,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            823,
+            160
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 GROUP FAIRNESS",
+        "text_level": 1,
+        "bbox": [
+            176,
+            180,
+            352,
+            196
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Without loss of generality, we consider the standard fair binary classification setup where we obtain inputs $\\boldsymbol { X } \\in \\mathcal { X } \\bar { \\subset } \\mathbb { R } ^ { d }$ , labels $Y \\in \\mathcal { Y } = \\{ 0 , 1 \\}$ , sensitive attribute $A \\in \\{ 0 , 1 \\}$ , and prediction score $\\hat { Y } \\in [ 0 , 1 ]$ from model $f : \\mathbb { R } ^ { d }  [ 0 , 1 ]$ . We will focus on demographic parity (DP) and equalized odds (EO) in this work, while our approach also encompasses other fairness metrics (detailed discussion in section 5). DP requires the predictions $\\hat { Y }$ to be independent of the sensitive attribute $A$ , that is, $P ( \\hat { Y } | A = 0 ) = P ( \\hat { Y } | A = 1 )$ . However, DP ignores the possible correlations between $Y$ and $A$ and could rule out the perfect predictor if $Y$ 6⊥⊥ $A$ . EO overcomes the limit of DP by conditioning on the label $Y$ . In particular, EO requires $\\hat { Y }$ and $A$ to be conditionally independent with respect to $Y$ , that is, $P ( \\hat { Y } | A = 1 , Y = y ) = P ( \\hat { Y } | A = 0 , Y = y )$ for $y \\in \\{ 0 , 1 \\}$ . Given the difficulty of optimizing the independency constraints, Madras et al. (2018) propose the following relaxed metrics: ",
+        "bbox": [
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+            212,
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+            376
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ab400496da5b724b20228fb6982b2836a7fbc7d5302b47f9acc4e89fcde24534.jpg",
+        "text": "$$\n\\Delta \\mathrm { D P } ( f ) = | \\mathbb { E } _ { x \\sim P _ { 0 } } f ( x ) - \\mathbb { E } _ { x \\sim P _ { 1 } } f ( x ) | \\Delta \\mathrm { E O } ( f ) = \\sum _ { y \\in \\{ 0 , 1 \\} } \\left| \\mathbb { E } _ { x \\sim P _ { 0 } ^ { y } } f ( x ) - \\mathbb { E } _ { x \\sim P _ { 1 } ^ { y } } f ( x ) \\right|\n$$",
+        "text_format": "latex",
+        "bbox": [
+            196,
+            381,
+            802,
+            419
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where we define $P _ { a } = P ( \\cdot | A = a )$ and $P _ { a } ^ { y } = P ( \\cdot | A = a , Y = y ) , a , $ $a , y \\in \\{ 0 , 1 \\}$ . We denote the joint distribution of $X$ and $Y$ by $P$ . Similar metrics have also been used in Agarwal et al. (2018), Wei et al. (2019), and Taskesen et al. (2020). One can formulate a penalized optimization problem to regularize the fairness measurement, for instance, ",
+        "bbox": [
+            173,
+            425,
+            825,
+            482
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dc30e30fae999becf13d02de022263bef17c03360992aa528102f08a438860da.jpg",
+        "text": "$$\n( \\mathrm { G a p \\ R e g u l a r i z a t i o n } ) ; \\quad \\operatorname* { m i n } _ { f } \\ \\mathbb { E } _ { ( x , y ) \\sim P } [ \\ell ( f ( x ) , y ) ] + \\lambda \\Delta \\mathrm { D P } ( f ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            284,
+            489,
+            712,
+            515
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\ell$ is the classification loss. In spite of its simplicity, our experiments show that small training values of $\\Delta \\mathrm { D P } ( f )$ do not necessarily generalize well at evaluation time (See section 6). To improve the generalizability, we introduce a data augmentation strategy via a dynamic form of group fairness metrics. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 DYNAMIC FORMULATION OF FAIRNESS: PATHS BETWEEN GROUPS ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            598,
+            759,
+            616
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For simplicity, we will first consider $\\Delta \\mathbf { D P }$ as the fairness metric, and extend our development to $\\Delta \\mathrm { E O }$ in section 5. $\\Delta \\mathbf { D P }$ provides a static measurement by quantifying the expected difference at $P _ { 0 }$ and $P _ { 1 }$ . In contrast, one can consider a dynamic metric that measures the change of $\\hat { Y }$ while transitioning gradually from $P _ { 0 }$ to $P _ { 1 }$ . To convert from the static to the dynamic formulations, we start with a simple Lemma that bridges two groups with an interpolator $T ( x _ { 0 } , x _ { 1 } , t )$ , which generates interpolated samples between $x _ { 0 }$ and $x _ { 1 }$ based on step $t$ . ",
+        "bbox": [
+            173,
+            630,
+            825,
+            717
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. Let $T : \\mathcal { X } ^ { 2 } \\times [ 0 , 1 ] \\to \\mathcal { X }$ be a function continuously differentiable w.r.t. t such that $T ( x _ { 0 } , x _ { 1 } , 0 ) = x _ { 0 }$ and $T ( x _ { 0 } , x _ { 1 } , 1 ) = x _ { 1 }$ . For any differentiable function $f$ , we have ",
+        "bbox": [
+            173,
+            720,
+            825,
+            751
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b9d95927d143b1a8f9d54bbdca93737131e620b24ff6bfee85ac369a2729c200.jpg",
+        "text": "$$\n\\Delta { \\bf D } { \\bf P } ( f ) = \\left| \\int _ { 0 } ^ { 1 } \\frac { d } { d t } \\int f ( \\underbrace { T ( x _ { 0 } , x _ { 1 } , t ) } _ { \\mathrm { i n t e r p o l a t i o n } } ) d P _ { 0 } ( x _ { 0 } ) d P _ { 1 } ( x _ { 1 } ) d t \\right| = : \\left| \\int _ { 0 } ^ { 1 } \\frac { d } { d t } \\mu _ { f } ( t ) d t \\right| ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            235,
+            756,
+            761,
+            815
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where we define $\\mu _ { f } ( t ) = \\mathbb { E } _ { x _ { 0 } \\sim P _ { 0 } , x _ { 1 } \\sim P _ { 1 } } { f ( T ( x _ { 0 } , x _ { 1 } , t ) ) }$ , the expected output of $f$ with respect to $T ( x _ { 0 } , x _ { 1 } , t )$ . ",
+        "bbox": [
+            173,
+            821,
+            826,
+            852
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Figure 2 provides an illustrative example of the idea. Lemma 1 relaxes the binary sensitive attribute into a continuous variable $t \\in [ 0 , 1 ]$ , where $\\mu _ { f }$ captures the behavior of $f$ while traveling from group 0 to group 1 along the path constructed with the interpolator $T$ . In particular, given two examples $x _ { 0 }$ and $x _ { 1 }$ drawn from each group, $T$ generates interpolated samples that change smoothly with respect to $t$ . ",
+        "bbox": [
+            173,
+            862,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For instance, given two racial backgrounds in the dataset, $\\mu _ { f }$ simulates how the prediction of $f$ changes while the data of one group smoothly transforms to another. We can then detect whether there are “unfair” changes in $\\mu _ { f }$ along the path. The dynamic formulation allows us to measure the sensitivity of $f$ with respect to a relaxed continuous sensitive attribute $t$ via the derivative $\\begin{array} { r } { \\frac { d } { d t } \\mu _ { f } ( t ) } \\end{array}$ . Ideally, if $f$ is invariant to the sensitive attribute, $\\begin{array} { r } { \\frac { d } { d t } \\mu _ { f } ( t ) } \\end{array}$ should be small along the path from $t = 0$ to 1. Importantly, a small $\\Delta \\mathbf { D P }$ does not imply $\\begin{array} { r } { | \\frac { d } { d t } \\mu _ { f } ( t ) | } \\end{array}$ is small for $t \\in [ 0 , 1 ]$ since the derivative could fluctuate as it can be seen in Figure 2. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/07eb0534a896a5eb81656f66e6cd0b9ea2531281232e9ba51b637dd628ff676a.jpg",
+        "image_caption": [
+            "Figure 2: The expected output $\\mu _ { f } ( t )$ gradually changes as $t \\to 1$ . Even when $\\Delta \\mathbf { D P }$ is small, $\\begin{array} { r } { | \\frac { d } { d t } \\mu _ { f } ( t ) | } \\end{array}$ could still be large along the path. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 SMOOTHNESS REGULARIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            294,
+            441,
+            308
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To make $f$ invariant to $t$ , we propose to regularize the derivative along the path: ",
+        "bbox": [
+            174,
+            320,
+            694,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0cb2538b6be508ee70475f3e8397fedc307d143282623161dd8ec86f6be72e0a.jpg",
+        "text": "$$\n\\mathrm { ( S m o o t h n e s s ~ R e g u l a r i z e r ) } ; \\quad R _ { T } ( f ) = \\int _ { 0 } ^ { 1 } \\left| \\frac { d } { d t } \\mu _ { f } ( t ) \\right| d t .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            344,
+            687,
+            382
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Interestingly, $R _ { T } ( f )$ is the arc length of the curve defined by $\\mu _ { f } ( t )$ for $t \\in [ 0 , 1 ]$ . Now, we can interpret the problem from a geometric point of view. The interpolator $T$ defines a curve $\\mu _ { f } ( t ) :$ $[ 0 , 1 ] \\to \\mathbb { R } .$ , and $\\Delta \\mathrm { D P } ( f ) = | \\bar { \\mu _ { f } } ( 0 ) - \\bar { \\mu _ { f } ( 1 ) } |$ is the Euclidean distance between points $t = 0$ and 1. $\\Delta \\mathrm { D P } ( f )$ fails to capture the behavior of $f$ while transitioning from $P _ { 0 }$ to $P _ { 1 }$ . In contrast, regularizing the arc length $R _ { T } ( f )$ favors a smooth transition from $t = 0$ to 1, which constrains the fluctuation of the function as the sensitive attributes change. By Jensen’s inequality, $\\Delta \\mathrm { D P } ( f ) \\le R _ { T } ( f )$ for any $f$ , which further justifies the validity of regularizing $\\Delta \\mathrm { D P } ( f )$ with $R _ { T } ( f )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "5 FAIR MIXUP: REGULARIZING MIXUP PATHS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            512,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "It remains to determine the interpolator $T$ . A good interpolater shall (1) generate meaningful interpolations, and (2) the derivative of $\\mu _ { f } ( . )$ with respect to $t$ should be easy to compute. In this section, we show that mixup (Zhang et al., 2018b), a powerful data augmentation strategy, is itself a valid interpolator that satisfies both criterions. ",
+        "bbox": [
+            173,
+            546,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Input Mixup We first adopt the standard mixup (Zhang et al., 2018b) by setting the interpolator as the linear interpolation in input space: $T ( x _ { 0 } , \\dot { x } _ { 1 } , t ) \\stackrel { - } { = } t x _ { 0 } + ( 1 - t ) x _ { 1 }$ . It can be verified that $T _ { \\mathrm { m i x u p } }$ satisfies the interpolator criterion defined in Lemma 1. The resulting smoothness regularizer has the following closed form expression1: ",
+        "bbox": [
+            173,
+            621,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4bf03cfdf650b4f2d4bf5a633c1f59215e81b99b653899013b8d018c271a4e54.jpg",
+        "text": "$$\nR _ { \\mathrm { m i x u p } } ^ { \\mathrm { { D P } } } ( f ) = \\int _ { 0 } ^ { 1 } \\left| \\int \\left. \\nabla _ { x } f ( t x _ { 0 } + ( 1 - t ) x _ { 1 } ) , x _ { 0 } - x _ { 1 } \\right. d P _ { 0 } ( x _ { 0 } ) d P _ { 1 } ( x _ { 1 } ) \\right| d t .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            245,
+            689,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The regularizer can be easily optimized by computing the Jacobian of $f$ on mixup samples. Jacobian regularization is a common approach to regularize neural networks (Drucker $\\&$ LeCun, 1992). For instance, regularizing the norm of the Jacobian can improve adversarial robustness (Chan et al., 2019; Hoffman et al., 2019). Here, we regularize the expected inner product between the Jacobian on mixup samples and the difference $x _ { 0 } - x _ { 1 }$ . ",
+        "bbox": [
+            173,
+            734,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Manifold Mixup One disadvantage of input mixup is that the curve is defined with mixup samples, which might not lie on the natural data manifold. Verma et al. (2019) propose Manifold Mixup, which generate the mixup samples in the latent space $\\mathcal { Z }$ . In particular, manifold mixup assumes a compositional hypothesis $f \\circ g$ where $g : \\mathcal { X }  \\mathcal { Z }$ is the feature encoder and the predictor $f : \\mathcal { Z } \\to \\mathcal { V }$ takes the encoded feature to perform prediction. Similarly, we can establish the equivalence between ",
+        "bbox": [
+            173,
+            823,
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+            893
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\Delta \\mathbf { D P }$ and manifold mixup: ",
+        "bbox": [
+            174,
+            103,
+            351,
+            118
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4cead4d2c54a86a2759b39fd28a8cfbb69e23351d7a334e05f98a5220f2f90e1.jpg",
+        "text": "$$\n\\Delta \\mathrm { D P } ( f \\circ g ) = \\left. \\int _ { 0 } ^ { 1 } \\frac { d } { d t } \\int f ( \\underbrace { t g ( x _ { 0 } ) + ( 1 - t ) g ( x _ { 1 } ) } _ { \\mathrm { M a n i f o l d M i x u p } } ) d P _ { 0 } ( x _ { 0 } ) d P _ { 1 } ( x _ { 1 } ) d t \\right. ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            254,
+            122,
+            741,
+            181
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "which results in the following smoothness regularizer: ",
+        "bbox": [
+            173,
+            184,
+            529,
+            199
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/341112ad106319a299f0dbda2d0c572fc994117f2e61b032d79e0bc6ba435f1e.jpg",
+        "text": "$$\nR _ { \\mathrm { m - m i x u p } } ^ { \\mathrm { D P } } ( f \\circ g ) = \\int _ { 0 } ^ { 1 } \\left| \\int \\left. \\nabla _ { z } f ( t g ( x _ { 0 } ) + ( 1 - t ) g ( x _ { 1 } ) ) , g ( x _ { 0 } ) - g ( x _ { 1 } ) \\right. d P _ { 0 } ( x _ { 0 } ) d P _ { 1 } ( x _ { 1 } ) \\right| d t .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            186,
+            203,
+            812,
+            241
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Previous works (Bojanowski et al., 2018; Berthelot et al., 2018) have showed that interpolations between a pair of latent features correspond to semantically meaningful, smooth interpolations in input space. By constructing a curve in the latent space, we can better capture the semantic changes while traveling from $P _ { 0 }$ to $P _ { 1 }$ . ",
+        "bbox": [
+            173,
+            251,
+            825,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Extensions and Implementation The derivations presented so far, can be easily extended to Equalized Odds (EO). In particular, Lemma 1 can be extended to $\\Delta \\mathrm { E O }$ by interpolating $P _ { 0 } ^ { y }$ and $P _ { 1 } ^ { \\bar { y } }$ for $y \\in \\{ 0 , 1 \\}$ : ",
+        "bbox": [
+            173,
+            321,
+            826,
+            366
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/767054284ce2838c01668c39dde1e4ad5adfc00865835873bf9c59aec0ae53dd.jpg",
+        "text": "$$\n\\Delta \\mathrm { E O } ( f ) = \\sum _ { y \\in \\{ 0 , 1 \\} } \\left| \\int _ { 0 } ^ { 1 } \\frac { d } { d t } \\int f ( T ( x _ { 0 } , x _ { 1 } , t ) ) d P _ { 0 } ^ { y } ( x _ { 0 } ) d P _ { 1 } ^ { y } ( x _ { 1 } ) d t \\right| .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            369,
+            722,
+            412
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The corresponding mixup regularizers can be obtained similarly by substituting $P _ { 0 }$ and $P _ { 1 }$ in $R _ { \\mathrm { m i x u p } }$ and $R _ { \\mathrm { m - m i x u p } }$ with $\\dot { \\Gamma } _ { 0 } ^ { y }$ and $P _ { 1 } ^ { \\bar { y } }$ : ",
+        "bbox": [
+            173,
+            416,
+            823,
+            446
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a600adc03939fdfb51f47883b172fd87dc4cca32d376dc7e3f3afc548eb1b3b1.jpg",
+        "text": "$$\nR _ { \\mathrm { m i x u p } } ^ { \\mathrm { E O } } ( f ) = \\sum _ { y \\in \\{ 0 , 1 \\} } \\int _ { 0 } ^ { 1 } \\Big | \\int \\langle \\nabla _ { x } f ( t x _ { 0 } + ( 1 - t ) x _ { 1 } ) , x _ { 0 } - x _ { 1 } \\rangle d P _ { 0 } ^ { y } ( x _ { 0 } ) d P _ { 1 } ^ { y } ( x _ { 1 } ) \\Big | d t .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            218,
+            450,
+            779,
+            494
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our formulation also encompasses other fairness metrics that quantify the expected difference between groups. This includes group fairness metrics such as accuracy equality which compares the mistreatment rate between groups (Berk et al., 2018). Similar to equation (1), we formulate a penalized optimization problem to enforce fairness via fair mixup: ",
+        "bbox": [
+            173,
+            498,
+            825,
+            555
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ccb7553d629f7cab25c20629fac5322cd597f772958772c5940584bd3f14c6c4.jpg",
+        "text": "$$\n( \\mathrm { F a i r } \\mathrm { M i x u p } ) { \\boldsymbol { : } } \\quad \\operatorname* { m i n } _ { f } \\mathbb { E } _ { ( { \\boldsymbol { x } } , { \\boldsymbol { y } } ) \\sim P } [ \\ell ( f ( { \\boldsymbol { x } } ) , { \\boldsymbol { y } } ) ] + \\lambda R _ { \\mathrm { m i x u p } } ( f ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            307,
+            559,
+            691,
+            584
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Implementation-wise, we follow Zhang et al. (2018b) where only one $t$ is sampled per batch to perform mixup. This strategy works well in practice and reduce the computational requirements. ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5.1 THEORETICAL ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            633,
+            392,
+            648
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To gain deeper insight, we analyze the optimal solution of fair mixup in a simple case. In particular, we consider the classification loss $\\ell ( f ( \\bar { x ) } , y ) = - y f ( x )$ and the following hypothesis class: ",
+        "bbox": [
+            174,
+            660,
+            823,
+            689
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/785c22ac85de89055014909e67e52696f276763bb7cc1647d36526614e21280f.jpg",
+        "text": "$$\n\\mathcal { H } = \\{ f | f ( x ) = \\left. v , \\Phi ( x ) \\right. , v \\in \\mathbb { R } ^ { m } , \\Phi : \\mathcal { X } \\to \\mathbb { R } ^ { m } \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            694,
+            671,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Define $m _ { \\pm } = \\mathbb { E } _ { x \\sim \\mathbb { P } _ { \\pm } } \\Phi ( x )$ , the label conditional mean embeddings, and $m _ { 0 } = \\mathbb { E } _ { x \\sim \\mathbb { P } _ { 0 } } \\Phi ( x )$ and $m _ { 1 } = \\mathbb { E } _ { x \\sim \\mathbb { P } _ { 1 } } \\Phi ( x )$ , the group mean embeddings. We then define the expected difference $\\delta _ { \\pm } = $ $m _ { + } - m _ { - }$ and $\\delta _ { 0 , 1 } = m _ { 0 } - m _ { 1 }$ . To derive an interpretable solution, we will consider the L2 variants of the penalized optimization problem. The following proposition gives the analytical solution when we regularize the model with $\\Delta \\mathbf { D P }$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2. (Gap Regularization) Consider the following minimization problem ",
+        "bbox": [
+            176,
+            787,
+            730,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4d55d21ba9fb626e38a88367e6f69bd726ccbd5d85d63fc7ac2f2b5a16263728.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { f \\in \\mathcal { H } } \\mathbb { E } _ { ( x , y ) \\sim P } [ \\ell ( f ( x ) , y ) ] + \\frac { \\lambda _ { 1 } } { 2 } \\Delta \\mathrm { D } \\mathbf { P } ( f ) ^ { 2 } + \\frac { \\lambda _ { 2 } } { 2 } | | f | | _ { \\mathcal { H } } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            808,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For a fixed embedding $\\Phi$ , the optimal solution $f ^ { * }$ corresponds to $v ^ { * }$ given by the following closed form: ",
+        "bbox": [
+            173,
+            844,
+            823,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4d867026648383ccf4c3fd63a06660263c54666f130d9fce45bfaf5c0b834cf6.jpg",
+        "text": "$$\nv ^ { \\ast } = \\frac { 1 } { \\lambda _ { 2 } } \\left( \\delta _ { \\pm } - p r o j _ { \\delta _ { 0 , 1 } } ^ { \\frac { \\lambda _ { 2 } } { \\lambda _ { 1 } } } ( \\delta _ { \\pm } ) \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            872,
+            606,
+            906
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where proj is the soft projection defined as p $\\begin{array} { r } { r o j _ { u } ^ { \\beta } ( x ) = \\frac { u \\otimes u } { | | u | | ^ { 2 } + \\beta } x } \\end{array}$ ",
+        "bbox": [
+            173,
+            909,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The solution $v ^ { * }$ can be interpreted as the projection of the label discriminating direction $\\delta _ { \\pm }$ on the subspace that is orthogonal to the group discriminating direction $\\delta _ { 0 , 1 }$ . By projecting to this orthogonal subspace, we can prevent the model from using group specific directions, that are unfair directions when performing prediction. Interestingly, the projection trick has been used in Zhang et al. (2018a), where they subtract the gradient of the model parameters in each update step with its projection on unfair directions. We then prove the optimal solution of fair mixup with the same setup as above. Similarly, we introduce an L2 variant of the fair mixup regularizer defined as follows: ",
+        "bbox": [
+            173,
+            103,
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+            202
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c62ca313c2dd422a031ffb73f0acb011cd2dd7bf4a4c3137d5f0791052b6684a.jpg",
+        "text": "$$\nR _ { \\mathrm { m i x u p } } ^ { \\mathrm { D P - 2 } } ( f ) = \\int _ { 0 } ^ { 1 } \\bigg | \\int \\left. \\nabla _ { x } f ( t x _ { 0 } + ( 1 - t ) x _ { 1 } ) , x _ { 0 } - x _ { 1 } \\right. d P _ { 0 } ( x _ { 0 } ) d P _ { 1 } ( x _ { 1 } ) \\bigg | ^ { 2 } d t ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            243,
+            207,
+            754,
+            244
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where we consider the squared absolute value of the derivative within the integral, in order to get a closed form solution. ",
+        "bbox": [
+            173,
+            250,
+            825,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3. (Fair Mixup) Consider the following minimization problem ",
+        "bbox": [
+            176,
+            281,
+            674,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3ce9fd8c4e32d94c1fe2898c9a51885895ea8eb6de1c8b5d86cd4e1e2436d24d.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { f \\in \\mathcal { H } } \\mathbb { E } _ { ( x , y ) \\sim P } [ \\ell ( f ( x ) , y ) ] + \\frac { \\lambda _ { 1 } } { 2 } R _ { \\mathrm { m i x u p } } ^ { \\mathrm { D P - 2 } } ( f ) + \\frac { \\lambda _ { 2 } } { 2 } | | f | | _ { \\mathcal { H } } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            303,
+            679,
+            335
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Let $m _ { t } = \\mathbb { E } _ { x _ { 0 } \\sim P _ { 0 } , x _ { 1 } \\sim P _ { 1 } } [ \\Phi ( t x _ { 0 } + ( 1 - t ) x _ { 1 } ) ]$ be the $t$ dependent mean embedding, and $\\dot { m } _ { t }$ its derivative with respect to $t$ . Let $D$ be a positive-semi definite matrix defined as follows: $D =$ $\\begin{array} { r } { \\int _ { 0 } ^ { 1 } { \\dot { m _ { t } } } \\otimes { \\dot { m } } _ { t } d t } \\end{array}$ . Given an embedding $\\Phi$ , the optimal solution $v ^ { * }$ has the following form: ",
+        "bbox": [
+            173,
+            340,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d046f3698b5a8103f9e73c80eb18b0209a7230e8e2af8c3c1c03fa4aa4c001b7.jpg",
+        "text": "$$\nv ^ { * } = ( \\lambda _ { 1 } D + \\lambda _ { 2 } I _ { m } ) ^ { - 1 } \\delta _ { \\pm } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            395,
+            589,
+            412
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Hence the optimal fair mixup classifier can be finally written as : ",
+        "bbox": [
+            173,
+            426,
+            599,
+            440
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/57b044f5d2b575b6b0af656450a5612a85b1478aaf48dc56bd66bd1d393b3ae8.jpg",
+        "text": "$$\nf ( x ) = \\left. \\delta _ { \\pm } , ( \\lambda _ { 1 } D + \\lambda _ { 2 } I _ { m } ) ^ { - 1 } \\Phi ( x ) \\right. ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "which means that fair mixup changes the geometry of the decision boundary via a new dot product in the feature space that ensures group fairness, instead of simply projecting on the subspace orthogonal to a single direction as in gap regularization. This dot product leads to a Mahalanobis distance in the feature space that is defined via the covariance of time derivatives of mean embeddings of intermediate densities between the groups. To understand this better, given two points $x _ { 0 }$ in group 0 and $x _ { 1 }$ in group 1, by the mean value theorem, there exists $x _ { c }$ such that: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fa7a7e29d36776eb1fcfe80496880e3f38e11f98238ab0e78f820aced4331c27.jpg",
+        "text": "$$\nf ( x _ { 0 } ) = f ( x _ { 1 } ) + \\langle \\nabla f ( x _ { c } ) , x _ { 0 } - x _ { 1 } \\rangle = f ( x _ { 1 } ) + \\langle \\delta _ { \\pm } , ( \\lambda _ { 1 } D + \\lambda _ { 2 } I ) ^ { - 1 } J \\Phi ( x _ { c } ) ( x _ { 0 } - x _ { 1 } ) \\rangle\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Note that $D$ provides the correct average conditioning for $J \\Phi ( x _ { c } ) ( x _ { 0 } - x _ { 1 } )$ , this can be seen from the expression of $\\dot { m } _ { t }$ $\\mathrm { D }$ is a covariance of $J \\Phi ( x _ { c } ) ( x _ { 0 } - x _ { 1 } ) )$ . This conditioned Jacobian ensures that the function does not fluctuate a lot between the groups, which matches our motivation. ",
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+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We now examine fair mixup with binary classification tasks on tabular benchmarks (Adult), visual recognition (CelebA), and language dataset (Toxicity). For evaluation, we show the trade-offs between average precision (AP) and fairness metrics (∆DP/∆EO) by varying the hyper-parameter $\\lambda$ in the objective. We evaluate both AP and fairness metrics on a testing set to assess the generalizability of learned models. For a fair comparison, we will compare fair mixup with baselines that optimize the fairness constraint at training time. In particular, we compare our method with (a) empirical risk minimization (ERM) that trains the model without regularization, (b) gap regularization, which directly regularizes the model as given in Equation (1), and (c) adversarial debiasing (Zhang et al., 2018a) introduced in section 2. Details about the baselines and experimental setups for each dataset can be found in appendix. ",
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+    {
+        "type": "text",
+        "text": "6.1 ADULT ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "UCI Adult dataset (Dua & Graff, 2017) contains information about over 40,000 individuals from the 1994 US Census. The task is to predict whether the income of a person is greater than $\\$ 50 k$ given attributes about the person. We consider gender as the sensitive attribute to measure the fairness of the algorithms. The models are two-layer ReLU networks with hidden size 200. We only evaluate input mixup for Adult dataset as the network is not deep enough to produce meaningful latent representations. We retrain each model 10 times and report the mean accuracy and fairness measurement. In each trial, the dataset is randomly randomly split into a training, validation, and testing set with partition $6 0 \\%$ , $2 0 \\%$ , and $2 0 \\%$ , respectively. The models are then selected via the performance on the validation set. ",
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+        "image_caption": [
+            "Figure 3: Adult Dataset. (a,b) The tradeoff between AP and $\\Delta \\mathrm { D P } / \\Delta \\mathrm { E O }$ . (c) Visualization of the mixup path for models that regularize $\\Delta \\mathbf { D P }$ with different algorithms. We plot the calibrated curve $\\mu _ { f } ^ { \\prime } ( t ) : = \\mu _ { f } ( t ) - \\mu _ { f } ( 0 )$ for a better visualization. In this case, $\\mu _ { f } ^ { \\prime } ( 0 ) = 0$ and $| \\mu _ { f } ^ { \\prime } ( 1 ) | = \\Delta \\mathrm { D P }$ for all the calibrated curves $\\mu _ { f } ^ { \\prime }$ . Therefore, we can compare the $\\Delta \\mathbf { D P }$ of each method with the absolute value of the last points $\\dot { \\boldsymbol { t } } = 1$ ). The flatness of the path is highly correlated with the $\\Delta \\mathbf { D P }$ . "
+        ],
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+        "image_caption": [
+            "Figure 4: CelebA Dataset. The tradeoff between AP and $\\Delta \\mathrm { D P } / \\Delta \\mathrm { E O }$ are shown in the first/second row for each task. Manifold mixup consistently outperforms the baseline across tasks. "
+        ],
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+    {
+        "type": "text",
+        "text": "Figures 3 (a) shows the tradeoff between AP and $\\Delta \\mathbf { D P }$ . We can see that fair mixup consistently achieves a better tradeoff compared to the baselines. We then show the tradeoff between AP and $\\Delta \\mathrm { E O }$ in figure 3 (b). For this metric, fair mixup performs slightly better than directly regularizing the EO gap. Interestingly, fair mixup even achieves a better AP compared to ERM, indicating that mixup regularization not only improves the generalization of fairness constraints but also overall accuracy. To understand the effect of fair mixup, we visualize the expected output $\\mu _ { f }$ along the path for each method (i.e $\\mu _ { f }$ as function of $t$ ). For a fair comparison, we select the models that have similar AP for the visualization. As we can see in figure 3 (c), the flatness of the path is highly correlated to $\\Delta \\mathbf { D P }$ . Traininig without any regularization leads to the largest derivative along the path, which eventually leads to large $\\Delta \\mathbf { D P }$ . All the fairness-aware algorithms regularize the slope to some extent, nevertheless, fair mixup achieves the shortest arc length and hence leads to the smallest $\\Delta \\mathbf { D P }$ . ",
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+        "page_idx": 6
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+        "img_path": "images/4a08b04e57db355f94da875c45227322763730847d175188284e9fc94c21a7a0.jpg",
+        "image_caption": [
+            "Figure 5: Visualization of calibrated paths on attractive classification task for $\\Delta \\mathbf { D P }$ regularized models. The flatness of both input and latent path plays an important role in regularizing $\\Delta \\mathbf { D P }$ . "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "6.2 CELEBA ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Next, we show that fair mixup generalizes well to high-dimensional tasks with the CelebA face attributes dataset (Liu et al.). CelebA contains over 200,000 images of celebrity faces, where each image is associated with 40 human-labeled binary attributes including gender. Among the attributes, we select attractive, smile, and wavy hair and use them to form three binary classification tasks while treating gender as the sensitive attribute2. The reason we choose these three attributes is that there exists in all these tasks, a sensitive group that has more positive samples than the other one. For each task, we train a ResNet-18 (He et al., 2016) along with two hidden layers for final prediction. To implement manifold fair mixup, we interpolate the representations before the average pooling layer. ",
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+    {
+        "type": "text",
+        "text": "The first row in figure 4 shows the tradeoff between AP and $\\Delta \\mathbf { D P }$ for each task. Again, fair mixup consistently outperforms the baselines by a large margin. We also observe that manifold mixup further boosts the performance for all the tasks. The tradeoffs between AP and $\\Delta \\mathrm { E O }$ are shown in the second row of figure 4. Again, both input mixup and manifold mixup yields well generalizing classifiers. To gain further insights, we plot the path in both input space and latent space in figure 5 (a) and (b) for the “attractive” attribute classification task. Fair mixup leads to a smoother path in both cases. Without mixup augmentation, gap regularization and adversarial debiasing present similar paths and both have larger $\\Delta \\mathbf { D P }$ . We also observe that the expected output $\\mu _ { f }$ in the latent path is almost linear with respect to the continuous sensitive attribute $t$ , manifold mixup being the curve with the smallest slope and hence smallest $\\Delta \\mathbf { D P }$ . ",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "6.3 TOXICITY CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Lastly, we consider comment toxicity classification with Jigsaw toxic comment dataset (Jigsaw, 2018). The data was initially released by Civil Comments platform, which was then extended to a public Kaggle challenge. The task is to predict whether a comment is toxic or not while being fair across groups. A subset of comments have been labeled with identity attributes, including gender and race. It has been shown that some of the identities (e.g., black) are correlated with the toxicity label. In this work, we consider race as the sensitive attribute and select the subset of comments that contain identities black or asian, as these two groups have the largest gap in terms of probability of being associated with a toxic comment. We use pretrained BERT embeddings (Devlin et al., 2019) to encode each comment into a vector of size 768. A three layer ReLU network is then trained to perform the prediction with the encoded feature. We directly adopt manifold mixup since input mixup is equivalent to manifold mixup by simply setting the encoder $g$ to BERT. Similarly, we retrain each model 10 times using randomly split training, validation, and testing sets, and report mean accuracy and fairness measurement. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figures 6 (a) and (b) show the tradeoff between AP and $\\Delta \\mathrm { D P } / \\Delta \\mathrm { E O }$ , respectively. Again, fair mixup consistently achieves a better tradeoff for both $\\Delta \\mathbf { D P }$ and $\\Delta \\mathrm { E O }$ . We then show the visualization of calibrated paths for $\\Delta \\mathbf { D P }$ -regularized models in Figure 6 (c). We can see that even with the powerful BERT embedding, all the baselines present fluctuated paths with similar patterns. In contrast, fair mixup introduces a nearly linear curve with a small slope, which eventually leads to the smallest $\\Delta \\mathbf { D P }$ . ",
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+    {
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+        "img_path": "images/e2af1e74704552b1d41bd209c55ee361e444a6f4773390df8b1bfebb5ea6f099.jpg",
+        "image_caption": [
+            "Figure 6: Toxic Classification (a,b) The tradeoff between AP and $\\Delta \\mathrm { D P } / \\Delta \\mathrm { E O }$ . (c) Visualization of the calibrated paths for models that regularize $\\Delta \\mathbf { D P }$ with different algorithms. Interestingly, fair mixup presents a nearly linear curve with small slope, while the baselines introduce “inverted-U” shaped curves. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose fair mixup, a data augmentation strategy to optimize fairness constraints. By bridging sensitive groups with interpolated samples, fair mixup consistently improves the generalizability of fairness constraints across benchmarks with different modalities. Interesting future directions include (1) generating interpolated samples that lie on the natural data manifold with generative models or via dynamic optimal transport paths between the groups (Benamou & Brenier, 2000), (2) extending fair mixup to other group fairness metrics such as accuracy equality, and (3) estimating the generalization of fairness constraints (Chuang et al., 2020). ",
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+    {
+        "type": "text",
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+            145
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016. ",
+        "bbox": [
+            174,
+            155,
+            823,
+            184
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. In International Conference on Learning Representations, 2018b. ",
+        "bbox": [
+            174,
+            193,
+            823,
+            222
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A PROOFS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            277,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 PROOF OF LEMMA 1 ",
+        "bbox": [
+            174,
+            132,
+            359,
+            147
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. Let $T : \\mathcal { X } ^ { 2 } \\times [ 0 , 1 ] \\to \\mathcal { X }$ be a function continuously differentiable w.r.t. t such that $T ( x _ { 0 } , x _ { 1 } , 0 ) = x _ { 0 }$ and $T ( x _ { 0 } , x _ { 1 } , 1 ) = x _ { 1 }$ . For any differentiable function $f$ , we have ",
+        "bbox": [
+            171,
+            157,
+            825,
+            189
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ee5e7a7d2f4cf6b7667eba9998214d3cab41256131206d11593631da836e88fb.jpg",
+        "text": "$$\n\\Delta \\mathrm { D P } ( f ) = \\left| \\int _ { 0 } ^ { 1 } \\frac { d } { d t } \\int f ( \\underbrace { T ( x _ { 0 } , x _ { 1 } , t ) } _ { \\mathrm { i n t e r p o l a t i o n } } ) d P _ { 0 } ( x _ { 0 } ) d P _ { 1 } ( x _ { 1 } ) d t \\right| .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            193,
+            696,
+            250
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proof. The result follows from the fundamental theorem of calculus. In particular, given an interpolator $T$ , we first rewrite the $\\Delta \\mathbf { D P }$ with the $T$ : ",
+        "bbox": [
+            173,
+            262,
+            821,
+            290
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e8870937a7726c1f7b6cef82beae9f4d5291b3e7b6f8842dbb8422914ec7fa07.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\Delta \\mathrm { D P } ( f ) = | \\mathbb { E } _ { x \\sim P _ { 0 } } f ( x ) - \\mathbb { E } _ { x \\sim P _ { 1 } } f ( x ) | } \\\\ & { \\qquad = | \\mathbb { E } _ { x _ { 0 } \\sim P _ { 0 } , x _ { 1 } \\sim P _ { 1 } } f ( x _ { 0 } ) - f ( x _ { 1 } ) | } \\\\ & { \\qquad = | \\mathbb { E } _ { x _ { 0 } \\sim P _ { 0 } , x _ { 1 } \\sim P _ { 1 } } f ( T ( x _ { 0 } , x _ { 1 } , 0 ) ) - \\mathbb { E } _ { x _ { 0 } \\sim P _ { 0 } , x _ { 1 } \\sim P _ { 1 } } f ( T ( x _ { 0 } , x _ { 1 } , 1 ) ) | . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            243,
+            292,
+            754,
+            349
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Not that $\\mathbb { E } _ { x _ { 0 } \\sim P _ { 0 } , x _ { 1 } \\sim P _ { 1 } } f ( T ( x _ { 0 } , x _ { 1 } , t ) )$ is a real-valued continuous function on $t \\in [ 0 , 1 ]$ . Therefore, we have the following equivalence via the fundamental theorem of calculus: ",
+        "bbox": [
+            168,
+            352,
+            825,
+            380
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/01f211037fa87723d3645aadd58763a68777e591ecff9829741da087f20b9022.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\Delta \\mathrm { D P } ( f ) = \\left| \\mathbb { E } _ { x _ { 0 } \\sim P _ { 0 } , x _ { 1 } \\sim P _ { 1 } } f ( T ( x _ { 0 } , x _ { 1 } , 0 ) ) - \\mathbb { E } _ { x _ { 0 } \\sim P _ { 0 } , x _ { 1 } \\sim P _ { 1 } } f ( T ( x _ { 0 } , x _ { 1 } , 1 ) ) \\right| . } \\\\ { \\displaystyle \\qquad = \\left| \\int _ { 0 } ^ { 1 } \\frac { d } { d t } \\mathbb { E } _ { x _ { 0 } \\sim P _ { 0 } , x _ { 1 } \\sim P _ { 1 } } f ( T ( x _ { 0 } , x _ { 1 } , t ) ) \\right| . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            241,
+            382,
+            754,
+            440
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2 PROOF OF PROPOSITION 2 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            472,
+            400,
+            487
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2. (Gap Regularization) Consider the following minimization problem ",
+        "bbox": [
+            174,
+            498,
+            728,
+            513
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1eddad0023153d8c426a252c8f0e4183d7a363c1e2b580dca0257e2b06d45999.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { f \\in \\mathcal { H } } \\mathbb { E } _ { ( x , y ) \\sim P } [ \\ell ( f ( x ) , y ) ] + \\frac { \\lambda _ { 1 } } { 2 } \\Delta \\mathrm { D } \\mathbf { P } ( f ) ^ { 2 } + \\frac { \\lambda _ { 2 } } { 2 } | | f | | _ { \\mathcal { H } } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            517,
+            678,
+            549
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For a fixed embedding $\\Phi$ , the optimal solution $f ^ { * }$ corresponds to $v ^ { * }$ given by following closed form: ",
+        "bbox": [
+            178,
+            553,
+            818,
+            569
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/311f3a5c39967c49e667250728ba2c4e1e0c28bf158e22b16c25e6ea954336d7.jpg",
+        "text": "$$\nv ^ { \\ast } = \\frac { 1 } { \\lambda _ { 2 } } \\left( \\delta _ { \\pm } - p r o j _ { \\delta _ { 0 , 1 } } ^ { \\frac { \\lambda _ { 2 } } { \\lambda _ { 1 } } } ( \\delta _ { \\pm } ) \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            571,
+            606,
+            607
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where proj is the soft projection defined as pro $\\begin{array} { r } { j _ { u } ^ { \\beta } ( x ) = \\frac { u \\otimes u } { | | u | | ^ { 2 } + \\beta } x } \\end{array}$ ",
+        "bbox": [
+            173,
+            611,
+            609,
+            630
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proof. The problem above can be written as follows: ",
+        "bbox": [
+            174,
+            643,
+            522,
+            659
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/73c25822909c5df0510a85b0dded1567596e84a5a60f98c860256799055d9e3a.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { v \\in \\mathbb { R } ^ { m } } \\mathcal { L } ( v ) : = - ( \\langle v , \\delta _ { \\pm } \\rangle ) + \\frac { \\lambda _ { 1 } } { 2 } | \\langle v , \\delta _ { 0 , 1 } \\rangle | ^ { 2 } + \\frac { \\lambda _ { 2 } } { 2 } | | v | | _ { 2 } ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            662,
+            679,
+            694
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Setting first order condition to zero ",
+        "bbox": [
+            174,
+            698,
+            406,
+            712
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/be626835ed51c60973edf74e142c62ca30979bd2bc3043fd2b4c7c27f4356939.jpg",
+        "text": "$$\n\\begin{array} { r } { \\nabla _ { v } \\mathcal { L } ( v ) = - \\delta _ { \\pm } + \\lambda _ { 1 } \\delta _ { 0 , 1 } \\otimes \\delta _ { 0 , 1 } v + \\lambda _ { 2 } v = 0 , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            717,
+            650,
+            734
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "we obtain ",
+        "bbox": [
+            174,
+            738,
+            240,
+            752
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/29c3db08e93c97b7ba5058c1e4eefbe1629dd79ec378e0364e10d361d5cfaf8c.jpg",
+        "text": "$$\n( \\lambda _ { 1 } \\delta _ { 0 , 1 } \\otimes \\delta _ { 0 , 1 } + \\lambda _ { 2 } I _ { m } ) v ^ { \\ast } = \\delta _ { \\pm } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            751,
+            607,
+            767
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "By inverting and applying the Sherman-Morrison Lemma, we have ",
+        "bbox": [
+            173,
+            768,
+            614,
+            784
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b0776530b2091990ab3073b3333a9ba99731a8a30d0f0de831669d641911b554.jpg",
+        "text": "$$\n\\begin{array} { l l l } { { v ^ { * } } } & { { = } } & { { \\displaystyle ( \\lambda _ { 1 } \\delta _ { 0 , 1 } \\otimes \\delta _ { 0 , 1 } + \\lambda _ { 2 } I _ { m } ) ^ { - 1 } \\delta _ { \\pm } } } \\\\ { { } } & { { = } } & { { \\displaystyle \\lambda _ { 1 } ^ { - 1 } \\left( \\frac { \\lambda _ { 2 } } { \\lambda _ { 1 } } I _ { m } + \\delta _ { 0 , 1 } \\otimes \\delta _ { 0 , 1 } \\right) ^ { - 1 } \\delta _ { \\pm } } } \\\\ { { } } & { { = } } & { { \\displaystyle \\lambda _ { 1 } ^ { - 1 } \\left( I _ { m } - \\frac { ( \\lambda _ { 1 } ) ^ { 2 } \\delta _ { 0 , 1 } \\otimes \\delta _ { 0 , 1 } } { 1 + | | \\delta _ { 0 , 1 } | | ^ { 2 } \\frac { \\lambda _ { 1 } } { \\lambda _ { 2 } } } \\right) \\delta _ { \\pm } } } \\\\ { { } } & { { = } } & { { \\displaystyle \\frac { 1 } { \\lambda _ { 2 } } \\left( I _ { m } - \\frac { \\delta _ { 0 , 1 } \\otimes \\delta _ { 0 , 1 } } { \\lambda _ { 2 } } \\right) \\delta _ { \\pm } . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            786,
+            645,
+            931
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Note the soft projection on a vector $u$ is defined as follows: ",
+        "bbox": [
+            173,
+            103,
+            562,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8f95f44e1f8ecef2c592c1ee7cfc13c1a5f7f7b8fbb188b9411bb92a55d8bc7b.jpg",
+        "text": "$$\np r o j _ { u } ^ { \\beta } ( x ) = \\frac { u \\otimes u } { | | u | | ^ { 2 } + \\beta } x .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            121,
+            583,
+            154
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "It follows that ",
+        "bbox": [
+            173,
+            155,
+            267,
+            170
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/69c76bd47ced71cd85ab097eaea85965d873e614a28fa1e7440e21e091cfd9f9.jpg",
+        "text": "$$\nv ^ { \\ast } = \\frac { 1 } { \\lambda _ { 2 } } \\left( \\delta _ { \\pm } - p r o j _ { \\delta _ { 0 , 1 } } ^ { \\frac { \\lambda _ { 2 } } { \\lambda _ { 1 } } } ( \\delta _ { \\pm } ) \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            166,
+            606,
+            200
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "which can be interpreted as the projection of the label discriminating direction $\\delta _ { \\pm }$ on the subspace that is orthogonal to the group discriminating direction $\\delta _ { 0 , 1 }$ . □ ",
+        "bbox": [
+            171,
+            200,
+            825,
+            229
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 PROOF OF PROPOSITION 3 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            244,
+            400,
+            260
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3. (Fair Mixup) Consider the following minimization problem ",
+        "bbox": [
+            174,
+            271,
+            673,
+            286
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2e82b9c385f69e24aa0de610786988849f090c0e9411be6ae96be206343280e2.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { f \\in \\mathcal { H } } \\mathbb { E } _ { ( x , y ) \\sim P } [ \\ell ( f ( x ) , y ) ] + \\frac { \\lambda _ { 1 } } { 2 } R _ { \\mathrm { m i x u p } } ^ { \\mathrm { D P - 2 } } ( f ) + \\frac { \\lambda _ { 2 } } { 2 } | | f | | _ { \\mathcal { H } } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            289,
+            679,
+            320
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Define $m _ { t } = \\mathbb { E } _ { x _ { 0 } \\sim P _ { 0 } , x _ { 1 } \\sim P _ { 1 } } [ \\Phi ( t x _ { 0 } + ( 1 - t ) x _ { 1 } ) ]$ be the $t$ dependent mean embedding, and $\\dot { m } _ { t }$ its derivative with respect to $t$ . Let $D$ be a positive-semi definite matrix defined as follows: $D =$ $\\begin{array} { r } { \\int _ { 0 } ^ { 1 } { \\dot { m _ { t } } } \\otimes { \\dot { m } } _ { t } d t } \\end{array}$ . Given an embedding $\\Phi$ , the optimal solution $v ^ { * }$ has the following form: ",
+        "bbox": [
+            174,
+            324,
+            826,
+            371
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/655c2d919f19da1d68289d0217169148a5923a70d0250cb9899fba6ddeeabfcc.jpg",
+        "text": "$$\nv ^ { * } = ( \\lambda _ { 1 } D + \\lambda _ { 2 } I _ { m } ) ^ { - 1 } \\delta _ { \\pm } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            375,
+            589,
+            393
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Proof. For $t \\in [ 0 , 1 ]$ , we note by $\\rho _ { t }$ the distribution of $T ( x _ { 0 } , x _ { 1 } , t )$ , for $x _ { 0 } \\sim P _ { 0 } , x _ { 1 } \\sim P _ { 1 }$ , and note $m _ { t } = \\mathbb { E } _ { x \\sim \\rho _ { t } } \\Phi ( x )$ , and $\\dot { m } _ { t }$ its time derivative. We consider the $\\ell _ { 2 }$ variant of $R _ { T } ( f )$ in the analysis: ",
+        "bbox": [
+            171,
+            406,
+            823,
+            436
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f9e597fb4c637b15fa45f537e1aa1f8c3589f3eee91f6e0a0b47f47055e67bc3.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { R _ { T } ^ { 2 } ( f ) } } & { { = } } & { { \\displaystyle \\int _ { 0 } ^ { 1 } \\left| \\frac { d } { d t } \\mu _ { f } ( t ) \\right| ^ { 2 } d t = \\displaystyle \\int _ { 0 } ^ { 1 } \\left| \\frac { d } { d t } \\left. v , m _ { t } \\right. \\right| ^ { 2 } d t } } \\\\ { { } } & { { = } } & { { \\displaystyle \\int _ { 0 } ^ { 1 } | \\langle v , \\dot { m } _ { t } \\rangle | ^ { 2 } d t = \\left. v , \\left( \\displaystyle \\int _ { 0 } ^ { 1 } \\dot { m } _ { t } \\otimes \\dot { m } _ { t } d t \\right) v \\right. , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            440,
+            702,
+            516
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We then expand $\\dot { m } _ { t }$ when $T$ is mixup: ",
+        "bbox": [
+            174,
+            516,
+            426,
+            531
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cf53c0c7ac94e3021c46a38f18a4c87a1eb57352286626b68048abd7a0bb7c61.jpg",
+        "text": "$$\n\\dot { m } _ { t } = \\frac { d } { d t } \\mathbb { E } _ { x _ { 0 } \\sim \\mathbb { P } _ { 0 } , x _ { 1 } \\sim \\mathbb { P } _ { 1 } } \\Phi ( t x _ { 0 } + ( 1 - t ) x _ { 1 } ) = \\mathbb { E } _ { x _ { 0 } \\sim \\mathbb { P } _ { 0 } , x _ { 1 } \\sim \\mathbb { P } _ { 1 } } J \\Phi ( t x _ { 0 } + ( 1 - t ) x _ { 1 } ) ( x _ { 0 } - x _ { 1 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            196,
+            535,
+            799,
+            564
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $J$ denotes the Jacobian. Note that $\\begin{array} { r } { D = \\int _ { 0 } ^ { 1 } \\dot { m _ { t } } \\otimes \\dot { m } _ { t } d t } \\end{array}$ , hence for the classification with fair mixup regularizer, the problem is equivalent to: ",
+        "bbox": [
+            174,
+            568,
+            820,
+            599
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3e2d343fb1b750ec1c021dacafb21a0f5c9eb5ac33b6bb9e7fb2a1bc1d63105e.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { v \\in \\mathbb { R } ^ { m } } \\mathcal { L } ( v ) : = - ( \\langle v , \\delta _ { \\pm } \\rangle ) + \\frac { \\lambda _ { 1 } } { 2 } \\langle v , D v \\rangle + \\frac { \\lambda _ { 2 } } { 2 } | | v | | _ { 2 } ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            602,
+            669,
+            633
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Setting first order condition we obtain $( \\lambda _ { 1 } D + \\lambda _ { 2 } I _ { m } ) v ^ { \\ast } = \\delta _ { \\pm }$ , which gives the optimal solution ",
+        "bbox": [
+            178,
+            645,
+            800,
+            661
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8635fa88c098edfe50e795f251323e71a40036739d80c93aee29b84110a1b249.jpg",
+        "text": "$$\nv ^ { * } = ( \\lambda _ { 1 } D + \\lambda _ { 2 } I _ { m } ) ^ { - 1 } \\delta _ { \\pm } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            664,
+            588,
+            683
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The corresponding optimal fair mixup classifier can be finally written as ",
+        "bbox": [
+            174,
+            685,
+            648,
+            700
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d0047c072fae0c32c353560f39081e7ab1984359aa0d5ccf975b699b8c9229ab.jpg",
+        "text": "$$\nf ( x ) = \\left. \\delta _ { \\pm } , ( \\lambda _ { 1 } D + \\lambda _ { 2 } I _ { m } ) ^ { - 1 } \\Phi ( x ) \\right. .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            702,
+            625,
+            722
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B EXPERIMENT DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            758,
+            398,
+            775
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Adult We follow the preprocessing procedure of Yurochkin et al. (2019) by removing some features in the dataset3. We then encode the discrete and quantized continuous attributes with one-hot encoding. We retrain each model 10 times with batch size 1000 and report the mean accuracy and fairness measurement. The models are selected via the performance on validation set. In each trial, the dataset is randomly split into training and testing set with partition $8 0 \\%$ and $2 0 \\%$ , respectively. The models are optimized with Adam optimizer (Kingma & Ba, 2014) with learning rate $\\mathrm { \\bar { 1 } } \\times e ^ { - \\mathrm { \\bar { 3 } } }$ For DP, we sample 500 datapoints for each $A \\in \\{ 0 , 1 \\}$ to form a batch. Similarly, for EO, we sample 250 datapoints for each $( A , Y )$ pair where $A , Y \\in \\{ 0 , 1 \\}$ . ",
+        "bbox": [
+            173,
+            789,
+            825,
+            902
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "CelebA Model-wise, we extract the feature of size 512 after the average pooling layer of ResNet18. A two-layer ReLU network with hidden size 512 is then trained to perform prediction. Percentage of positive-labeled datapoints for attractive, wavy hair, and smiling that is male are $2 2 . 7 \\%$ , $1 8 . 3 6 \\%$ , and $3 4 . 6 \\%$ , respectively. We use the original validation set of CelebA to perform model selection and report the accuracy and fairness metrics on the testing set. The visualization paths are also plotted with respect to the testing data. To implement manifold mixup, we interpolate the spatial features before the average pooling layer. Similarly, all the models are optimized with Adam optimizer with learning rate $1 \\times e ^ { - 3 }$ . ",
+        "bbox": [
+            173,
+            103,
+            825,
+            215
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Toxicity Classification We download the Jigsaw toxic comment dataset from Kaggle website4. Percentage of positive-labeled datapoints for black and asian are $1 8 . 8 \\%$ and $6 . 4 \\%$ , respectively, which together results in a dataset of size 22835. We retrain each model 10 times with batch size 200 and report the mean accuracy and fairness measurement. The models are selected via the performance on validation set. The batch-sampling and data splitting procedure is the same as the one for Adult dataset. The models are again optimized with Adam optimizer with learning rate $1 \\times e ^ { - 3 }$ . ",
+        "bbox": [
+            174,
+            229,
+            825,
+            314
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C ADDITIONAL EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            334,
+            444,
+            351
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.1 TRAINING PERFORMANCE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            366,
+            400,
+            380
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In Figure 7, we show the training performance for Adult dataset. As expected, GapReg outperforms Fair mixup on the training set since it directly optimizes the fairness metric. The results also support our motivation: the constraints that are satisfied during training might not generalize at evaluation time. ",
+        "bbox": [
+            173,
+            391,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/38d30429260f23fa3c4cb2c80e1c0feaa90205a73b7d9a5233050439ab5c215b.jpg",
+        "image_caption": [
+            "Figure 7: Training Performance on Adult Dataset. The tradeoff between AP and $\\Delta \\mathrm { D P } / \\Delta \\mathrm { E O }$ on training set. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            271,
+            459,
+            725,
+            604
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.2 EVALUATION METRIC ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            683,
+            369,
+            696
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The relaxed evaluation metric could overestimate the performance when the predicted confidence is significantly different between groups. For instance, a classifier $f$ can be completely unfair while satisfying this condition: $f ( x ) { \\overset { } { = } } 1$ w.p. $60 \\%$ , 0 w.p. $40 \\%$ on $P _ { 0 }$ , and $f ( x ) \\stackrel { - } { = } 0 . 6$ w.p. $100 \\%$ on $P _ { 1 }$ . This satisfies this expectation-based condition. However, it is highly unfair if we binarize the prediction by setting the threshold $= 0 . 5$ . ",
+        "bbox": [
+            173,
+            708,
+            826,
+            779
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To overcome this issue, let $f _ { t }$ be the binarized predictor $f _ { t } ( x ) = \\mathbb { 1 } ( f ( x ) \\geq t )$ , we evaluate the model with average $\\Delta \\mathrm { D P } ( \\overline { { \\Delta \\mathrm { D P } } } )$ defined as follows: ",
+        "bbox": [
+            176,
+            785,
+            823,
+            815
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/809f942bbd7cf5549d0783e9339b7086aa23f6d0716830d0693892d616d6c001.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\overline { { \\Delta \\mathrm { D P } } } ( f ) = \\displaystyle \\frac { 1 } { | T | } \\sum _ { t \\in \\mathcal { T } } \\left| \\mathbb { E } _ { \\boldsymbol { x } \\sim P _ { 0 } } f _ { t } ( \\boldsymbol { x } ) - \\mathbb { E } _ { \\boldsymbol { x } \\sim P _ { 1 } } f _ { t } ( \\boldsymbol { x } ) \\right| ; } \\\\ & { \\overline { { \\Delta \\mathrm { E O } } } ( f ) = \\displaystyle \\frac { 1 } { | T | } \\sum _ { t \\in \\mathcal { T } } \\sum _ { y \\in \\{ 0 , 1 \\} } \\left| \\mathbb { E } _ { \\boldsymbol { x } \\sim P _ { 0 } ^ { y } } f _ { t } ( \\boldsymbol { x } ) - \\mathbb { E } _ { \\boldsymbol { x } \\sim P _ { 1 } ^ { y } } f _ { t } ( \\boldsymbol { x } ) \\right| , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            821,
+            694,
+            901
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $\\tau$ is a set of threshold values. $\\overline { { \\Delta \\mathsf { D P } } }$ averages the $\\Delta \\mathbf { D P }$ with binarized predictions derived via different thresholds. For instance, by averaging the $\\Delta \\mathbf { D P }$ with thersholds $\\mathcal { T } = [ 0 . 1 , 0 . 2 , \\cdot \\cdot \\cdot , 0 . 9 ] ,$ , $\\overline { { \\Delta \\mathrm { D P } } } = 0 . 5$ instead of 0 for the example above, which captures the unfairness between groups. We report $\\overline { { \\Delta \\mathsf { D P } } }$ for each methods with thersholds $\\mathcal { T } = [ 0 . 1 , \\bar { 0 } . 2 , \\cdot \\cdot \\cdot . , 0 . 9 ]$ in Figure 8. Similarly, fair mixup exhibits the best tradeoff comparing to the baselines for demographic parity. For equalized odds, the performances of fair mixup and GapReg are similar, where fair mixup achieves a better tradeoff when $\\overline { { \\Delta \\mathrm { E O } } }$ is small. ",
+        "bbox": [
+            173,
+            102,
+            826,
+            203
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/5ee5e26b6b027b544d99a89cf0f68d7011e47bc42ed3d946f586dec59c345bb7.jpg",
+        "image_caption": [
+            "Figure 8: Average ∆DP and ∆EO on Adult Dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            236,
+            215,
+            758,
+            383
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C.3 SMALLER MODEL SIZE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            446,
+            380,
+            462
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "To examine the effect of model size, we reduce the hidden size from 200 to 50 and show the result in Figure 9. Overall, the performance does not vary significantly after reducing the model size. We can again observe that fair mixup outperform the baselines for $\\Delta \\mathbf { D P }$ . Similar to the results in section C.2, the performances of fair mixup and GapReg are similar, where fair mixup achieves a better tradeoff when $\\Delta \\mathrm { E O }$ is small. ",
+        "bbox": [
+            173,
+            472,
+            826,
+            541
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/472cd71dd38bb056ebae1b91a67bd5645ae14173cedc6b390a003a86962f3536.jpg",
+        "image_caption": [
+            "Figure 9: Reducing the Model Size on Adult Dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            271,
+            555,
+            727,
+            700
+        ],
+        "page_idx": 14
+    }
+]

parse/train/FjnCWyzXu8m/FjnCWyzXu8m.md

@@ -0,0 +1,420 @@
+# Rethinking supervised learning: insights from biological learning and from calling it by its name
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+The renaissance of artificial neural networks was catalysed by the success of classification models, tagged by the community with the broader term supervised learning. The extraordinary results gave rise to a hype loaded with ambitious promises and overstatements. Soon the community realised that the success owed much to the availability of thousands of labelled examples and supervised learning went, for many, from glory to shame: Some criticised deep learning as a whole and others proclaimed that the way forward had to be “alternatives” to supervised learning: predictive, unsupervised, semi-supervised and, more recently, self-supervised learning. However, these seem all brand names, rather than actual categories of a theoretically grounded taxonomy. Moreover, the call to banish supervised learning was motivated by the questionable claim that humans learn with little or no supervision and are capable of robust out-of-distribution generalisation. Here, we review insights about learning and supervision in nature, revisit the notion that learning and generalization are not possible without supervision or inductive biases and argue that we will make better progress if we just call it by its name.
+
+# 16 1 Introduction
+
+17 The re-emergence of deep learning during the last decade due to the noteworthy achievements   
+18 of artificial neural networks (ANN) built up a sort of philosophy that nearly anything could be   
+19 automatically learnt from data without human intervention, in contrast to the previous approaches:   
+20 [hand designing good feature extractors, engineering skill and domain expertise]   
+21 can all be avoided if good features can be learned automatically using a general  
+22 purpose learning procedure. This is the key advantage of deep learning (LeCun   
+23 et al., 2015).   
+24 Read in hindsight, this claim was clearly an overstatement. The success of deep learning has   
+25 required iterative hand design of network architectures and techniques that demanded collective, high   
+26 engineering skill and large doses of interdisciplinary domain expertise. Furthermore, deep learning   
+27 owes much to the immense computational power poured into training artificial networks (Amodei &   
+28 Hernandez, 2018; Schwartz et al., 2019) and to the human effort of manually collecting and labelling   
+29 thousands of images and other data modalities (Russakovsky et al., 2015; Cao et al., 2018). However,   
+30 the gist of the claim has permeated machine learning research and is pervasive up to these days.   
+31 The realisation that the success of deep learning was largely due to the availability of huge labelled data   
+32 sets prompted various reactions: some authors strongly questioned the usefulness of the algorithms   
+33 (Marcus, 2018); some delved into the question of whether neural networks generalise beyond or   
+34 simply memorise the training examples (Zhang et al., 2017; Arpit et al., 2017); and some proposed   
+35 new research horizons that can be overly ambitious and potentially misleading: “learning a class   
+36 from a single labelled example”, based on the statement that “humans learn new concepts with very   
+37 little supervision, [but] the standard supervised deep learning paradigm does not offer a satisfactory   
+38 solution for learning new concepts rapidly from little data” (Vinyals et al., 2016). As a consequence,   
+39 multiple research programmes, with various brand names, followed up with the aim of minimising or   
+40 removing the need for “supervision” to train neural networks: few-shot, one-shot, zero-shot, predictive,   
+41 unsupervised, semi-supervised and self-supervised learning are only a few popular examples.   
+42 Exploring alternatives to classification and improving the efficiency of learning algorithms should   
+43 indeed be a priority of machine learning research. As a matter of fact, related approaches have   
+44 been subject of study since long before the explosion of deep learning (Hinton & Sejnowski, 1999;   
+45 Chapelle et al., 2006). However, the current publication and discussion trends in the field denote   
+46 overambitious promises that are in part based on misconceptions and overstatements about biological   
+47 learning, and amplified by overselling nomenclature. While much of the research output derived from   
+48 these programmes does provide us with useful techniques and insight, it leaves behind a landscape of   
+49 confusing terminology and tangled research directions that are hard to navigate and lead many astray.   
+50 In this paper, we reflect upon fundamental concepts in machine learning such as supervision, in  
+51 ductive biases and generalisation, which in spite of resting on theoretical grounds, are at the core   
+52 of misconceptions and overstatements about deep learning commonly seen in the literature. First,   
+53 we review aspects from biological learning, and compare them to the traits often (mis)attributed to   
+54 human learning and generalisation in the machine learning literature (Section 2). Second, we revisit   
+55 insights from classical statistical learning and critically review the terminology and current trends in   
+56 deep learning research (Section 3). Altogether, we aim at tempering certain claims and promises of   
+57 deep learning, helping mitigate the confusion over the terminology and suggesting desirable—in our   
+58 opinion—directions and changes in machine learning research.
+
+# 59 2 Supervision in biological learning
+
+60 The link between artificial intelligence—specifically artificial neural networks (Rosenblatt, 1958;   
+61 Fukushima & Miyake, 1982)—and biological learning systems is intrinsic to the field, as one long  
+62 term goal of artificial intelligence is to mirror the capabilities of human intelligence. However, these   
+63 capabilities are, in our view, often overestimated. One example is the argument that intelligence in   
+64 nature evolves without supervision and is capable of robust out-of-distribution generalisation. In   
+65 particular, it is often claimed that humans and other animals learn to visually categorise objects with   
+66 little or no supervision from a few examples (Vinyals et al., 2016; Marcus, 2018; Morgenstern et al.,   
+67 2019). In what follows, we will discuss three aspects of biological learning to argue against this   
+68 view, so as to gain insights that better inform our progress in machine learning: first, we will discuss   
+69 how generalisation requires exposure to relevant training data; second, we will review the variety of   
+70 supervised signals that the brain has access to; third, we will comment on the role of evolution and   
+71 brain development.
+
+# 72 2.1 Generalisation requires exposure to relevant training data
+
+73 In the argument that machine learning models should generalise from a few examples, there seems   
+74 to be a promise or aspiration that future better methods will be able to perform robust visual object   
+75 categorisation—for instance—among many object classes after being trained on one or a few examples   
+76 per class. While a primary objective is to develop techniques that efficiently extract the maximum   
+77 possible information from the available examples, we should also remind ourselves that no machine   
+78 learning algorithm can robustly learn anything that cannot be inferred from the data it has been trained   
+79 on. Although this may seem to contradict certain current trends and statements in the literature, we   
+80 should also bear in mind that learning in nature is not different.   
+81 First, the amount of data that animals and humans in particular are exposed to is often underestimated.   
+82 A biological brain continuously receives, processes and integrates multimodal inputs from various   
+83 sensors—images (light), sound, smell, etc. Humans do not learn to recognise objects by looking   
+84 at photos from ImageNet, but are rather exposed to a continuous flow of visual stimuli with slow   
+85 changes of the viewing angle and lighting conditions. Furthermore, the stimuli are coherent across   
+86 modalities, we are allowed to interact with the objects and we even receive multiple supervision   
+87 signals, as we discuss later.   
+88 The exposure to so much training data makes the human visual system remarkably robust, but still its   
+89 capabilities are optimised for the tasks it needs to perform and largely determined by the training data   
+90 distribution—and years of evolution, as we will discuss below. For instance, a well-studied property   
+91 of human vision is that our face recognition ability is severely impaired if faces are presented upside   
+92 down (Yin, 1969; Valentine, 1988). Setting aside the specific complexity of face processing in the   
+93 brain, a compelling explanation for this impairment is that we are simply not used to seeing and   
+94 recognising inverted faces. More generally, while human perception of objects is largely invariant   
+95 under certain conditions (Biederman & Bar, 1999), object recognition is sensitive to changes in view   
+96 angle (Tarr et al., 1998), especially when we see objects from unfamiliar viewpoints (Edelman &   
+97 Bülthoff, 1992; Bülthoff & Newell, 2006; Milivojevic, 2012).   
+98 Furthermore, although better than the one-shot or few-shot generalisation of current ANNs, humans   
+99 also have limited ability to recognise truly novel classes (Morgenstern et al., 2019). Interestingly,   
+100 experiments with certain novel classes of objects known as Greebles showed that, with sufficient   
+101 training, humans can acquire expertise in recognising new objects from different viewpoints, even   
+102 making use of an area of the brain—the fusiform face area—that typically responds strongly with   
+103 face stimuli (Gauthier et al., 1999). This provides evidence that recognition from multiple viewpoints   
+104 is possible but only developed after exposure to similar conditions, that is relevant data. This is   
+105 reminiscent of the effectiveness of data augmentation in deep learning, compared to more naïve   
+106 regularisation methods (Hernández-García & König, 2018).   
+107 The need for exposure to relevant stimuli challenges the notion that humans are capable of strong   
+108 out-of-distribution generalisation. Rather, it seems that the transfer learning capabilities of humans   
+109 are limited to relatively small changes in the data distribution. A compelling example is our difficulty   
+110 to learn new languages: someone who natively speaks or has learnt Spanish will be able to transfer a   
+111 significant amount of knowledge if they are to learn Italian, due to the overlap in the data distribution,   
+112 but they will have very little to transfer for learning Kanien’kéha or Mandarin.
+
+# 113 2.2 Supervised signals for the brain
+
+114 Another commonly found argument has it that children—animals in general—learn robust object   
+115 recognition without supervision: “a child can generalize the concept of ‘giraffe’ from a single picture   
+116 in a book” (Vinyals et al., 2016). First of all, we should mention the role of evolution (expanded in   
+117 Section 2.3), which can be interpreted as a pre-trained model, optimised through millions of years of   
+118 data with natural selection as a supervisory signal (Zador, 2019). Second, there is abundant evidence   
+119 to argue against the very claim that children—and adults—learn in fully unsupervised fashion.   
+120 Obviously, the kind of supervision that humans make use of is not that of classification algorithms—   
+121 we do not see a class label on top of every object we look at. However, we receive supervision   
+122 from multiple sources. Even though not for every visual stimulus, children do frequently receive   
+123 information about the object classes they see. For instance, parents would point at objects and name   
+124 them, then we learn how to read, and generally play a crucial role as teachers in language development   
+125 (Kuhl, 2007). Non-human animals such as zebra finches learning to sing have also been found to rely   
+126 on feedback (supervision) from the female adult and not just imitation Carouso-Peck & Goldstein   
+127 (2019). Furthermore, humans usually follow guided hierarchical learning: children do not directly   
+128 learn to tell apart breeds of dogs, but rather start with umbrella terms and then progressively learn   
+129 down the class hierarchy (Bornstein & Arterberry, 2010; Spriet et al., 2021). Gopnik (2021) has   
+130 asserted that “we learn more from other people than we do from any other source” and Hasson et al.   
+131 (2020) mention other examples of supervision from social cues, that is from other humans, such as   
+132 learning to recognise individual faces, produce grammatical sentences, read and write; as well as   
+133 from embodiment and action, such as learning to balance the body while walking or grasping objects.   
+134 In all these actions, we can identify a supervisory signal that surely influences learning in the brain   
+135 (Shapiro, 2012; Gopnik et al., 2020).   
+136 While these supervision signals largely differ from what is most commonly considered supervised   
+137 learning in machine learning, we can still draw some parallels with human learning. We learn to   
+138 categorise many concepts and objects as children, but most people carry on learning new categories as   
+139 adults. For example, some people put effort in improving their understanding of the natural world by   
+140 learning to recognise and name trees, plants or birds. Those who have engaged in such an endeavour   
+141 may have noticed that the learning process is easier and faster if we count upon the expert knowledge   
+142 of a friend or of technology such as iNaturalist (Van Horn et al., 2018). Another example: those   
+143 who have—or attempted to learn—a new language as an adult may have realised that whereas it is   
+144 possible to learn the meaning of a new word by repeated exposure to it in multiple contexts, it is   
+145 certainly easier if we look up the ground truth definition in a dictionary or, even easier, if there exists   
+146 a direct mapping to a word in our native language. Summing up, not only does supervision facilitate   
+147 learning, but human beings actively seek for it.   
+148 Besides this kind of explicit supervision, the brain certainly makes use of more subtle, implicit   
+149 supervised signals, such as temporal stability (Becker, 1999; Wyss et al., 2003): The light that enters   
+150 the retina, and the sound waves that reach the cochlea, are not random signals from a sequence of   
+151 rapidly changing arbitrary photos or noise, but highly coherent and regular flows of slowly changing   
+152 stimuli, especially at the higher, semantical level (Kording et al., 2004). At the very least, this is how   
+153 we perceive it and if such a smooth perception turns out to be a consequence rather than a cause, then   
+154 it should be a by-product of a long process of evolution that would be worth taking into account.
+
+# 155 2.3 The role of evolution and brain development
+
+156 In the previous sections, we have discussed some misconceptions or overstatements about how   
+157 humans learn and generalise that are often found in the machine literature. Namely, that humans   
+158 are able to generalise from a few examples and that this occurs with little or no supervision. Still,   
+159 the commonplace comparison of artificial neural networks with human learning and the brain often   
+160 misses a fundamental component of biology, recently brought to the fore by Zador (2019) and Hasson   
+161 et al. (2020), although considered since the early days of artificial intelligence (Turing, 1968): the   
+162 role that millions of years of evolution have played in developing the nervous systems of organisms   
+163 in nature, including the human brain.   
+164 The most common way of training artificial neural networks, especially in machine learning research,   
+165 is from tabula rasa, that is from randomly initialised parameters1. In contrast, a large part of the   
+166 brain connectivity is encoded genetically and certain properties and behaviour are known to be   
+167 innate, that is developed without prior exposure to stimuli (Farroni et al., 2005; Spriet et al., 2021).   
+168 Importantly, evolution not only provides innate behaviour, but also determines what cannot be learnt,   
+169 or relevant constraints—scientists who have trained animals in the laboratory for psychological   
+170 or neuroscientific studies are well aware that tasks have to be carefully adapted to the ecological   
+171 behaviour and limitations of the animal, determined by evolution.   
+172 Taking into account the role of evolution, we can draw conclusions that relate to the claims discussed   
+173 in the previous sections. If our brains are the product of millions of years of exposure to relevant   
+174 stimuli and adaptation, is it really fair to say that humans are capable of robust out-of-distribution   
+175 generalisation and that we learn from from a few examples? If evolution has largely determined   
+176 what our brain can and cannot learn, providing as with a “pre-trained model”, is it really fair to   
+177 say that humans learn in a unsupervised fashion? This questions are relevant for machine learning   
+178 research: if we take biological learning as motivation for artificial intelligence, should we not temper   
+179 our expectations of what learning algorithms should aspire to? And, therefore, would it not be worth   
+180 reconsidering some research programmes?   
+181 On the flip side, insights from evolutionary theory are likely to be a fruitful source of inspiration   
+182 for machine learning (Hasson et al., 2020; Zador, 2019). As we have observed, training a neural   
+183 network from scratch may be more similar to a simulation of evolution than to the process by which   
+184 an adult learns a new concept. As a shortcut to simulating evolution, neuroscience is a rich source   
+185 of inspiration of constraints and inductive biases that determine learning in the biological brain and   
+186 can potentially inform machine learning (Hassabis et al., 2017; Lindsey et al., 2019). For instance,   
+187 simulating properties of the primary visual cortex in the early layers of an artificial neural network   
+188 has been shown to improve adversarial robustness (Dapello et al., 2020; Malhotra et al., 2020).
+
+Besides evolution, the focus on the capabilities of adults often makes us miss another important aspect of biological learning, particularly important in humans: the role of learning in infancy and brain development. While learning occurs too in adulthood, childhood is a particularly important and
+
+192 active time for learning (Atkinson, 2002; Gelman & Meyer, 2011). In fact, sensitive or critical periods   
+193 for learning in infancy have been described or hypothesised, for example for vision (Harwerth et al.,   
+194 1986) and language development (Lenneberg, 1967). Machine learning papers that draw motivation   
+195 from the alleged generalisation capabilities of humans often underestimate the amount of input stimuli   
+196 and supervision that infants receive (Gopnik, 2020). However, childhood can be regarded as period   
+197 dedicated almost exclusively to learn, not only formally from parents and teachers, but also through   
+198 playing, which plays a critical role in cognitive development Burghardt (2005); Pelz & Kidd (2020).   
+199 Finally, the fact that humans—and other cognitively advanced animals, such as corvid birds, which   
+200 also exhibit cultural learning—have a comparatively long childhood period, has led Uomini et al.   
+201 (2020) to recently proposed that extended parenting is pivotal in the evolution of cognition. This   
+202 adds to the discussion on the undervalued role of supervision. In sum, we propose machine learning   
+203 research can benefit from drawing inspiration from both evolutionary biology and the literature on   
+204 developmental psychology, brain development and life history and learning (Gopnik et al., 2020).
+
+# 205 3 Supervision in machine learning
+
+If we open a machine learning textbook (Murphy, 2012; Abu-Mostafa et al., 2012; Goodfellow et al., 2016), we will most surely find a taxonomy of learning algorithms with a clear distinction between supervised and unsupervised learning. However, while this separation can be useful, the boundaries are certainly not clear. As a matter of fact, if we take a look at the deep learning literature of the past years, we will also find abundant work on some variants supposedly in between—semi-supervised learning, self-supervised learning, etc.—whose definitions are all but clear.
+
+# 3.1 Catastrophic forgetting of old concepts
+
+If we recall a classical result in statistical learning theory and inference, the no free lunch theorem (Wolpert, 1996), no learning algorithm is better than any other at classifying unobserved data points, when averaged over all possible data distributions. Therefore, we need to constrain the distributions or, in other words, introduce prior knowledge—that is supervision. Recently, Locatello et al. (2018) obtained a related result for the case of unsupervised learning of disentangled representations: without inductive biases for both the models and the data sets, unsupervised disentanglement learning is fundamentally impossible. These results are purely theoretical and have limited impact on real world applications (Giraud-Carrier & Provost, 2005), precisely because in practice we use multiple inductive biases and implicit supervision, even when we do so-called unsupervised learning.
+
+In a strict sense, even the classical, purely unsupervised methods, such as independent component analysis or nearest neighbours classifiers, make use of inductive biases, such as independence or minimum distance, respectively. Without inductive bias, learning is not possible: purely unsupervised learning is an illusion. While this is not news, the terminology used in the recent and current machine learning literature seems to reject supervision and neglect these nuances, evidencing that the field suffers catastrophic forgetting of well-established notions.
+
+# 3.2 The brands of alt-supervised learning
+
+Particularly in deep learning and computer vision, the term supervised learning has adopted, in practice, the meaning of classification of examples annotated by humans, that is models trained on examples labelled according to, for instance, the object classes. This is yet another instance of catastrophic forgetting—or, at best, abuse—of well-established concepts. It should not be necessary to recall that, first of all, supervised learning is a broader category than classification, which includes also regression and ranking, among other learning modalities. Second, even if we narrow our view to classification only, supervised learning is not restricted to learning from examples annotated by humans. Goodfellow et al. (2016) did not overlook this in their definition of supervised learning: “In many cases the outputs y may be difficult to collect automatically and must be provided by a human ‘supervisor,’ but the term still applies even when the training set targets were collected automatically”.
+
+239 In turn, the term unsupervised learning is now used for any model that does not use manually collected   
+240 labels, regardless of what other kind of supervision it may use. Further, the term semi-supervised   
+241 learning generally refers in practice to models that are trained with a fraction of the labels, but are   
+242 tested on the same classification benchmarks. Finally, the term self-supervised learning has recently   
+43 gained much popularity, referring to models that are trained on tasks other than the standard task   
+44 defined by classification labels.   
+245 Some of the methods proposed under these categories are certainly useful—that is not the subject of   
+246 criticism of this work—but the terminology is overly confusing and unnecessary. A newcomer would   
+247 easily fall into a scientific rabbit hole trying to discern the meaning of each of these names through   
+248 publications—not to mention if they incorporated social media discussions into their endeavour. By   
+249 way of illustration, the authors of this paper have witnessed how a recurrent question by students who   
+250 learn about recent deep learning methods is whether there is any difference between self-supervised   
+251 and unsupervised learning. Are students missing something fundamental? The following anecdotal   
+252 recall of influential keynote talks at artificial intelligence conferences should shed some light on part   
+253 of the origins of this confusion: In December 2016, Prof. Yann LeCun titled his NeurIPS keynote   
+254 presentation “Predictive Learning”, to refer to “what many people mean by unsupervised learning”   
+255 (LeCun, 2016). A few years later, in his keynote presentation at ISSCC in February 2019, he spoke   
+256 about similar ideas, but this time the title was “Self-Supervised Learning” (LeCun, 2019). In social   
+257 media, he wrote: “I now call it ‘self-supervised learning’, because ‘unsupervised’ is both a loaded   
+258 and confusing term” . Students may be getting things rather right.
+
+Is there then a fundamental difference—a theoretically grounded one—between the deep learning methods labelled as unsupervised learning and more recently self-supervised learning? We argue that these are mostly brand names that reflect trends in the field, adding noise to the scientific progress and leading many astray. Therefore, we propose that, given the recent progress, the field of machine learning research would benefit from an exercise of self-reflection and from an effort to devise a rigorous taxonomy of the variety of methods. From a theoretical point of view, both the conventional classification models and the recent wave of self-supervised tasks can all be formalised as sub-categories of supervised learning.
+
+# 3.3 Supervision comes in different flavours
+
+In Section 2.2, we have seen examples of different forms of supervision used by humans and other animals. In machine learning, the field focused for many years on a few loss functions, such as classification and simple forms of regression. The relatively recent explosion of deep learning has brought about the development of several libraries for automatic differentiation (Baydin et al., 2017), which in turn have enabled the proposal of multiple loss functions and learning tasks with various types of supervision that can easily be optimised numerically by stochastic gradient descent and artificial neural networks. This has certainly opened promising and already fruitful avenues to incorporate richer forms of supervision and inductive biases other than classification, some inspired by biological learning, into machine learning algorithms.
+
+277 A currently popular example is image data augmentation: Although until recently it was seen as a   
+278 naïve technique to simply create additional training data, data augmentation actually encodes rich prior   
+279 knowledge about human visual perception, in the case of computer vision. This is why it outperforms   
+280 explicit regularisation methods, which provide less effective inductive biases Hernández-García &   
+281 König (2018), and was used in “semi-supervised” tasks Laine & Aila (2016). The rich information   
+282 embedded in image transformations has been used to encourage invariant outputs under different   
+283 augmentations through contrastive losses (Ye et al., 2019), and even at intermediate representations,   
+284 inspired by the invariance in the visual cortex (Hernández-García et al., 2019), although these methods   
+285 were not branded as self-supervision. The use of this term for losses based on data augmentation   
+286 was further popularised after the success of similar methods such as SimCLR (Chen et al., 2020).   
+287 Beyond data augmentation invariance, the zoo of self-supervised learning tasks in computer vision is   
+288 rich and diverse: classifying the rotation applied to image patches (Gidaris et al., 2018), predicting   
+289 image colourisation (Larsson et al., 2017), classifying the relative position of two image patches   
+290 (Doersch et al., 2015), or even solving full jigsaw puzzles (Noroozi & Favaro, 2016) (Jing & Tian   
+291 (2020) recently performed an extensive review).   
+292 The current trend is to refer to these methods as self-supervised learning, but similar methods were   
+293 referred to in the past as semi-supervised, unsupervised, and even predictive learning, as we have seen.   
+294 A look at the papers reveals that these terms have been used mostly interchangeably. The terms self  
+295 and semi- and unsupervised learning imply that less supervision is used, but it would be misleading to   
+296 seriously argue that the tasks are devoid of supervision. Most of these techniques make use of a wide   
+297 range of surrogate tasks with supervisory signals defined by humans. In fact, they could have been   
+298 called hyper-supervised2 learning. Here, we contend that these methods are all variants of supervised   
+299 learning, only that supervision comes in different flavours, both in biological and machine learning,   
+300 and we should call it by its name and ideally develop a rigorous taxonomy.
+
+# 01 4 Discussion
+
+In this paper, we have discussed some of the overambitious promises of the deep learning hype, namely that machine learning models should be able to generalise to unseen distributions, from a few examples, without human intervention or supervision. These claims have often been motivated by alleged generalisation capabilities of humans. In order to assess these motivations, we have first reviewed, in Section 2, some often overlooked characteristics of biological learning relevant to machine learning research. In particular, we have argued that humans and other animals receive extensive and diverse input stimuli as well as multiple supervisory signals, including the long history of evolution and cultural transmission. In the light of these insights from biological learning, we have then, in Section 3, critically reviewed the various terms that are currently used to refer to supposed alternatives to supervised learning: semi-, self- and unsupervised learning, among others. In sum, we pointed out that all these approaches are in fact supervised learning—though not necessarily classification—and the machine learning (research) community would benefit from using more rigorous, less overselling nomenclature, and from devising a more rigorous taxonomy.
+
+15 Supervision is not evil. It is at the core of statistical learning theory: learning is impossible without   
+316 inductive biases or supervision. But supervision comes in different flavours, not only as classification   
+17 labels. Neither is deep learning some sort of exceptional solution to learn without human intervention   
+318 and supervision, nor is it a hopeless model class because it requires large data sets (Marcus, 2018).   
+319 The human visual system is exposed to a lot of stimuli too. One exceptional advantage of deep   
+320 learning is precisely that it is possible to effectively optimise different learning objectives, almost   
+321 end-to-end, from large collections of nearly naturalistic sensory signals, such as digital images (Saxe   
+322 et al., 2020). While other models are known to scale poorly as the amount of data increases, neural   
+323 networks excel at fitting the training data and interpolating on unseen examples (Belkin et al., 2019;   
+324 Hasson et al., 2020). This is a feature, not a bug. But we will make better progress if we exploit   
+325 these advantages of deep learning without neglecting that supervision will always be necessary—the   
+26 critical goal is how to best incorporate it and exploit it.
+
+In this regard, we argue that deep learning needs more supervision, and not less. A major focus of the deep learning community in the last decade has been image object classification. This has brought about unprecedented progress and unveiled the limitations of having classification as chief task and class labels as main supervisory signal. For example, deep classifiers have been found to learn spurious features that are highly discriminative for the classification task but with little true generalisation power and clearly not aligned with perceptual features (Jo & Bengio, 2017; Wang et al., 2019; Geirhos et al., 2020). In fact, this mismatch has been argued to be at the root of adversarial vulnerability (Ilyas et al., 2019) and seems to be the consequence of training highly expressive, over-parameterised models in heavily unconstrained tasks. This can be addressed with meaningful constraints, that is more and richer supervision, possibly inspired by human perception and biological learning. For example, combining a classification loss with a similarity loss inspired by the invariance in the visual cortex yields more robust representations without detriment to categorisation (HernándezGarcía et al., 2019), and simulating the properties of the primary visual cortex may improve the adversarial robustness of neural networks (Dapello et al., 2020). Expanding in this direction leads to biologically-inspired, multi-task and representation learning, and away from just classification.
+
+# 342 5 Conclusions for future research directions
+
+The chief goal of this paper is rather descriptive than prescriptive. We have aimed to identify and describe aspects of the current trends in machine learning research that could be improved, in the hope of inspiring future work that effectively address them. Nonetheless, throughout the paper we have made suggestions that may help mitigate the confusion with the terminology, clarify research directions and ultimately bring about scientific progress in machine learning research. We outline these suggestions here to conclude the paper.
+
+349 We have drawn parallels from cognitive neuroscience to contend that learning in nature also requires   
+350 abundant data and supervision in multiple forms. Even evolution can be regarded as an optimisation   
+351 process where natural selection is the supervisory signal. We have argued, as others have before, that   
+352 these insights from biology, neuroscience and developmental psychology, among other fields, offer a   
+353 great opportunity for machine learning research to draw inspiration and calibrate its compass.   
+354 As we have discussed, research in deep learning has departed from pure classification and has been   
+355 exploring new learning tasks and ways of training artificial neural networks. Nonetheless, in some   
+356 fields such as computer vision, the ultimate benchmark to assess the value of a method is still the   
+357 accuracy on classification data sets, such as ImageNet, even though there is evidence of overfitting   
+358 the test set. While object recognition will remain an important benchmark, as deep learning is   
+359 well suited to learn representations, we should develop methods to assess the quality of the learnt   
+360 representations for tasks other than classification. In this regard, we encourage researchers to evaluate   
+361 their models with tests that are still not widespread, such as the suitability for transfer learning,   
+362 adversarial robustness, comparison with brain measurements, behavioural tasks, etc.   
+363 We have also argued that the field would benefit from an effort to devise a rigorous taxonomy of   
+364 learning methods that sheds light on the ocean of methods proposed in the past years. The terms   
+365 self-, semi- and unsupervised learning have been used interchangeably and this is often a source of   
+366 confusion for students and newcomers. While confusing terminology is natural in a rapidly growing,   
+367 the time might have come for distilling the progress of the past years into rigorous nomenclature that   
+368 better survive the test of time.   
+369 Finally, we recall that most of the learning theory has been developed for simple loss functions such   
+370 as binary classification or mean squared error regression, but certain methods successfully used in   
+371 practice today escape the available theory. Given the success of this kind of more complex supervised   
+372 objectives, the study of these methods from a theoretical point of view might be a fruitful direction   
+373 for future work.
+
+# 374 Broader Impact
+
+Since this article does not present a new method or results from data sets, potential risks of “bias in the data” or “failure of the system” do not apply. As a critical review of current trends in the field and cite multiple research articles, some researchers could potentially feel addressed and affected by our mentions. We declare that we do not intend to negatively affect any individual researcher and we have only referred to individuals directly in the case of well-established scientist with a reputation. Our goal has been in any case to potentially improve scientific progress through a constructive reflection.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See the beginning of Section 5.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 5.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [N/A]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [N/A]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [N/A]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [N/A]   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+        "text": "Rethinking supervised learning: insights from biological learning and from calling it by its name ",
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+        "text": "The renaissance of artificial neural networks was catalysed by the success of classification models, tagged by the community with the broader term supervised learning. The extraordinary results gave rise to a hype loaded with ambitious promises and overstatements. Soon the community realised that the success owed much to the availability of thousands of labelled examples and supervised learning went, for many, from glory to shame: Some criticised deep learning as a whole and others proclaimed that the way forward had to be “alternatives” to supervised learning: predictive, unsupervised, semi-supervised and, more recently, self-supervised learning. However, these seem all brand names, rather than actual categories of a theoretically grounded taxonomy. Moreover, the call to banish supervised learning was motivated by the questionable claim that humans learn with little or no supervision and are capable of robust out-of-distribution generalisation. Here, we review insights about learning and supervision in nature, revisit the notion that learning and generalization are not possible without supervision or inductive biases and argue that we will make better progress if we just call it by its name. ",
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+        "text": "17 The re-emergence of deep learning during the last decade due to the noteworthy achievements   \n18 of artificial neural networks (ANN) built up a sort of philosophy that nearly anything could be   \n19 automatically learnt from data without human intervention, in contrast to the previous approaches:   \n20 [hand designing good feature extractors, engineering skill and domain expertise]   \n21 can all be avoided if good features can be learned automatically using a general  \n22 purpose learning procedure. This is the key advantage of deep learning (LeCun   \n23 et al., 2015).   \n24 Read in hindsight, this claim was clearly an overstatement. The success of deep learning has   \n25 required iterative hand design of network architectures and techniques that demanded collective, high   \n26 engineering skill and large doses of interdisciplinary domain expertise. Furthermore, deep learning   \n27 owes much to the immense computational power poured into training artificial networks (Amodei &   \n28 Hernandez, 2018; Schwartz et al., 2019) and to the human effort of manually collecting and labelling   \n29 thousands of images and other data modalities (Russakovsky et al., 2015; Cao et al., 2018). However,   \n30 the gist of the claim has permeated machine learning research and is pervasive up to these days.   \n31 The realisation that the success of deep learning was largely due to the availability of huge labelled data   \n32 sets prompted various reactions: some authors strongly questioned the usefulness of the algorithms   \n33 (Marcus, 2018); some delved into the question of whether neural networks generalise beyond or   \n34 simply memorise the training examples (Zhang et al., 2017; Arpit et al., 2017); and some proposed   \n35 new research horizons that can be overly ambitious and potentially misleading: “learning a class   \n36 from a single labelled example”, based on the statement that “humans learn new concepts with very   \n37 little supervision, [but] the standard supervised deep learning paradigm does not offer a satisfactory   \n38 solution for learning new concepts rapidly from little data” (Vinyals et al., 2016). As a consequence,   \n39 multiple research programmes, with various brand names, followed up with the aim of minimising or   \n40 removing the need for “supervision” to train neural networks: few-shot, one-shot, zero-shot, predictive,   \n41 unsupervised, semi-supervised and self-supervised learning are only a few popular examples.   \n42 Exploring alternatives to classification and improving the efficiency of learning algorithms should   \n43 indeed be a priority of machine learning research. As a matter of fact, related approaches have   \n44 been subject of study since long before the explosion of deep learning (Hinton & Sejnowski, 1999;   \n45 Chapelle et al., 2006). However, the current publication and discussion trends in the field denote   \n46 overambitious promises that are in part based on misconceptions and overstatements about biological   \n47 learning, and amplified by overselling nomenclature. While much of the research output derived from   \n48 these programmes does provide us with useful techniques and insight, it leaves behind a landscape of   \n49 confusing terminology and tangled research directions that are hard to navigate and lead many astray.   \n50 In this paper, we reflect upon fundamental concepts in machine learning such as supervision, in  \n51 ductive biases and generalisation, which in spite of resting on theoretical grounds, are at the core   \n52 of misconceptions and overstatements about deep learning commonly seen in the literature. First,   \n53 we review aspects from biological learning, and compare them to the traits often (mis)attributed to   \n54 human learning and generalisation in the machine learning literature (Section 2). Second, we revisit   \n55 insights from classical statistical learning and critically review the terminology and current trends in   \n56 deep learning research (Section 3). Altogether, we aim at tempering certain claims and promises of   \n57 deep learning, helping mitigate the confusion over the terminology and suggesting desirable—in our   \n58 opinion—directions and changes in machine learning research. ",
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+        "text": "59 2 Supervision in biological learning ",
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+        "text": "60 The link between artificial intelligence—specifically artificial neural networks (Rosenblatt, 1958;   \n61 Fukushima & Miyake, 1982)—and biological learning systems is intrinsic to the field, as one long  \n62 term goal of artificial intelligence is to mirror the capabilities of human intelligence. However, these   \n63 capabilities are, in our view, often overestimated. One example is the argument that intelligence in   \n64 nature evolves without supervision and is capable of robust out-of-distribution generalisation. In   \n65 particular, it is often claimed that humans and other animals learn to visually categorise objects with   \n66 little or no supervision from a few examples (Vinyals et al., 2016; Marcus, 2018; Morgenstern et al.,   \n67 2019). In what follows, we will discuss three aspects of biological learning to argue against this   \n68 view, so as to gain insights that better inform our progress in machine learning: first, we will discuss   \n69 how generalisation requires exposure to relevant training data; second, we will review the variety of   \n70 supervised signals that the brain has access to; third, we will comment on the role of evolution and   \n71 brain development. ",
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+        "text": "72 2.1 Generalisation requires exposure to relevant training data ",
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+        "text": "73 In the argument that machine learning models should generalise from a few examples, there seems   \n74 to be a promise or aspiration that future better methods will be able to perform robust visual object   \n75 categorisation—for instance—among many object classes after being trained on one or a few examples   \n76 per class. While a primary objective is to develop techniques that efficiently extract the maximum   \n77 possible information from the available examples, we should also remind ourselves that no machine   \n78 learning algorithm can robustly learn anything that cannot be inferred from the data it has been trained   \n79 on. Although this may seem to contradict certain current trends and statements in the literature, we   \n80 should also bear in mind that learning in nature is not different.   \n81 First, the amount of data that animals and humans in particular are exposed to is often underestimated.   \n82 A biological brain continuously receives, processes and integrates multimodal inputs from various   \n83 sensors—images (light), sound, smell, etc. Humans do not learn to recognise objects by looking   \n84 at photos from ImageNet, but are rather exposed to a continuous flow of visual stimuli with slow   \n85 changes of the viewing angle and lighting conditions. Furthermore, the stimuli are coherent across   \n86 modalities, we are allowed to interact with the objects and we even receive multiple supervision   \n87 signals, as we discuss later.   \n88 The exposure to so much training data makes the human visual system remarkably robust, but still its   \n89 capabilities are optimised for the tasks it needs to perform and largely determined by the training data   \n90 distribution—and years of evolution, as we will discuss below. For instance, a well-studied property   \n91 of human vision is that our face recognition ability is severely impaired if faces are presented upside   \n92 down (Yin, 1969; Valentine, 1988). Setting aside the specific complexity of face processing in the   \n93 brain, a compelling explanation for this impairment is that we are simply not used to seeing and   \n94 recognising inverted faces. More generally, while human perception of objects is largely invariant   \n95 under certain conditions (Biederman & Bar, 1999), object recognition is sensitive to changes in view   \n96 angle (Tarr et al., 1998), especially when we see objects from unfamiliar viewpoints (Edelman &   \n97 Bülthoff, 1992; Bülthoff & Newell, 2006; Milivojevic, 2012).   \n98 Furthermore, although better than the one-shot or few-shot generalisation of current ANNs, humans   \n99 also have limited ability to recognise truly novel classes (Morgenstern et al., 2019). Interestingly,   \n100 experiments with certain novel classes of objects known as Greebles showed that, with sufficient   \n101 training, humans can acquire expertise in recognising new objects from different viewpoints, even   \n102 making use of an area of the brain—the fusiform face area—that typically responds strongly with   \n103 face stimuli (Gauthier et al., 1999). This provides evidence that recognition from multiple viewpoints   \n104 is possible but only developed after exposure to similar conditions, that is relevant data. This is   \n105 reminiscent of the effectiveness of data augmentation in deep learning, compared to more naïve   \n106 regularisation methods (Hernández-García & König, 2018).   \n107 The need for exposure to relevant stimuli challenges the notion that humans are capable of strong   \n108 out-of-distribution generalisation. Rather, it seems that the transfer learning capabilities of humans   \n109 are limited to relatively small changes in the data distribution. A compelling example is our difficulty   \n110 to learn new languages: someone who natively speaks or has learnt Spanish will be able to transfer a   \n111 significant amount of knowledge if they are to learn Italian, due to the overlap in the data distribution,   \n112 but they will have very little to transfer for learning Kanien’kéha or Mandarin. ",
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+        "text": "113 2.2 Supervised signals for the brain ",
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+        "text": "114 Another commonly found argument has it that children—animals in general���learn robust object   \n115 recognition without supervision: “a child can generalize the concept of ‘giraffe’ from a single picture   \n116 in a book” (Vinyals et al., 2016). First of all, we should mention the role of evolution (expanded in   \n117 Section 2.3), which can be interpreted as a pre-trained model, optimised through millions of years of   \n118 data with natural selection as a supervisory signal (Zador, 2019). Second, there is abundant evidence   \n119 to argue against the very claim that children—and adults—learn in fully unsupervised fashion.   \n120 Obviously, the kind of supervision that humans make use of is not that of classification algorithms—   \n121 we do not see a class label on top of every object we look at. However, we receive supervision   \n122 from multiple sources. Even though not for every visual stimulus, children do frequently receive   \n123 information about the object classes they see. For instance, parents would point at objects and name   \n124 them, then we learn how to read, and generally play a crucial role as teachers in language development   \n125 (Kuhl, 2007). Non-human animals such as zebra finches learning to sing have also been found to rely   \n126 on feedback (supervision) from the female adult and not just imitation Carouso-Peck & Goldstein   \n127 (2019). Furthermore, humans usually follow guided hierarchical learning: children do not directly   \n128 learn to tell apart breeds of dogs, but rather start with umbrella terms and then progressively learn   \n129 down the class hierarchy (Bornstein & Arterberry, 2010; Spriet et al., 2021). Gopnik (2021) has   \n130 asserted that “we learn more from other people than we do from any other source” and Hasson et al.   \n131 (2020) mention other examples of supervision from social cues, that is from other humans, such as   \n132 learning to recognise individual faces, produce grammatical sentences, read and write; as well as   \n133 from embodiment and action, such as learning to balance the body while walking or grasping objects.   \n134 In all these actions, we can identify a supervisory signal that surely influences learning in the brain   \n135 (Shapiro, 2012; Gopnik et al., 2020).   \n136 While these supervision signals largely differ from what is most commonly considered supervised   \n137 learning in machine learning, we can still draw some parallels with human learning. We learn to   \n138 categorise many concepts and objects as children, but most people carry on learning new categories as   \n139 adults. For example, some people put effort in improving their understanding of the natural world by   \n140 learning to recognise and name trees, plants or birds. Those who have engaged in such an endeavour   \n141 may have noticed that the learning process is easier and faster if we count upon the expert knowledge   \n142 of a friend or of technology such as iNaturalist (Van Horn et al., 2018). Another example: those   \n143 who have—or attempted to learn—a new language as an adult may have realised that whereas it is   \n144 possible to learn the meaning of a new word by repeated exposure to it in multiple contexts, it is   \n145 certainly easier if we look up the ground truth definition in a dictionary or, even easier, if there exists   \n146 a direct mapping to a word in our native language. Summing up, not only does supervision facilitate   \n147 learning, but human beings actively seek for it.   \n148 Besides this kind of explicit supervision, the brain certainly makes use of more subtle, implicit   \n149 supervised signals, such as temporal stability (Becker, 1999; Wyss et al., 2003): The light that enters   \n150 the retina, and the sound waves that reach the cochlea, are not random signals from a sequence of   \n151 rapidly changing arbitrary photos or noise, but highly coherent and regular flows of slowly changing   \n152 stimuli, especially at the higher, semantical level (Kording et al., 2004). At the very least, this is how   \n153 we perceive it and if such a smooth perception turns out to be a consequence rather than a cause, then   \n154 it should be a by-product of a long process of evolution that would be worth taking into account. ",
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+        "text": "155 2.3 The role of evolution and brain development ",
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+        "text": "156 In the previous sections, we have discussed some misconceptions or overstatements about how   \n157 humans learn and generalise that are often found in the machine literature. Namely, that humans   \n158 are able to generalise from a few examples and that this occurs with little or no supervision. Still,   \n159 the commonplace comparison of artificial neural networks with human learning and the brain often   \n160 misses a fundamental component of biology, recently brought to the fore by Zador (2019) and Hasson   \n161 et al. (2020), although considered since the early days of artificial intelligence (Turing, 1968): the   \n162 role that millions of years of evolution have played in developing the nervous systems of organisms   \n163 in nature, including the human brain.   \n164 The most common way of training artificial neural networks, especially in machine learning research,   \n165 is from tabula rasa, that is from randomly initialised parameters1. In contrast, a large part of the   \n166 brain connectivity is encoded genetically and certain properties and behaviour are known to be   \n167 innate, that is developed without prior exposure to stimuli (Farroni et al., 2005; Spriet et al., 2021).   \n168 Importantly, evolution not only provides innate behaviour, but also determines what cannot be learnt,   \n169 or relevant constraints—scientists who have trained animals in the laboratory for psychological   \n170 or neuroscientific studies are well aware that tasks have to be carefully adapted to the ecological   \n171 behaviour and limitations of the animal, determined by evolution.   \n172 Taking into account the role of evolution, we can draw conclusions that relate to the claims discussed   \n173 in the previous sections. If our brains are the product of millions of years of exposure to relevant   \n174 stimuli and adaptation, is it really fair to say that humans are capable of robust out-of-distribution   \n175 generalisation and that we learn from from a few examples? If evolution has largely determined   \n176 what our brain can and cannot learn, providing as with a “pre-trained model”, is it really fair to   \n177 say that humans learn in a unsupervised fashion? This questions are relevant for machine learning   \n178 research: if we take biological learning as motivation for artificial intelligence, should we not temper   \n179 our expectations of what learning algorithms should aspire to? And, therefore, would it not be worth   \n180 reconsidering some research programmes?   \n181 On the flip side, insights from evolutionary theory are likely to be a fruitful source of inspiration   \n182 for machine learning (Hasson et al., 2020; Zador, 2019). As we have observed, training a neural   \n183 network from scratch may be more similar to a simulation of evolution than to the process by which   \n184 an adult learns a new concept. As a shortcut to simulating evolution, neuroscience is a rich source   \n185 of inspiration of constraints and inductive biases that determine learning in the biological brain and   \n186 can potentially inform machine learning (Hassabis et al., 2017; Lindsey et al., 2019). For instance,   \n187 simulating properties of the primary visual cortex in the early layers of an artificial neural network   \n188 has been shown to improve adversarial robustness (Dapello et al., 2020; Malhotra et al., 2020). ",
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+        "text": "Besides evolution, the focus on the capabilities of adults often makes us miss another important aspect of biological learning, particularly important in humans: the role of learning in infancy and brain development. While learning occurs too in adulthood, childhood is a particularly important and ",
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+        "text": "192 active time for learning (Atkinson, 2002; Gelman & Meyer, 2011). In fact, sensitive or critical periods   \n193 for learning in infancy have been described or hypothesised, for example for vision (Harwerth et al.,   \n194 1986) and language development (Lenneberg, 1967). Machine learning papers that draw motivation   \n195 from the alleged generalisation capabilities of humans often underestimate the amount of input stimuli   \n196 and supervision that infants receive (Gopnik, 2020). However, childhood can be regarded as period   \n197 dedicated almost exclusively to learn, not only formally from parents and teachers, but also through   \n198 playing, which plays a critical role in cognitive development Burghardt (2005); Pelz & Kidd (2020).   \n199 Finally, the fact that humans—and other cognitively advanced animals, such as corvid birds, which   \n200 also exhibit cultural learning—have a comparatively long childhood period, has led Uomini et al.   \n201 (2020) to recently proposed that extended parenting is pivotal in the evolution of cognition. This   \n202 adds to the discussion on the undervalued role of supervision. In sum, we propose machine learning   \n203 research can benefit from drawing inspiration from both evolutionary biology and the literature on   \n204 developmental psychology, brain development and life history and learning (Gopnik et al., 2020). ",
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+        "text": "205 3 Supervision in machine learning ",
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+        "text": "If we open a machine learning textbook (Murphy, 2012; Abu-Mostafa et al., 2012; Goodfellow et al., 2016), we will most surely find a taxonomy of learning algorithms with a clear distinction between supervised and unsupervised learning. However, while this separation can be useful, the boundaries are certainly not clear. As a matter of fact, if we take a look at the deep learning literature of the past years, we will also find abundant work on some variants supposedly in between—semi-supervised learning, self-supervised learning, etc.—whose definitions are all but clear. ",
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+        "text": "3.1 Catastrophic forgetting of old concepts ",
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+        "text": "If we recall a classical result in statistical learning theory and inference, the no free lunch theorem (Wolpert, 1996), no learning algorithm is better than any other at classifying unobserved data points, when averaged over all possible data distributions. Therefore, we need to constrain the distributions or, in other words, introduce prior knowledge—that is supervision. Recently, Locatello et al. (2018) obtained a related result for the case of unsupervised learning of disentangled representations: without inductive biases for both the models and the data sets, unsupervised disentanglement learning is fundamentally impossible. These results are purely theoretical and have limited impact on real world applications (Giraud-Carrier & Provost, 2005), precisely because in practice we use multiple inductive biases and implicit supervision, even when we do so-called unsupervised learning. ",
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+        "text": "In a strict sense, even the classical, purely unsupervised methods, such as independent component analysis or nearest neighbours classifiers, make use of inductive biases, such as independence or minimum distance, respectively. Without inductive bias, learning is not possible: purely unsupervised learning is an illusion. While this is not news, the terminology used in the recent and current machine learning literature seems to reject supervision and neglect these nuances, evidencing that the field suffers catastrophic forgetting of well-established notions. ",
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+        "text": "3.2 The brands of alt-supervised learning ",
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+        "text": "Particularly in deep learning and computer vision, the term supervised learning has adopted, in practice, the meaning of classification of examples annotated by humans, that is models trained on examples labelled according to, for instance, the object classes. This is yet another instance of catastrophic forgetting—or, at best, abuse—of well-established concepts. It should not be necessary to recall that, first of all, supervised learning is a broader category than classification, which includes also regression and ranking, among other learning modalities. Second, even if we narrow our view to classification only, supervised learning is not restricted to learning from examples annotated by humans. Goodfellow et al. (2016) did not overlook this in their definition of supervised learning: “In many cases the outputs y may be difficult to collect automatically and must be provided by a human ‘supervisor,’ but the term still applies even when the training set targets were collected automatically”. ",
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+        "text": "239 In turn, the term unsupervised learning is now used for any model that does not use manually collected   \n240 labels, regardless of what other kind of supervision it may use. Further, the term semi-supervised   \n241 learning generally refers in practice to models that are trained with a fraction of the labels, but are   \n242 tested on the same classification benchmarks. Finally, the term self-supervised learning has recently   \n43 gained much popularity, referring to models that are trained on tasks other than the standard task   \n44 defined by classification labels.   \n245 Some of the methods proposed under these categories are certainly useful—that is not the subject of   \n246 criticism of this work—but the terminology is overly confusing and unnecessary. A newcomer would   \n247 easily fall into a scientific rabbit hole trying to discern the meaning of each of these names through   \n248 publications—not to mention if they incorporated social media discussions into their endeavour. By   \n249 way of illustration, the authors of this paper have witnessed how a recurrent question by students who   \n250 learn about recent deep learning methods is whether there is any difference between self-supervised   \n251 and unsupervised learning. Are students missing something fundamental? The following anecdotal   \n252 recall of influential keynote talks at artificial intelligence conferences should shed some light on part   \n253 of the origins of this confusion: In December 2016, Prof. Yann LeCun titled his NeurIPS keynote   \n254 presentation “Predictive Learning”, to refer to “what many people mean by unsupervised learning”   \n255 (LeCun, 2016). A few years later, in his keynote presentation at ISSCC in February 2019, he spoke   \n256 about similar ideas, but this time the title was “Self-Supervised Learning” (LeCun, 2019). In social   \n257 media, he wrote: “I now call it ‘self-supervised learning’, because ‘unsupervised’ is both a loaded   \n258 and confusing term” . Students may be getting things rather right. ",
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+        "text": "Is there then a fundamental difference—a theoretically grounded one—between the deep learning methods labelled as unsupervised learning and more recently self-supervised learning? We argue that these are mostly brand names that reflect trends in the field, adding noise to the scientific progress and leading many astray. Therefore, we propose that, given the recent progress, the field of machine learning research would benefit from an exercise of self-reflection and from an effort to devise a rigorous taxonomy of the variety of methods. From a theoretical point of view, both the conventional classification models and the recent wave of self-supervised tasks can all be formalised as sub-categories of supervised learning. ",
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+        "text": "3.3 Supervision comes in different flavours ",
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+        "text": "In Section 2.2, we have seen examples of different forms of supervision used by humans and other animals. In machine learning, the field focused for many years on a few loss functions, such as classification and simple forms of regression. The relatively recent explosion of deep learning has brought about the development of several libraries for automatic differentiation (Baydin et al., 2017), which in turn have enabled the proposal of multiple loss functions and learning tasks with various types of supervision that can easily be optimised numerically by stochastic gradient descent and artificial neural networks. This has certainly opened promising and already fruitful avenues to incorporate richer forms of supervision and inductive biases other than classification, some inspired by biological learning, into machine learning algorithms. ",
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+        "text": "277 A currently popular example is image data augmentation: Although until recently it was seen as a   \n278 naïve technique to simply create additional training data, data augmentation actually encodes rich prior   \n279 knowledge about human visual perception, in the case of computer vision. This is why it outperforms   \n280 explicit regularisation methods, which provide less effective inductive biases Hernández-García &   \n281 König (2018), and was used in “semi-supervised” tasks Laine & Aila (2016). The rich information   \n282 embedded in image transformations has been used to encourage invariant outputs under different   \n283 augmentations through contrastive losses (Ye et al., 2019), and even at intermediate representations,   \n284 inspired by the invariance in the visual cortex (Hernández-García et al., 2019), although these methods   \n285 were not branded as self-supervision. The use of this term for losses based on data augmentation   \n286 was further popularised after the success of similar methods such as SimCLR (Chen et al., 2020).   \n287 Beyond data augmentation invariance, the zoo of self-supervised learning tasks in computer vision is   \n288 rich and diverse: classifying the rotation applied to image patches (Gidaris et al., 2018), predicting   \n289 image colourisation (Larsson et al., 2017), classifying the relative position of two image patches   \n290 (Doersch et al., 2015), or even solving full jigsaw puzzles (Noroozi & Favaro, 2016) (Jing & Tian   \n291 (2020) recently performed an extensive review).   \n292 The current trend is to refer to these methods as self-supervised learning, but similar methods were   \n293 referred to in the past as semi-supervised, unsupervised, and even predictive learning, as we have seen.   \n294 A look at the papers reveals that these terms have been used mostly interchangeably. The terms self  \n295 and semi- and unsupervised learning imply that less supervision is used, but it would be misleading to   \n296 seriously argue that the tasks are devoid of supervision. Most of these techniques make use of a wide   \n297 range of surrogate tasks with supervisory signals defined by humans. In fact, they could have been   \n298 called hyper-supervised2 learning. Here, we contend that these methods are all variants of supervised   \n299 learning, only that supervision comes in different flavours, both in biological and machine learning,   \n300 and we should call it by its name and ideally develop a rigorous taxonomy. ",
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+        "type": "text",
+        "text": "01 4 Discussion ",
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+        "text": "In this paper, we have discussed some of the overambitious promises of the deep learning hype, namely that machine learning models should be able to generalise to unseen distributions, from a few examples, without human intervention or supervision. These claims have often been motivated by alleged generalisation capabilities of humans. In order to assess these motivations, we have first reviewed, in Section 2, some often overlooked characteristics of biological learning relevant to machine learning research. In particular, we have argued that humans and other animals receive extensive and diverse input stimuli as well as multiple supervisory signals, including the long history of evolution and cultural transmission. In the light of these insights from biological learning, we have then, in Section 3, critically reviewed the various terms that are currently used to refer to supposed alternatives to supervised learning: semi-, self- and unsupervised learning, among others. In sum, we pointed out that all these approaches are in fact supervised learning—though not necessarily classification—and the machine learning (research) community would benefit from using more rigorous, less overselling nomenclature, and from devising a more rigorous taxonomy. ",
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+        "text": "15 Supervision is not evil. It is at the core of statistical learning theory: learning is impossible without   \n316 inductive biases or supervision. But supervision comes in different flavours, not only as classification   \n17 labels. Neither is deep learning some sort of exceptional solution to learn without human intervention   \n318 and supervision, nor is it a hopeless model class because it requires large data sets (Marcus, 2018).   \n319 The human visual system is exposed to a lot of stimuli too. One exceptional advantage of deep   \n320 learning is precisely that it is possible to effectively optimise different learning objectives, almost   \n321 end-to-end, from large collections of nearly naturalistic sensory signals, such as digital images (Saxe   \n322 et al., 2020). While other models are known to scale poorly as the amount of data increases, neural   \n323 networks excel at fitting the training data and interpolating on unseen examples (Belkin et al., 2019;   \n324 Hasson et al., 2020). This is a feature, not a bug. But we will make better progress if we exploit   \n325 these advantages of deep learning without neglecting that supervision will always be necessary—the   \n26 critical goal is how to best incorporate it and exploit it. ",
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+        "text": "In this regard, we argue that deep learning needs more supervision, and not less. A major focus of the deep learning community in the last decade has been image object classification. This has brought about unprecedented progress and unveiled the limitations of having classification as chief task and class labels as main supervisory signal. For example, deep classifiers have been found to learn spurious features that are highly discriminative for the classification task but with little true generalisation power and clearly not aligned with perceptual features (Jo & Bengio, 2017; Wang et al., 2019; Geirhos et al., 2020). In fact, this mismatch has been argued to be at the root of adversarial vulnerability (Ilyas et al., 2019) and seems to be the consequence of training highly expressive, over-parameterised models in heavily unconstrained tasks. This can be addressed with meaningful constraints, that is more and richer supervision, possibly inspired by human perception and biological learning. For example, combining a classification loss with a similarity loss inspired by the invariance in the visual cortex yields more robust representations without detriment to categorisation (HernándezGarcía et al., 2019), and simulating the properties of the primary visual cortex may improve the adversarial robustness of neural networks (Dapello et al., 2020). Expanding in this direction leads to biologically-inspired, multi-task and representation learning, and away from just classification. ",
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+        "type": "text",
+        "text": "342 5 Conclusions for future research directions ",
+        "text_level": 1,
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+        "text": "The chief goal of this paper is rather descriptive than prescriptive. We have aimed to identify and describe aspects of the current trends in machine learning research that could be improved, in the hope of inspiring future work that effectively address them. Nonetheless, throughout the paper we have made suggestions that may help mitigate the confusion with the terminology, clarify research directions and ultimately bring about scientific progress in machine learning research. We outline these suggestions here to conclude the paper. ",
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+        "text": "349 We have drawn parallels from cognitive neuroscience to contend that learning in nature also requires   \n350 abundant data and supervision in multiple forms. Even evolution can be regarded as an optimisation   \n351 process where natural selection is the supervisory signal. We have argued, as others have before, that   \n352 these insights from biology, neuroscience and developmental psychology, among other fields, offer a   \n353 great opportunity for machine learning research to draw inspiration and calibrate its compass.   \n354 As we have discussed, research in deep learning has departed from pure classification and has been   \n355 exploring new learning tasks and ways of training artificial neural networks. Nonetheless, in some   \n356 fields such as computer vision, the ultimate benchmark to assess the value of a method is still the   \n357 accuracy on classification data sets, such as ImageNet, even though there is evidence of overfitting   \n358 the test set. While object recognition will remain an important benchmark, as deep learning is   \n359 well suited to learn representations, we should develop methods to assess the quality of the learnt   \n360 representations for tasks other than classification. In this regard, we encourage researchers to evaluate   \n361 their models with tests that are still not widespread, such as the suitability for transfer learning,   \n362 adversarial robustness, comparison with brain measurements, behavioural tasks, etc.   \n363 We have also argued that the field would benefit from an effort to devise a rigorous taxonomy of   \n364 learning methods that sheds light on the ocean of methods proposed in the past years. The terms   \n365 self-, semi- and unsupervised learning have been used interchangeably and this is often a source of   \n366 confusion for students and newcomers. While confusing terminology is natural in a rapidly growing,   \n367 the time might have come for distilling the progress of the past years into rigorous nomenclature that   \n368 better survive the test of time.   \n369 Finally, we recall that most of the learning theory has been developed for simple loss functions such   \n370 as binary classification or mean squared error regression, but certain methods successfully used in   \n371 practice today escape the available theory. Given the success of this kind of more complex supervised   \n372 objectives, the study of these methods from a theoretical point of view might be a fruitful direction   \n373 for future work. ",
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+        "type": "text",
+        "text": "374 Broader Impact ",
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+        "text": "Since this article does not present a new method or results from data sets, potential risks of “bias in the data” or “failure of the system” do not apply. As a critical review of current trends in the field and cite multiple research articles, some researchers could potentially feel addressed and affected by our mentions. We declare that we do not intend to negatively affect any individual researcher and we have only referred to individuals directly in the case of well-established scientist with a reputation. Our goal has been in any case to potentially improve scientific progress through a constructive reflection. ",
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+        "text": "References ",
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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] See the beginning of Section 5.   \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 5.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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parse/train/HyxoX6EKvB/HyxoX6EKvB.md

@@ -0,0 +1,354 @@
+# REFLECTION-BASED WORD ATTRIBUTE TRANSFER
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We propose a word attribute transfer framework based on reflection to obtain a word vector with an inverted target attribute for a given word in a word embedding space. Word embeddings based on Pointwise Mutual Information (PMI) represent such analogic relations as ${ \overrightarrow { k i n g } } - { \overrightarrow { m a n } } + { \overrightarrow { w o m a n } } \approx { \overrightarrow { q u e e n } }$ . These relations can be used for changing a word’s attribute from king to queen by changing its gender. This attribute transfer can be performed by subtracting a difference vector $\overrightarrow { m a n } - \overrightarrow { w o m a n }$ from $\overrightarrow { k i n g }$ when we have explicit knowledge of the gender of given word king. However, this knowledge cannot be developed for various words and attributes in practice. For transferring queen into king in this analogy-based manner, we need to know that queen denotes a female and add the difference vector to it. In this work, we transfer such binary attributes based on an assumption that such transfer mapping will become identity mapping when we apply it twice. We introduce a framework based on reflection mapping that satisfies this property; queen should be transferred back to king with the same mapping as the transfer from king to queen. Experimental results show that the proposed method can transfer the word attributes of the given words, and does not change the words that do not have the target attributes.
+
+# 1 INTRODUCTION
+
+Distributed representation (Hinton et al., 1984) is a kind of data representation that can capture data similarities in a vector space. In natural language processing, various studies have been conducted on word embeddings (Mikolov et al., 2013a;b; Pennington et al., 2014; Peters et al., 2018; Bojanowski et al., 2017). Word embedding models, such as skip-gram with negative sampling (SGNS) (Mikolov et al., 2013b) or GloVe (Pennington et al., 2014), capture some analogic relations, such as $\overrightarrow { k i n g } -$ $\overrightarrow { m a n } + \overrightarrow { w o m a n } \approx \overrightarrow { q u e e n }$ . Previous work offer theoretical explanation based on Pointwise Mutual Information (PMI; Church & Hanks (1990)) for maintaining the analogic relations in word vectors (Levy & Goldberg, 2014b; Arora et al., 2016; Gittens et al., 2017; Ethayarajh et al., 2019; Allen & Hospedales, 2019).
+
+These relations can be used for transferring a certain attribute of a word, such as changing king into queen by transferring the gender. This task, which is called word attribute transfer, enables us to rewrite He is a boy as She is a girl. Word attribute transfer is expected to be applicable for natural language inference and data augmentation in natural language processing. The above analogic relations can be used, including adding difference vector $\overrightarrow { w o m a n } - \overrightarrow { m a n }$ to $\overrightarrow { k i n g }$ to transfer $\overrightarrow { k i n g }$ to $\overrightarrow { q u e e n }$ . This operation requires the explicit knowledge whether an input word is male or female; we have to add a difference vector to a male word and subtract it from a female word for a gender transfer. We also have to avoid changing words without any gender attributes, such as is and $a$ in the example above. Thus, analogy-based word attribute transfer requires explicit knowledge of word attributes, such as king is male, queen is female, and is has no gender attribute. Developing such knowledge is very difficult for various words and attributes in practice.
+
+In this study, we propose a novel framework based on reflection, which enables word attribute transfer by a single reflection-based mapping for a certain attribute. Reflection in geometry is a mapping that exchanges the locations of two vectors in a Euclidean space by a hyperplane called a mirror, which satisfies the above desired property: working as identity mapping when it is applied twice and when it is applied to vectors on the mirror. We apply this reflection mapping to the problem of word attribute transfer by estimating an appropriate mirror that maps word pairs with a binary target attribute (e.g., male and female) and keeps the other words without that attribute using training data. We also extend this approach by introducing parameterized mirrors, which work as different mirrors based on the given input words, to overcome a limitation using a single fixed mirror to represent complex transfer mappings for different words. Experimental results show that the reflection-based method enables such transfers, achieves comparable performance to analogy-based methods with explicit attribute knowledge, even though our proposed method does not use such knowledge.
+
+The following are the contributions of this paper:
+
+• We propose a novel representation learning framework that obtains a vector with an inverted attribute in embedding space without explicit attribute knowledge of the given word. Our proposed reflection-based word attribute transfer enables us to transfer word attributes in up to $76 \%$ for words with target attributes and to avoid changing words without target attributes in over $9 9 \%$ in our experiments.
+
+# 2 WORD ATTRIBUTE TRANSFER
+
+![](images/d7258f940354ae874b04c913c0b9c4e30a2bfb5fa96d1d2c84de158be4df3acb.jpg)  
+Figure 1: Given word vector ${ \bf v } _ { x }$ and attribute one-hot vector $\mathbf { z }$ , word attribute transfer predicts word vector $\mathbf { v } _ { t }$ , which is the inverted attribute of ${ \bf v } _ { x }$ .
+
+Let $x$ denote a word and let ${ \bf v } _ { x }$ denote its vector representation. Here we assume that ${ \bf v } _ { x }$ is learned in advance with an embedding model such as SGNS. In this task, we have two inputs, word $x$ and onehot vector $\mathbf { z }$ , representing a certain target attribute, and one output, return word $t$ with the inverted attribute of $x$ for $\mathbf { z }$ . Let $\mathcal { A }$ denote a set of a triplet $( x , t , \mathbf { z } )$ , e.g., $( m a n , w o m a n , \mathbf { z } _ { \mathrm { g e n d e r } } ) \in \mathcal { A }$ . Let $\mathcal { N }$ denote a set of words without attribute $\mathbf { z }$ , e.g., apple $\in \mathcal { N }$ , when $\mathbf { z }$ represents gender. The purpose of this task is to transfer ${ \bf v } _ { x }$ to $\mathbf { v } _ { t }$ by transfer function $f _ { \mathbf { Z } }$ that inverts attribute $\mathbf { z }$ of ${ \bf v } _ { x }$ . In other words, output $\mathbf { v } _ { y }$ should be close to the vector of corresponding target word $\mathbf { v } _ { t }$ , which is typically the nearest neighbor of $\mathbf { v } _ { t }$ in the word embedding space.
+
+$$
+\mathbf { v } _ { t } \approx \mathbf { v } _ { y } = f _ { \mathbf { Z } } ( \mathbf { v } _ { x } ) .
+$$
+
+Note that mapping $f _ { \mathbf { Z } }$ transfers word $x$ if it has target attribute $\mathbf { z }$ ; otherwise $f _ { \mathbf { Z } }$ works as identity mapping. For instance with $\mathbf { z } _ { g e n d e r }$ , given input word man, gender attribute transfer $f _ { \mathbf { Z } _ { g e n d e r } } ( \mathbf { v } _ { m a n } )$ should result in a vector close to $\mathbf { v } _ { w o m a n }$ . Given input word apple as $x$ , the results should be $\mathbf { v } _ { a p p l e }$ .
+
+# 3 ANALOGY-BASED WORD ATTRIBUTE TRANSFER
+
+Analogy is a general idea for realizing attribute transfer. Several PMI-based word embedding methods (Mikolov et al., 2013c; Linzen, 2016) tackled to embed words into word embedding space to capture the analogic relations. An embedded vector with SGNS or GloVe captures analogic relations (Levy & Goldberg, $2 0 1 4 \mathrm { a }$ ; Mikolov et al., 2013c; Linzen, 2016). For instance, ${ \mathbf { v } } _ { q u e e n }$ is near the vector obtained on the right side of Eq. 2. By rearranging Eq. 2, Eq. 3 is obtained:
+
+$$
+\begin{array} { r } { \mathbf { v } _ { q u e e n } \approx \mathbf { v } _ { k i n g } - \mathbf { v } _ { m a n } + \mathbf { v } _ { w o m a n } , } \\ { \mathbf { \tau } } \\ { \approx \mathbf { v } _ { k i n g } - ( \mathbf { v } _ { m a n } - \mathbf { v } _ { w o m a n } ) . } \end{array}
+$$
+
+We can transfer the gender attribute by subtracting difference vector ${ \bf v } _ { m a n } - { \bf v } _ { w o m a n }$ from input word vectors, e.g., $\mathbf { v } _ { k i n g }$ . The analogy-based transfer function is
+
+$$
+f _ { \mathbf { Z } } ( \mathbf { v } _ { x } ) = { \left\{ \begin{array} { l l } { \mathbf { v } _ { x } - \mathbf { d } } & { { \mathrm { ~ i f ~ } } x \in { \mathcal { M } } , } \\ { \mathbf { v } _ { x } + \mathbf { d } } & { { \mathrm { ~ i f ~ } } x \in { \mathcal { F } } , } \end{array} \right. }
+$$
+
+where $\mathbf { d }$ is a difference vector of the given word pair such as man and woman, $\mathcal { M }$ is a set of words having a target attribute, and $\mathcal { F }$ is a set of words having an inverse attribute, for example, $m a n \in \mathcal { M }$ and $w o m a n \in { \mathcal { F } }$ for gender attributes. Eq. 4 indicates that the operation changes depending on whether input word $x$ belongs to $\mathcal { M }$ or $\mathcal { F }$ . For gender words, we subtract difference vector $\mathbf { d }$ if $x$ is male, and add it if $x$ is female. Therefore, we need such explicit knowledge. However, this knowledge cannot be developed for various words and attributes in practice.
+
+# 4 REFLECTION-BASED WORD ATTRIBUTE TRANSFER
+
+# 4.1 IDEALIZED TRANSFER WITHOUT EXPLICIT KNOWLEDGE
+
+What is an idealized transfer function $\phi _ { \mathbf { Z } }$ for the word attribute transfer? The following are the idealized natures of such a transfer function:
+
+$$
+\begin{array} { r } { \mathbf { v } _ { m } = \phi _ { \mathbf { Z } } ( \mathbf { v } _ { w } ) , } \\ { \mathbf { v } _ { w } = \phi _ { \mathbf { Z } } ( \mathbf { v } _ { m } ) , } \end{array}
+$$
+
+where $m \in \mathcal { M }$ and $w \in { \mathcal { F } }$ . This function $\phi _ { \mathbf { Z } }$ enables to transfer a word without explicit knowledge. Function $\phi _ { \mathbf { Z } }$ transfers ${ \bf v } _ { m }$ to ${ \bf v } _ { w }$ and ${ \bf v } _ { w }$ to ${ \bf v } _ { m }$ without such explicit knowledge as $m \in \mathcal { M }$ and $w \in { \mathcal { F } }$ . By combining Eqs. 5 and 6, we obtain the following formula:
+
+$$
+\forall m \in \mathcal { M } , \qquad \mathbf { v } _ { m } = \phi _ { \mathbf { Z } } \big ( \phi _ { \mathbf { Z } } ( \mathbf { v } _ { m } \big ) \big ) ,
+$$
+
+and
+
+$$
+\forall w \in \mathcal { F } , \qquad \mathbf { v } _ { w } = \phi _ { \mathbf { Z } } \big ( \phi _ { \mathbf { Z } } ( \mathbf { v } _ { w } ) \big ) .
+$$
+
+Hence, the idealized transfer function is a mapping that becomes an identity mapping when we apply it twice for any $\mathbf { v }$ . Such a mapping is called involution in geometry. For example, $\phi \colon \mathbf { v } \mapsto - \mathbf { v }$ is one example of an involution. Note that the identity map itself, such as $\phi \colon \mathbf { v } \mapsto \mathbf { v }$ , is excluded from the involution.
+
+# 4.2 REFLECTION
+
+A reflection is an involution that reverses the location between two vectors in a Euclidean space through an affine hyperplane (mirror). Reflection is an idealized function because every point returns to its original location when reflection is applied twice:
+
+$$
+\forall \mathbf { v } \in \mathbb { R } ^ { n } , \qquad \mathbf { v } = R e f _ { \mathbf { a } , \mathbf { c } } ( R e f _ { \mathbf { a } , \mathbf { c } } ( \mathbf { v } ) { \bf \phi } ) .
+$$
+
+Given vector $\mathbf { v }$ in Euclidean space $\mathbb { R } ^ { n }$ , the formula for the reflection in the mirror is given:
+
+$$
+R e f _ { \mathbf { a } , \mathbf { c } } ( \mathbf { v } ) = \mathbf { v } - 2 { \frac { ( \mathbf { v } - \mathbf { c } ) \cdot \mathbf { a } } { \mathbf { a } \cdot \mathbf { a } } } \mathbf { a } ,
+$$
+
+where $\mathbf { a } \in \mathbb { R } ^ { n }$ is a vector orthogonal to the mirror (normal vector) and $\mathbf { c } \in \mathbb { R } ^ { n }$ is a point through which the mirror passes. a and c are parameters that determine the mirror.
+
+# 4.3 REFLECTION-BASED WORD ATTRIBUTE TRANSFER
+
+We apply reflection to the word attribute transfer to invert a specific attribute of an input word without its explicit attribute knowledge. We learn a mirror (hyperplane) in a pre-trained embedding space using training word pairs with a common (binary) attribute $\mathbf { z }$ (Fig. 2). Here since the mirror is uniquely determined by two parameter vectors, a and $\mathbf { c }$ , we estimate a and c from target attribute $\mathbf { z }$ using fully connected multi-layer perceptrons:
+
+$$
+\begin{array} { r } { \mathbf { a } = M L P ( \mathbf { z } ) , } \\ { \mathbf { c } = M L P ( \mathbf { z } ) . } \end{array}
+$$
+
+Transferred vector $\mathbf { v } _ { y }$ is obtained by inverting attribute $\mathbf { z }$ of ${ \bf v } _ { x }$ by reflection:
+
+$$
+\begin{array} { r } { \mathbf { v } _ { y } = R e f _ { \mathbf { a } , \mathbf { c } } ( \mathbf { v } _ { x } ) . } \end{array}
+$$
+
+![](images/4b41c91e239af6f7ce1b9bc1201af457426e51e5f181fbb02278f0acf591f13a.jpg)  
+Figure 2: Reflection-based word attribute transfer examples.
+
+![](images/5dec7c1bb005195aa8edc93418d25df229d78394ea104c2d37ea207d3d975c87.jpg)  
+Figure 3: Mirror estimation methods
+
+# 4.4 PARAMETERIZED MIRRORS
+
+Reflection with a mirror by Eqs. 11 and 12 assumes a single mirror depending only on z. Previous discussion assumed that there will be pairs sharing a stable attribute such king and queen. However, often gendered words don’t come in pairs, and gender is far from a stable attribute. For example, actress may be feminine, but actor is clearly neutral in many cases (Fig. 3). Thus, actor isn’t as obvious a masculine counterpart as king. In fact, it is known that there are biases in gender words in the embedding space (Zhao et al., 2018; Kaneko & Bollegala, 2019). This phenomenon can occur not only with the gender attribute, but also with other attributes. With this assumption of a single mirror, the mirror must be a hyperplane that goes through the midpoints for all word vector pairs. However, the vector pairs shown on the left of Fig. 3 cannot be transferred well since the single mirror does not satisfy this constraint due to the bias of the embedding space. To solve this problem, we introduce different mirrors for different words. We propose parameterized mirrors determined by input vector ${ \bf v } _ { x }$ in addition to attribute $\mathbf { z }$ . The following are the definitions of the mirror parameters:
+
+$$
+\begin{array} { r } { \mathbf { a } = M L P ( [ \mathbf { z } ; \mathbf { v } _ { x } ] ) , } \\ { \mathbf { \ c } = M L P ( [ \mathbf { z } ; \mathbf { v } _ { x } ] ) , } \end{array}
+$$
+
+where $[ \cdot ; \cdot ]$ indicates the vector concatenation in the column. The parameterized mirrors are expected to work flexibly on different words. For instance, as shown in Fig. 3, suppose we learned the mirror (the blue line) that transfers $\mathbf { v } _ { h e r o }$ to $\mathbf { v } _ { h e r o i n e }$ in advance. If input word vector $\mathbf { v } _ { a c t o r }$ resembles $\mathbf { v } _ { h e r o }$ , a mirror that is similar to the one for $\mathbf { v } _ { h e r o }$ should be derived and used for the attribute transfer.
+
+# 4.5 LOSS FUNCTION
+
+Loss function $\mathcal { L }$ is defined:
+
+$$
+\mathcal { L } ( \boldsymbol { \Theta } ) = \frac { 1 } { | \mathcal { A } | } \sum _ { ( x _ { i } , t _ { i } , \mathbf { Z } _ { i } ) \in \mathcal { A } } ( \mathbf { v } _ { y _ { i } } - \mathbf { v } _ { t _ { i } } ) ^ { 2 } + \frac { 1 } { | \mathcal { N } | } \sum _ { x _ { j } \in \mathcal { N } } ( \mathbf { v } _ { y _ { j } } - \mathbf { v } _ { x _ { j } } ) ^ { 2 } ,
+$$
+
+where $\begin{array} { r } { \frac { 1 } { | \mathcal { A } | } \sum _ { ( x _ { i } , t _ { i } , \mathbf { Z } _ { i } ) \in \mathcal { A } } ( \mathbf { v } _ { y _ { i } } - \mathbf { v } _ { t _ { i } } ) ^ { 2 } } \end{array}$ is a term that draws target word vector $\mathbf { v } _ { t _ { i } }$ closer to corresponding transferred vector $\mathbf { v } _ { y _ { i } }$ and $\begin{array} { r } { \frac { 1 } { | \mathcal { N } | } \sum _ { x _ { j } \in \mathcal { N } } ( \mathbf { v } _ { y _ { j } } - \mathbf { v } _ { x _ { j } } ) ^ { 2 } } \end{array}$ is a term that prevents words without a target attribute from being moved by transfer function $f _ { \mathbf { Z } } . \Theta$ represents the set of all the trainable parameters. The parameters in the proposed model are the MLP weights used to determine mirror hyperplanes via a and $\mathbf { c }$ . We iteratively update $\Theta$ to minimize $\mathcal { L }$ :
+
+$$
+\begin{array} { r } { \Theta _ { t + 1 } \gets \underset { \Theta _ { t } } { \operatorname { a r g m i n } } \mathcal { L } ( \Theta _ { t } ) , } \end{array}
+$$
+
+where $t$ is the number of parameter updates at that time.
+
+# 5 EXPERIMENT
+
+We evaluated the performance of the proposed reflection-based word attribute transfer using data with some different attributes.
+
+# 5.1 EXPERIMENTAL SETUP
+
+We used three different datasets of word pairs with three binary attributes: Male-Female 1, SingularPlural and Capital-Country, shown in Table 1.These word pairs were collected from analogy test sets (Mikolov et al., 2013a; Gladkova et al., 2016), the Internet and Nguyen et al. (2017) for antonyms of noun2. Since these datasets are very small, we added Gaussian noise $\sigma = 0 . 1$ ) to every input vector ${ \bf v } _ { x }$ during training to avoid overfit. Random noise was applied independently to every sample in every iteration. For a non-attribute dataset $\mathcal { N }$ , we sampled words from the three-million-word vocabulary of the word embedding model. We sampled from 4 to 50 words for training $0 ~ \leq$ $| \mathcal { N } _ { \mathrm { t r a i n } } | \leq 5 0 )$ and 1000 words for the test $\lvert N _ { \mathrm { t e s t } } \rvert = \bar { 1 } 0 0 0 )$ . We used a mixed dataset that included both $| \mathcal { N } _ { \mathrm { t r a i n } } |$ and $| \mathcal { A } _ { \mathrm { t r a i n } } |$ . Note that $\mathcal { N } _ { \mathrm { t e s t } }$ was sampled and excluded words from $\mathcal { N } _ { \mathrm { t r a i n } }$ and $\mathcal { A } _ { \mathrm { t r a i n } }$ . We had no $\mathcal { N } _ { \mathrm { v a l } }$ because the tuning was conducted with only $| \mathcal { A } _ { \mathrm { v a l } } |$ . We used word2vec (Mikolov et al., 2013b) 3 and GloVe (Pennington et al., 2014) 4 as the pre-trained embedding model. The embedded vector dimension is $n = 3 0 0$ .
+
+Table 1: Statistics of binary attribute word pair datasets (in the number of word pairs)   
+
+<table><tr><td>Dataset A</td><td>Train</td><td>Val</td><td>Test</td><td>Total</td></tr><tr><td>Male-Female (MF)</td><td>29</td><td>12</td><td>12</td><td>53</td></tr><tr><td>Singular-Plural (SP)</td><td>90</td><td>25</td><td>25</td><td>140</td></tr><tr><td>Capital-Country (CC)</td><td>59</td><td>25</td><td>25</td><td>109</td></tr><tr><td>Antonym (AN)</td><td>1354</td><td>290</td><td>290</td><td>1934</td></tr></table>
+
+# 5.2 EVALUATION METRICS
+
+We measured the accuracy and stability performances of the word attribute transfer. The accuracy measures how many input words in $\mathcal { A } _ { \mathrm { t e s t } }$ were transferred correctly to the corresponding target words. The stability score measures how many words in $\mathcal { N } _ { \mathrm { t e s t } }$ are not mapped to other words. For example, in a gender transfer, given man, the transfer is regarded as correct if woman is the closest word to the transferred vector; otherwise it is incorrect. Given apple, the transfer is regarded as correct if apple is the closest word to the transferred vector; otherwise its stability is incorrect. Here we used cosine similarity to measure the similarity of output vector $\mathbf { v } _ { y }$ and target vector $\mathbf { v } _ { t }$ . The accuracy and stability scores are calculated by the following formula:
+
+$$
+\delta ( { \bf v } _ { y } , t ) = \left\{ \begin{array} { l l } { 1 } & { \mathrm { ~ i f ~ } \quad \underset { k \in \mathcal { V } } { \arg \operatorname* { m a x } } \frac { { \bf v } _ { y } \cdot { \bf v } _ { k } } { \| { \bf v } _ { y } \| \| { \bf V } _ { k } \| } = t , } \\ { 0 } & { \mathrm { ~ o t h e r w i s e , } } \end{array} \right.
+$$
+
+$$
+\mathrm { A c c u r a c y } = \frac { 1 } { | \mathcal { A } _ { \mathrm { t e s t } } | } \sum _ { ( x _ { i } , t _ { i } , \mathbf { Z } _ { i } ) \in \mathcal { A } _ { \mathrm { t e s t } } } \delta ( \mathbf { v } _ { y _ { i } } , t _ { i } ) ,
+$$
+
+$$
+\mathrm { S t a b i l i t y } = \frac { 1 } { | \mathcal N _ { \mathrm { t e s t } } | } \sum _ { \substack { x _ { i } \in \mathcal N _ { \mathrm { t e s t } } } } \delta ( \mathbf v _ { y _ { i } } , x _ { i } ) ,
+$$
+
+where $\nu$ is the vocabulary of the word embedding model.
+
+# 5.3 METHODS AND CONFIGURATIONS
+
+In our experiment, we compared our proposed method with the following baseline methods:
+
+REF Reflection-based word attribute transfer with a single mirror. We used a fully connected 2-layer MLP with 300 hidden units and ReLU activations to estimate a and c.
+
+REF+PM Reflection-based word attribute transfer with parameterized mirrors. We used the same MLP as the REF.
+
+MLP Fully connected MLP: $\mathbf v _ { y } = M L P ( [ \mathbf v _ { x } ; \mathbf z ] )$ . The highest accuracy models are a 2-layer MLP for Capital-Country and 3-layer MLP for the other datasets.
+
+DIFF Analogy-based word attribute transfer with a difference vector: $\mathbf { d } = \mathbf { v } _ { m } - \mathbf { v } _ { w }$ , where $m$ and $w$ are in the training data of $\mathcal { A }$ . We chose d because it achieved the best accuracy in the validation data of $\mathcal { A }$ . We determined whether to add or subtract $\mathbf { d }$ to ${ \bf v } _ { x }$ based on attribute knowledge (Eq. 4).
+
+DIFF $^ +$ Analogy-based word attribute transfer with a difference vector regardless of the attribute knowledge. d was obtained in the same way as the DIFF. We added $\mathbf { d }$ to ${ \bf v } _ { x }$ for any input $x$ : $f _ { \mathbf { Z } } ( \mathbf { v } _ { x } ) = \mathbf { v } _ { x } + \mathbf { d }$ , $\forall \mathbf { v } _ { x } \in \mathbb { R } ^ { n }$ .
+
+DIFF − Analogy-based word attribute transfer with a difference vector regardless of the attribute knowledge. d was obtained in the same way as the DIFF. We subtracted $\mathbf { d }$ from ${ \bf v } _ { x }$ for any input $x$ : ${ \bf { \bar { f } } } _ { \mathbf { Z } } ( { \bf v } _ { x } ) = { \bf v } _ { x } - \mathbf { d } , \quad \forall { \bf v } _ { x } \in \mathbb { R } ^ { n } .$ .
+
+MEANDIFF Analogy-based word attribute transfer with a mean difference vector $\bar { \bf d }$ : $\bar { \textbf { d } } =$ 1|Atrain| P(mi,wi)∈Atrain (vmi − vwi ). We determined whether to add or subtract d¯ to vx based on the attribute knowledge (Eq.4).
+
+MEANDIFF $^ +$ Analogy-based word attribute transfer with a mean difference vector regardless of the attribute knowledge: $f _ { \mathbf { Z } } ( \mathbf { v } _ { x } ) = \mathbf { v } _ { x } + \bar { \mathbf { d } }$ , $\forall \mathbf { v } _ { x } \in \mathbb { R } ^ { n }$ . $\bar { \bf d }$ was obtained in the same way as the MEANDIFF.
+
+MEANDIFF − Analogy-based word attribute transfer with a mean difference vector regardless of the attribute knowledge: $f _ { \mathbf { Z } } ( \mathbf { v } _ { x } ) = \mathbf { v } _ { x } - \bar { \mathbf { d } }$ , $\forall \mathbf { v } _ { x } \in \mathbb { R } ^ { n }$ . d¯ was obtained in the same way as the MEANDIFF.
+
+Based on the tuning, we used the Adam optimizer (Kingma & Ba, 2015) with a learning rate of $\alpha = 1 0 ^ { - 4 }$ (the other hyperparameters were the same as the original one (Kingma & Ba, 2015)), and a batchsize of 62 for male-female, and 32 for the others. These hyperparameters were identical for the learning-based methods: REF, $\boldsymbol { \mathrm { R E F + P M } }$ , or MLP. We did not use such regularization methods as dropout (Srivastava et al., 2014) or batch normalization (Ioffe & Szegedy, 2015) because they did not show any improvement in our pilot test.
+
+# 5.4 ACCURACY AND STABILITY
+
+Table 2 and 3 shows the transfer accuracy and stability score results. Both experiments using GloVe or word2vec obtained similar results. $\boldsymbol { \mathrm { R E F + P M } }$ achieved the best accuracy among the methods that did not use explicit attribute knowledge. This means that reflection can be used for word attribute transfers even without attribute knowledge. In the stability evaluation, reflection-based methods (REF, $\mathsf { R E F } + \mathsf { P M } )$ and the analogy-based methods with a mean difference vector (MEANDIFF $-$ , MEANDIFF $^ +$ ) achieved high stability. In particular, reflection-based transfers achieved outstanding stability scores exceeding $9 9 \%$ . The stability of DIFF $^ +$ and DIFF $-$ was much lower than the other methods. Although MEANDIFF − and MEANDIFF + achieved high stability, their accuracy results were very low. Interestingly, reflection-based transfer with parameterized mirrors $( \mathrm { R E F } + \mathrm { P M } )$ achieved high performance in both accuracy and stability. For example, the accuracy of RE $\boldsymbol { \mathbf { \ell } } + \mathbf { P M }$ was $4 1 . 6 7 \%$ , and the stability was $9 9 . 9 \%$ in Male-Female (MF), and the accuracy was $58 \%$ and the stability was $9 9 . 4 0 ~ \%$ in Capital-Country (CC). These results show that the proposed method transfers an input word if it has a target attribute and does not transfer an input word even though it does not use explicit attribute knowledge on the input words. MLP worked poorly both in accuracy and stability. In the antonym (AN), while the transfer accuracy by the proposed method was a bit lower than that by MLP, the stability of the proposed method was $100 \%$ and that of MLP was really poor (almost $0 \%$ ).
+
+Table 2: Results in accuracy and stability scores (word2vec).   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Knowledge</td><td colspan="4">Accuracy (%)</td><td colspan="4">Stability (%)</td></tr><tr><td>MF</td><td>SP</td><td>CC</td><td>AN</td><td>MF</td><td>SP</td><td>CC</td><td>AN</td></tr><tr><td>REF</td><td></td><td>20.83</td><td>0.00</td><td>36.00</td><td>0.00</td><td>99.80</td><td>100.00</td><td>99.80</td><td>100.00</td></tr><tr><td>REF+PM</td><td></td><td>41.67</td><td>22.00</td><td>58.00</td><td>28.79</td><td>99.90</td><td>99.40</td><td>99.40</td><td>100.00</td></tr><tr><td>MLP</td><td></td><td>8.33</td><td>4.00</td><td>12.00</td><td>35.86</td><td>2.20</td><td>0.00</td><td>2.70</td><td>1.90</td></tr><tr><td>DIFF+</td><td></td><td>25.00</td><td>2.00</td><td>32.00</td><td>-</td><td>72.10</td><td>77.90</td><td>53.90</td><td>-</td></tr><tr><td>DIFF-</td><td></td><td>25.00</td><td>2.00</td><td>30.00</td><td>=</td><td>49.60</td><td>78.20</td><td>56.30</td><td>1</td></tr><tr><td>MEANDIFF +</td><td></td><td>4.17</td><td>0.00</td><td>22.00</td><td>=</td><td>98.60</td><td>99.40</td><td>87.60</td><td></td></tr><tr><td>MEANDIFF</td><td></td><td>8.33</td><td>0.00</td><td>14.00</td><td>=</td><td>97.20</td><td>99.30</td><td>92.40</td><td></td></tr><tr><td>DIFF</td><td></td><td>62.50</td><td>4.00</td><td>64.00</td><td></td><td>1</td><td>=</td><td>=</td><td></td></tr><tr><td>MEANDIFF</td><td></td><td>12.50</td><td>0.00</td><td>36.00</td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+Table 3: Results in accuracy and stability scores (GloVe).   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Knowledge</td><td colspan="4">Accuracy (%)</td><td colspan="4">Stability (%)</td></tr><tr><td>MF</td><td>SP</td><td>CC</td><td>AN</td><td>MF</td><td>SP</td><td>CC</td><td>AN</td></tr><tr><td>REF</td><td></td><td>12.50</td><td>2.00</td><td>26.00</td><td>0.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td></tr><tr><td>REF+PM</td><td></td><td>45.83</td><td>50.00</td><td>76.00</td><td>33.54</td><td>99.70</td><td>99.10</td><td>99.20</td><td>100.00</td></tr><tr><td>MLP</td><td></td><td>4.17</td><td>10.00</td><td>18.00</td><td>36.72</td><td>5.10</td><td>7.00</td><td>5.20</td><td>1.20</td></tr><tr><td>DIFF+</td><td></td><td>25.00</td><td>2.00</td><td>26.00</td><td>-</td><td>99.30</td><td>94.20</td><td>99.30</td><td>1</td></tr><tr><td>DIFF-</td><td></td><td>25.00</td><td>2.00</td><td>24.00</td><td>=</td><td>100.60</td><td>99.90</td><td>99.50</td><td>=</td></tr><tr><td>MEANDIFF +</td><td></td><td>0.00</td><td>0.00</td><td>22.00</td><td></td><td>100.00</td><td>100.00</td><td>100.00</td><td>一</td></tr><tr><td>MEANDIFF</td><td></td><td>0.00</td><td>0.00</td><td>0.00</td><td></td><td>100.00</td><td>100.00</td><td>100.00</td><td></td></tr><tr><td>DIFF</td><td></td><td>50.00</td><td>4.00</td><td>44.00</td><td></td><td>=</td><td>=</td><td></td><td></td></tr><tr><td>MEANDIFF</td><td></td><td>0.00</td><td>0.00</td><td>0.00</td><td></td><td></td><td>-</td><td></td><td></td></tr></table>
+
+We investigated the relation between the size of $| \mathcal { N } _ { \mathrm { t r a i n } } |$ and the stability of learning-based methods by conducting an additional experiment by varying $| \mathcal { N } _ { \mathrm { t r a i n } } |$ from 0 to 50. The stability scores by MLP did not improve (Table 4). On the other hand, REF and $\boldsymbol { \mathrm { R E F } } + \boldsymbol { \mathrm { P M } }$ achieved high stability scores with just $| \bar { \mathcal { N } } _ { \mathrm { t r a i n } } | = 4$ and maintained the accuracy.
+
+# 5.5 TRANSFER EXAMPLE
+
+Table 5 shows examples of a gender transfer at the sentence level, where the attribute transfer was applied to words in sentence $\bar { X } = \{ x _ { 1 } , x _ { 2 } , \ldots \}$ . Here since such words as $a$ and . are not in the vocabulary of the original word embedding model, we omitted them from the inputs. MLP made many wrong transfers on words without gender attributes, e.g., the became By Katie Klingsporn, was became she, when became Doughty Evening Chronicle, and woman became girlfriend. $\mathrm { D I F F ^ { + } }$ can transfer if $x$ is female, e.g., it transferred from woman to man, but it could not transfer grandfather and boy. Similarly, DIFF − failed to transfer from female to male. In addition, since the stability of these methods was low, they erroneously transferred. For example, in DIFF $^ -$ , the and when became she. REF + PM can selectively transfer words with a gender attribute without using explicit gender information. For example, when woman was given, $R e f ( \mathbf { v } _ { w o m a n } )$ became man without knowledge that woman is a female word, and when man was given, it became woman. When non-attribute word married was given, $R e f ( \mathbf { v } _ { m a r r i e d } )$ became married without knowledge that married has no gender attribute. When we applied the reflection-based transfer twice, the transferred word returned to its original word, e.g., $R e f ( R e f ( \mathbf { v } _ { w o m a n } ) )$ gives woman.
+
+Table 4: Relation among size of $| \mathcal { N } _ { \mathrm { t r a i n } } |$ and stability of learning-based methods.   
+
+<table><tr><td rowspan="2" colspan="2"></td><td colspan="4">Accuracy (%)</td><td colspan="4">Stability (%)</td></tr><tr><td colspan="4">Wtrainl</td><td colspan="4">|Wtrainl</td></tr><tr><td></td><td></td><td>0</td><td>4</td><td>10</td><td>50</td><td>0</td><td>4</td><td>10</td><td>50</td></tr><tr><td rowspan="3">MF</td><td>REF</td><td>16.67</td><td>20.83</td><td>20.83</td><td>20.83</td><td>98.30</td><td>98.80</td><td>99.40</td><td>99.80</td></tr><tr><td>REF+PM</td><td>41.67</td><td>45.83</td><td>20.83</td><td>41.67</td><td>38.30</td><td>98.40</td><td>100.00</td><td>99.90</td></tr><tr><td>MLP</td><td>4.17</td><td>8.33</td><td>8.33</td><td>8.33</td><td>0.00</td><td>0.30</td><td>0.30</td><td>2.20</td></tr><tr><td rowspan="3">SP</td><td>REF</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>99.90</td><td>99.90</td><td>99.90</td><td>100.90</td></tr><tr><td>REF+PM</td><td>12.00</td><td>22.00</td><td>18.00</td><td>18.00</td><td>98.40</td><td>99.40</td><td>99.30</td><td>99.80</td></tr><tr><td>MLP</td><td>4.00</td><td>4.00</td><td>2.00</td><td>2.00</td><td>0.00</td><td>0.00</td><td>0.10</td><td>3.40</td></tr><tr><td rowspan="3">CC</td><td>REF</td><td>36.00</td><td>36.00</td><td>36.00</td><td>34.00</td><td>99.80</td><td>99.80</td><td>99.80</td><td>100.00</td></tr><tr><td>REF+PM</td><td>58.00</td><td>56.00</td><td>58.00</td><td>54.00</td><td>73.80</td><td>99.70</td><td>99.40</td><td>99.40</td></tr><tr><td>MLP</td><td>6.00</td><td>6.00</td><td>8.00</td><td>12.00</td><td>0.00</td><td>0.30</td><td>0.50</td><td>2.70</td></tr><tr><td rowspan="3">AN</td><td>REF</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>99.90</td><td>99.90</td><td>99.90</td><td>99.80</td></tr><tr><td>REF+PM</td><td>21.72</td><td>27.24</td><td>28.62</td><td>28.79</td><td>95.30</td><td>99.20</td><td>99.50</td><td>99.80</td></tr><tr><td>MLP</td><td>34.14</td><td>35.00</td><td>34.31</td><td>35.86</td><td>0.00</td><td>0.01</td><td>0.02</td><td>1.90</td></tr></table>
+
+Table 5: Transfer results when sentence $X = \{ \mathrm { t h e , . . . , b o y } \}$ was given. Out-of-vocabulary words $a$ and . were not given as input.   
+
+<table><tr><td>X</td><td>the woman was married when your grandfather was (a) boy (.)</td></tr><tr><td>Ref(x)</td><td>the man was married when your grandmother was (a) girl (.)</td></tr><tr><td>Ref(Ref(x))</td><td>the woman was married when your grandfather was (a) boy (.)</td></tr><tr><td>MLP</td><td>By_Katie_Klingsporn girlfriend she fiancee Doughty_Evening_Chronicle ma&#x27;am daughter she (a) mother (.)</td></tr><tr><td>DIFF+</td><td>the man was married when your grandfather was (a) boy (.)</td></tr><tr><td>DIFF-</td><td>she woman was married she your grandmother was (a) girl (.)</td></tr></table>
+
+We can transfer some different attributes of words with reflection-based transfer one-by-one. Table 6 shows that the words having different target attributes were transferred by each reflection-based transfer in the order of Male-Female, Singular-Plural, and Country-Capital. Given actress for a Male-Female transfer, it was transferred to actor and to actors for Singular-Plural. Given Tokyo for Male-Female, Singular-Plural, and Antonym, it was not transferred, but it was transferred to Japan for Country-Capital. Given rich for Male-Female, Singular-Plural, and Capital-Country, it was not transferred, but it was transferred to poor for Antonym.
+
+Table 6: Transfer of different attributes with reflection-based word attribute transfer with parameterized mirrors.   
+
+<table><tr><td>X</td><td>the rich actress and the poor actor want to stay the beautiful city in Tokyo.</td></tr><tr><td>+Male-Female</td><td>the rich actor and the poor actress want to stay the beautiful city in Tokyo.</td></tr><tr><td>+ Singular-Plural</td><td></td></tr><tr><td></td><td>the rich actors and the poor actresses want to stay the beautiful cities in Tokyo.</td></tr><tr><td>+Capital-Country</td><td>the rich actors and the poor actresses want to stay the beautiful citie in Japan.</td></tr><tr><td>+ Antonym</td><td>the poor actors and the rich actresses want to stay the beautiful cities in Japan.</td></tr></table>
+
+# 6 RELATED WORK
+
+The embedded vectors obtained by SGNS (Mikolov et al., 2013a;b) and GloVe (Pennington et al., 2014) have analogic relations. The theory of analogic relations in word embeddings has been widely discussed: Levy & Goldberg (2014b); Arora et al. (2016); Gittens et al. (2017); Ethayarajh et al. (2019); Allen & Hospedales (2019); Linzen (2016). Levy & Goldberg (2014b) offer the explanation that SGNS factorizes a shifted PMI matrix. Allen & Hospedales (2019) and Ethayarajh et al. (2019) argued that they proved the existence of such analogic relations without strong assumptions. In our work, we focus on the analogic relations in a word embedding space and propose a novel framework to obtain a word vector with inverted attributes. Style transfers (Niu et al., 2018; Prabhumoye et al., 2018; Jain et al., 2019; Logeswaran et al., 2018; Dai et al., 2019; Zhang et al., 2018) resemble our task. In a style transfer, the text style of the input sentences is changed. For instance, Jain et al. (2019) transferred from formal to informal sentences. Logeswaran et al. (2018) transferred sentences by controlling such attributes as mood and tense. These style transfer tasks use sentence pairs; our word attribute transfer task uses word pairs. Style transfer changes sentence styles, but our task changes the word attributes (contents). Soricut & Och (2015) studied the problem of morphological transformation based on character information. Our work aims more general attribute transfer such as gender transfer and country-capital and is not limited to the morphological transformation.
+
+# 7 CONCLUSION AND FUTURE WORK
+
+We proposed a novel representation learning framework based on reflection to invert a certain attribute of a word vector. We proposed a reflection-based method for word attribute transfers without relying on the explicit attribute knowledge of an input word, which is necessary for a simple analogy-based transfer. Experimental results showed that our proposed method can transfer the word attributes if the input word has a target attribute. If not, reflection does not transfer the word.
+
+Future work includes applications to other transfer tasks: sentence by sentence transfer, such Niu et al. (2018); Prabhumoye et al. (2018); Jain et al. (2019), and entity prediction on an analogic graph embedding space (Liu et al., 2017), in the field of computer vision, visual analogy (Reed et al., 2015), or style transfer (Zhu et al., 2017; Liao et al., 2017) with GANs (Radford et al., 2016; Goodfellow et al., 2014) because their latent space holds analogic relations (Radford et al., 2016).
+
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+Scott E. Reed, Yi Zhang, Yuting Zhang, and Honglak Lee. Deep Visual AnalogyMaking. In Advances in Neural Information Processing Systems 28: Annual Conference on Neural Information Processing Systems 2015, December 7-12, 2015, Montreal, Quebec, Canada, pp. 1252–1260, 2015. URL http://papers.nips.cc/paper/ 5845-deep-visual-analogy-making.
+
+Radu Soricut and Franz Josef Och. Unsupervised Morphology Induction Using Word Embeddings. In NAACL HLT 2015, The 2015 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Denver, Colorado, USA, May 31 - June 5, 2015, pp. 1627–1637, 2015. URL https://www.aclweb.org/anthology/ N15-1186/.
+
+Nitish Srivastava, Geoffrey E. Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. J. Mach. Learn. Res., 15(1): 1929–1958, 2014. URL http://dl.acm.org/citation.cfm?id=2670313.
+
+Zhirui Zhang, Shuo Ren, Shujie Liu, Jianyong Wang, Peng Chen, Mu Li, Ming Zhou, and Enhong Chen. Style Transfer as Unsupervised Machine Translation. CoRR, abs/1808.07894, 2018. URL http://arxiv.org/abs/1808.07894.
+
+Jieyu Zhao, Yichao Zhou, Zeyu Li, Wei Wang, and Kai-Wei Chang. Learning Gender-Neutral Word Embeddings. In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, Brussels, Belgium, October 31 - November 4, 2018, pp. 4847–4853, 2018. URL https://www.aclweb.org/anthology/D18-1521/.
+
+Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A. Efros. Unpaired Image-to-Image Translation Using Cycle-Consistent Adversarial Networks. In IEEE International Conference on Computer Vision, ICCV 2017, Venice, Italy, October 22-29, 2017, pp. 2242–2251, 2017. doi: 10.1109/ICCV.2017.244. URL https://doi.org/10.1109/ICCV.2017.244.
+
+# A TOP THREE ACCURACY AND STABILITY
+
+Table 7: Male-Female   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Knowledge</td><td colspan="4">Accuracy (%)</td><td colspan="4">Stability (%)</td></tr><tr><td>Mean@3</td><td>@1</td><td>@2</td><td>@3</td><td>Mean@3</td><td>@1</td><td>@2</td><td>@3</td></tr><tr><td>REF</td><td></td><td>50.00</td><td>20.83</td><td>62.50</td><td>66.67</td><td>99.80</td><td>99.80</td><td>99.80</td><td>99.80</td></tr><tr><td>REF+PM</td><td></td><td>55.56</td><td>41.67</td><td>58.33</td><td>66.67</td><td>99.13</td><td>99.00</td><td>99.20</td><td>99.20</td></tr><tr><td>MLP</td><td></td><td>20.83</td><td>8.33</td><td>20.83</td><td>33.33</td><td>2.20</td><td>2.20</td><td>2.20</td><td>2.20</td></tr><tr><td>DIFF+</td><td></td><td>31.94</td><td>25.00</td><td>33.33</td><td>37.50</td><td>75.43</td><td>72.10</td><td>75.80</td><td>78.40</td></tr><tr><td>DIFF-</td><td></td><td>31.94</td><td>25.00</td><td>33.33</td><td>37.50</td><td>55.80</td><td>49.60</td><td>57.30</td><td>60.50</td></tr><tr><td>MEANDIFF +</td><td></td><td>23.61</td><td>4.17</td><td>33.33</td><td>33.33</td><td>98.93</td><td>98.60</td><td>99.10</td><td>99.10</td></tr><tr><td>MEANDIFF</td><td></td><td>23.61</td><td>8.33</td><td>29.17</td><td>33.33</td><td>97.63</td><td>97.20</td><td>97.80</td><td>97.90</td></tr><tr><td>DIFF</td><td>√</td><td>68.05</td><td>62.50</td><td>66.66</td><td>75.00</td><td>-</td><td>1</td><td>-</td><td>1</td></tr><tr><td>MEANDIFF</td><td>1</td><td>47.22</td><td>12.50</td><td>62.50</td><td>66.67</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></table>
+
+Table 8: Singular-Plural   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Knowledge</td><td colspan="4">Accuracy (%)</td><td colspan="4">Stability (%)</td></tr><tr><td>Mean@3</td><td>@1</td><td>@2</td><td>@3</td><td>Mean@3</td><td>@1</td><td>@2</td><td>@3</td></tr><tr><td>REF</td><td></td><td>50.00</td><td>0.00</td><td>72.00</td><td>78.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td></tr><tr><td>REF+PM</td><td></td><td>56.67</td><td>22.00</td><td>72.00</td><td>76.00</td><td>99.70</td><td>99.40</td><td>99.80</td><td>99.90</td></tr><tr><td>MLP</td><td></td><td>7.33</td><td>4.00</td><td>8.00</td><td>10.00</td><td>0.13</td><td>0.00</td><td>0.20</td><td>0.20</td></tr><tr><td>DIFF+</td><td></td><td>36.67</td><td>2.00</td><td>50.00</td><td>58.00</td><td>80.27</td><td>77.90</td><td>79.70</td><td>83.20</td></tr><tr><td>DIFF-</td><td></td><td>36.67</td><td>2.00</td><td>52.00</td><td>56.00</td><td>80.27</td><td>78.20</td><td>80.70</td><td>81.90</td></tr><tr><td>MEANDIFF +</td><td></td><td>42.00</td><td>0.00</td><td>56.00</td><td>70.00</td><td>99.47</td><td>99.40</td><td>99.40</td><td>99.60</td></tr><tr><td>MEANDIFF -</td><td></td><td>39.33</td><td>0.00</td><td>56.00</td><td>62.00</td><td>99.50</td><td>99.30</td><td>99.60</td><td>99.60</td></tr><tr><td>DIFF</td><td>√</td><td>49.33</td><td>4.00</td><td>70.00</td><td>74.00</td><td>1</td><td>-</td><td>1</td><td>1</td></tr><tr><td>MEANDIFF</td><td>√</td><td>52.00</td><td>0.00</td><td>76.00</td><td>80.00</td><td>1</td><td>=</td><td>-</td><td>-</td></tr></table>
+
+Table 9: Capital-Country   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Knowledge</td><td colspan="4">Accuracy (%)</td><td colspan="4">Stability (%)</td></tr><tr><td>Mean@3</td><td>@1</td><td>@2</td><td>@3</td><td>Mean@3</td><td>@1</td><td>@2</td><td>@3</td></tr><tr><td>REF</td><td></td><td>66.67</td><td>36.00</td><td>78.00</td><td>86.00</td><td>99.80</td><td>99.80</td><td>99.80</td><td>99.80</td></tr><tr><td>REF+PM</td><td></td><td>72.67</td><td>58.00</td><td>74.00</td><td>86.00</td><td>99.53</td><td>99.40</td><td>99.60</td><td>99.60</td></tr><tr><td>MLP</td><td></td><td>35.33</td><td>12.00</td><td>40.00</td><td>54.00</td><td>2.80</td><td>2.70</td><td>2.80</td><td>2.90</td></tr><tr><td>DIFF+</td><td></td><td>39.33</td><td>32.00</td><td>42.00</td><td>44.00</td><td>64.87</td><td>53.90</td><td>69.60</td><td>71.10</td></tr><tr><td>DIFF</td><td></td><td>34.67</td><td>30.00</td><td>36.00</td><td>38.00</td><td>58.63</td><td>56.30</td><td>58.90</td><td>60.70</td></tr><tr><td>MEANDIFF +</td><td></td><td>36.00</td><td>22.00</td><td>42.00</td><td>44.00</td><td>89.33</td><td>87.60</td><td>89.70</td><td>90.70</td></tr><tr><td>MEANDIFF-</td><td></td><td>34.67</td><td>14.00</td><td>44.00</td><td>46.00</td><td>93.07</td><td>92.40</td><td>93.10</td><td>93.70</td></tr><tr><td>DIFF</td><td>√</td><td>80.00</td><td>64.00</td><td>86.00</td><td>90.00</td><td>-</td><td>-</td><td>-</td><td>1</td></tr><tr><td>MEANDIFF</td><td>1</td><td>70.67</td><td>36.00</td><td>86.00</td><td>90.00</td><td>=</td><td>-</td><td>-</td><td>-</td></tr></table>
+
+Table 10: Antonym   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Knowledge</td><td colspan="4">Accuracy (%)</td><td colspan="4">Stability (%)</td></tr><tr><td>Mean@3</td><td>@1</td><td>@2</td><td>@3</td><td>Mean@3</td><td>@1</td><td>@2</td><td>@3</td></tr><tr><td>REF</td><td></td><td>0.94</td><td>0.00</td><td>1.19</td><td>16.21</td><td>99.97</td><td>99.90</td><td>100.00</td><td>100.00</td></tr><tr><td>REF+PM</td><td></td><td>38.56</td><td>28.79</td><td>41.38</td><td>45.52</td><td>99.93</td><td>99.80</td><td>100.00</td><td>100.00</td></tr><tr><td>MLP</td><td></td><td>41.38</td><td>35.86</td><td>42.59</td><td>45.69</td><td>1..97</td><td>1.90</td><td>2.00</td><td>2.00</td></tr></table>
+
+# B VISUALIZATION OF PARAMETERIZED MIRRORS
+
+Figures below visualize PCA results of a obtained for the test words. We normalized the L2 norm of a to 1 $( \frac { \mathbf { a } } { \lVert \mathbf { a } \rVert } )$ . Corresponding word pairs are connected by solid lines. Figs. 4 and 5 suggest not only the mirror parameters of paired words are similar to each other but also the parameters with the attribute form a cluster — words with the same attribute has similar mirror parameter a. The mirrors of paired words are close to each other in the same attribute (Fig. 4 and 5). Some MF pairs in Fig. 5 are placed away from the cluster of the MF words. This may come from missing principal components due to the small data size used for PCA. Figs. 6, 7, 8, 9, 10, 11, 12, and 13 are the detailed PCA results for four different attributes: MF, SP, CC, and AN. These results show that a reflection transfers a paired word each other by using a similar mirror. For example, rich and poor use almost the same mirror (Fig. 9). On the other hand, different mirrors are used for different word pairs since the mirrors are parameterized. These results shows the effect of the mirror as described in section 4.4.
+
+![](images/786866719e752120c080bcbcca2472a8fa6198330a10cbf7cb7107e46f7e6c45.jpg)  
+Figure 4: A PCA result of a (word2vec).
+
+![](images/87bbae84df859c8cedc508264a9f16e82a8e52bf4ef2c7e2881954da13947ebc.jpg)  
+Figure 5: A PCA result of a (GloVe).
+
+![](images/3a16a3d5f9aeae75fe79399dd9a0de03e620603e20747108fc9ecaf439fdca14.jpg)  
+Figure 6: Male-Female (word2vec)
+
+![](images/028b7a5a4d0e9d42e17b9c8f8b49f949de875485693c188f88a69690d71dddc1.jpg)  
+Figure 7: Singular-Plural (word2vec)
+
+![](images/2e61fbd48df681210a7065e038778434d49bc23d8c225b79e76148c229bc3800.jpg)  
+Figure 8: Capital-Country (word2vec)
+
+![](images/5f6a179ee60381eba5262a3ceaa8a9fe7c05976f4050571f9bb44f3f008913da.jpg)  
+Figure 9: Antonym (word2vec)
+
+![](images/e48e2dec7a093b8d622deaf6b992ade918e682e9aa59d2dccf89851b162157b1.jpg)  
+Figure 10: Male-Female (GloVe)
+
+![](images/56e0d4c4e8fc38ff738ff48ea9b51fe47a995ddebc53f3b37269a7e651b3581d.jpg)  
+Figure 11: Singular-Plural (GloVe)
+
+![](images/ca7242aaa1ea2d5e327de2696960c27280d651ea2e1fea3fabdadc51337cdabf.jpg)  
+Figure 12: Capital-Country (GloVe)
+
+![](images/f2d0836f02185abbc6a71731fec5a2233e99faeaf783994e832748d9de2896c2.jpg)  
+Figure 13: Antonym (GloVe)

parse/train/HyxoX6EKvB/HyxoX6EKvB_content_list.json

@@ -0,0 +1,1786 @@
+[
+    {
+        "type": "text",
+        "text": "REFLECTION-BASED WORD ATTRIBUTE TRANSFER ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            787,
+            122
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            145,
+            400,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            452,
+            209,
+            544,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose a word attribute transfer framework based on reflection to obtain a word vector with an inverted target attribute for a given word in a word embedding space. Word embeddings based on Pointwise Mutual Information (PMI) represent such analogic relations as ${ \\overrightarrow { k i n g } } - { \\overrightarrow { m a n } } + { \\overrightarrow { w o m a n } } \\approx { \\overrightarrow { q u e e n } }$ . These relations can be used for changing a word’s attribute from king to queen by changing its gender. This attribute transfer can be performed by subtracting a difference vector $\\overrightarrow { m a n } - \\overrightarrow { w o m a n }$ from $\\overrightarrow { k i n g }$ when we have explicit knowledge of the gender of given word king. However, this knowledge cannot be developed for various words and attributes in practice. For transferring queen into king in this analogy-based manner, we need to know that queen denotes a female and add the difference vector to it. In this work, we transfer such binary attributes based on an assumption that such transfer mapping will become identity mapping when we apply it twice. We introduce a framework based on reflection mapping that satisfies this property; queen should be transferred back to king with the same mapping as the transfer from king to queen. Experimental results show that the proposed method can transfer the word attributes of the given words, and does not change the words that do not have the target attributes. ",
+        "bbox": [
+            232,
+            238,
+            766,
+            482
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            507,
+            336,
+            522
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Distributed representation (Hinton et al., 1984) is a kind of data representation that can capture data similarities in a vector space. In natural language processing, various studies have been conducted on word embeddings (Mikolov et al., 2013a;b; Pennington et al., 2014; Peters et al., 2018; Bojanowski et al., 2017). Word embedding models, such as skip-gram with negative sampling (SGNS) (Mikolov et al., 2013b) or GloVe (Pennington et al., 2014), capture some analogic relations, such as $\\overrightarrow { k i n g } -$ $\\overrightarrow { m a n } + \\overrightarrow { w o m a n } \\approx \\overrightarrow { q u e e n }$ . Previous work offer theoretical explanation based on Pointwise Mutual Information (PMI; Church & Hanks (1990)) for maintaining the analogic relations in word vectors (Levy & Goldberg, 2014b; Arora et al., 2016; Gittens et al., 2017; Ethayarajh et al., 2019; Allen & Hospedales, 2019). ",
+        "bbox": [
+            173,
+            537,
+            825,
+            666
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "These relations can be used for transferring a certain attribute of a word, such as changing king into queen by transferring the gender. This task, which is called word attribute transfer, enables us to rewrite He is a boy as She is a girl. Word attribute transfer is expected to be applicable for natural language inference and data augmentation in natural language processing. The above analogic relations can be used, including adding difference vector $\\overrightarrow { w o m a n } - \\overrightarrow { m a n }$ to $\\overrightarrow { k i n g }$ to transfer $\\overrightarrow { k i n g }$ to $\\overrightarrow { q u e e n }$ . This operation requires the explicit knowledge whether an input word is male or female; we have to add a difference vector to a male word and subtract it from a female word for a gender transfer. We also have to avoid changing words without any gender attributes, such as is and $a$ in the example above. Thus, analogy-based word attribute transfer requires explicit knowledge of word attributes, such as king is male, queen is female, and is has no gender attribute. Developing such knowledge is very difficult for various words and attributes in practice. ",
+        "bbox": [
+            173,
+            672,
+            825,
+            833
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this study, we propose a novel framework based on reflection, which enables word attribute transfer by a single reflection-based mapping for a certain attribute. Reflection in geometry is a mapping that exchanges the locations of two vectors in a Euclidean space by a hyperplane called a mirror, which satisfies the above desired property: working as identity mapping when it is applied twice and when it is applied to vectors on the mirror. We apply this reflection mapping to the problem of word attribute transfer by estimating an appropriate mirror that maps word pairs with a binary target attribute (e.g., male and female) and keeps the other words without that attribute using training data. We also extend this approach by introducing parameterized mirrors, which work as different mirrors based on the given input words, to overcome a limitation using a single fixed mirror to represent complex transfer mappings for different words. Experimental results show that the reflection-based method enables such transfers, achieves comparable performance to analogy-based methods with explicit attribute knowledge, even though our proposed method does not use such knowledge. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            188
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The following are the contributions of this paper: ",
+        "bbox": [
+            173,
+            194,
+            496,
+            208
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We propose a novel representation learning framework that obtains a vector with an inverted attribute in embedding space without explicit attribute knowledge of the given word. Our proposed reflection-based word attribute transfer enables us to transfer word attributes in up to $76 \\%$ for words with target attributes and to avoid changing words without target attributes in over $9 9 \\%$ in our experiments. ",
+        "bbox": [
+            215,
+            220,
+            825,
+            296
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 WORD ATTRIBUTE TRANSFER ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            315,
+            457,
+            332
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/d7258f940354ae874b04c913c0b9c4e30a2bfb5fa96d1d2c84de158be4df3acb.jpg",
+        "image_caption": [
+            "Figure 1: Given word vector ${ \\bf v } _ { x }$ and attribute one-hot vector $\\mathbf { z }$ , word attribute transfer predicts word vector $\\mathbf { v } _ { t }$ , which is the inverted attribute of ${ \\bf v } _ { x }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            341,
+            352,
+            650,
+            430
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Let $x$ denote a word and let ${ \\bf v } _ { x }$ denote its vector representation. Here we assume that ${ \\bf v } _ { x }$ is learned in advance with an embedding model such as SGNS. In this task, we have two inputs, word $x$ and onehot vector $\\mathbf { z }$ , representing a certain target attribute, and one output, return word $t$ with the inverted attribute of $x$ for $\\mathbf { z }$ . Let $\\mathcal { A }$ denote a set of a triplet $( x , t , \\mathbf { z } )$ , e.g., $( m a n , w o m a n , \\mathbf { z } _ { \\mathrm { g e n d e r } } ) \\in \\mathcal { A }$ . Let $\\mathcal { N }$ denote a set of words without attribute $\\mathbf { z }$ , e.g., apple $\\in \\mathcal { N }$ , when $\\mathbf { z }$ represents gender. The purpose of this task is to transfer ${ \\bf v } _ { x }$ to $\\mathbf { v } _ { t }$ by transfer function $f _ { \\mathbf { Z } }$ that inverts attribute $\\mathbf { z }$ of ${ \\bf v } _ { x }$ . In other words, output $\\mathbf { v } _ { y }$ should be close to the vector of corresponding target word $\\mathbf { v } _ { t }$ , which is typically the nearest neighbor of $\\mathbf { v } _ { t }$ in the word embedding space. ",
+        "bbox": [
+            173,
+            484,
+            825,
+            597
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fa7ef3d6c94bd27368dc14353d7db665dcccf3bdba20dee4a01e52ce8cfab19c.jpg",
+        "text": "$$\n\\mathbf { v } _ { t } \\approx \\mathbf { v } _ { y } = f _ { \\mathbf { Z } } ( \\mathbf { v } _ { x } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            415,
+            603,
+            550,
+            621
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Note that mapping $f _ { \\mathbf { Z } }$ transfers word $x$ if it has target attribute $\\mathbf { z }$ ; otherwise $f _ { \\mathbf { Z } }$ works as identity mapping. For instance with $\\mathbf { z } _ { g e n d e r }$ , given input word man, gender attribute transfer $f _ { \\mathbf { Z } _ { g e n d e r } } ( \\mathbf { v } _ { m a n } )$ should result in a vector close to $\\mathbf { v } _ { w o m a n }$ . Given input word apple as $x$ , the results should be $\\mathbf { v } _ { a p p l e }$ . ",
+        "bbox": [
+            174,
+            627,
+            825,
+            671
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 ANALOGY-BASED WORD ATTRIBUTE TRANSFER ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            689,
+            609,
+            707
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Analogy is a general idea for realizing attribute transfer. Several PMI-based word embedding methods (Mikolov et al., 2013c; Linzen, 2016) tackled to embed words into word embedding space to capture the analogic relations. An embedded vector with SGNS or GloVe captures analogic relations (Levy & Goldberg, $2 0 1 4 \\mathrm { a }$ ; Mikolov et al., 2013c; Linzen, 2016). For instance, ${ \\mathbf { v } } _ { q u e e n }$ is near the vector obtained on the right side of Eq. 2. By rearranging Eq. 2, Eq. 3 is obtained: ",
+        "bbox": [
+            173,
+            720,
+            825,
+            792
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c70f7271cf1ec2728dd8cb022d5c9205e031aeeb048d9fcf43a26fc8a48f5ed4.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { v } _ { q u e e n } \\approx \\mathbf { v } _ { k i n g } - \\mathbf { v } _ { m a n } + \\mathbf { v } _ { w o m a n } , } \\\\ { \\mathbf { \\tau } } \\\\ { \\approx \\mathbf { v } _ { k i n g } - ( \\mathbf { v } _ { m a n } - \\mathbf { v } _ { w o m a n } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            799,
+            611,
+            849
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We can transfer the gender attribute by subtracting difference vector ${ \\bf v } _ { m a n } - { \\bf v } _ { w o m a n }$ from input word vectors, e.g., $\\mathbf { v } _ { k i n g }$ . The analogy-based transfer function is ",
+        "bbox": [
+            171,
+            854,
+            825,
+            883
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e3d8454c7f3719e189735961690833af9c8389dc6b551ec7b225fb1a43138e40.jpg",
+        "text": "$$\nf _ { \\mathbf { Z } } ( \\mathbf { v } _ { x } ) = { \\left\\{ \\begin{array} { l l } { \\mathbf { v } _ { x } - \\mathbf { d } } & { { \\mathrm { ~ i f ~ } } x \\in { \\mathcal { M } } , } \\\\ { \\mathbf { v } _ { x } + \\mathbf { d } } & { { \\mathrm { ~ i f ~ } } x \\in { \\mathcal { F } } , } \\end{array} \\right. }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            362,
+            892,
+            599,
+            928
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { d }$ is a difference vector of the given word pair such as man and woman, $\\mathcal { M }$ is a set of words having a target attribute, and $\\mathcal { F }$ is a set of words having an inverse attribute, for example, $m a n \\in \\mathcal { M }$ and $w o m a n \\in { \\mathcal { F } }$ for gender attributes. Eq. 4 indicates that the operation changes depending on whether input word $x$ belongs to $\\mathcal { M }$ or $\\mathcal { F }$ . For gender words, we subtract difference vector $\\mathbf { d }$ if $x$ is male, and add it if $x$ is female. Therefore, we need such explicit knowledge. However, this knowledge cannot be developed for various words and attributes in practice. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            188
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 REFLECTION-BASED WORD ATTRIBUTE TRANSFER ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            207,
+            632,
+            224
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.1 IDEALIZED TRANSFER WITHOUT EXPLICIT KNOWLEDGE ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            238,
+            602,
+            252
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "What is an idealized transfer function $\\phi _ { \\mathbf { Z } }$ for the word attribute transfer? The following are the idealized natures of such a transfer function: ",
+        "bbox": [
+            173,
+            263,
+            823,
+            292
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ded2c11aa051a89f0a02c5f55559326eac53555cf6e0e7e33ebe526babe940c4.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { v } _ { m } = \\phi _ { \\mathbf { Z } } ( \\mathbf { v } _ { w } ) , } \\\\ { \\mathbf { v } _ { w } = \\phi _ { \\mathbf { Z } } ( \\mathbf { v } _ { m } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            428,
+            296,
+            535,
+            344
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $m \\in \\mathcal { M }$ and $w \\in { \\mathcal { F } }$ . This function $\\phi _ { \\mathbf { Z } }$ enables to transfer a word without explicit knowledge. Function $\\phi _ { \\mathbf { Z } }$ transfers ${ \\bf v } _ { m }$ to ${ \\bf v } _ { w }$ and ${ \\bf v } _ { w }$ to ${ \\bf v } _ { m }$ without such explicit knowledge as $m \\in \\mathcal { M }$ and $w \\in { \\mathcal { F } }$ . By combining Eqs. 5 and 6, we obtain the following formula: ",
+        "bbox": [
+            174,
+            348,
+            825,
+            390
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4381d9237616e0d1eb3a6f6e8d5d5dda75d2ef55844bdf181e7be053762c70cf.jpg",
+        "text": "$$\n\\forall m \\in \\mathcal { M } , \\qquad \\mathbf { v } _ { m } = \\phi _ { \\mathbf { Z } } \\big ( \\phi _ { \\mathbf { Z } } ( \\mathbf { v } _ { m } \\big ) \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            374,
+            395,
+            620,
+            412
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and ",
+        "bbox": [
+            173,
+            417,
+            200,
+            431
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5d873f143cb0b4d9d016120011ab80765a622a4ad373ba80d94711670f061ed4.jpg",
+        "text": "$$\n\\forall w \\in \\mathcal { F } , \\qquad \\mathbf { v } _ { w } = \\phi _ { \\mathbf { Z } } \\big ( \\phi _ { \\mathbf { Z } } ( \\mathbf { v } _ { w } ) \\big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            436,
+            616,
+            454
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Hence, the idealized transfer function is a mapping that becomes an identity mapping when we apply it twice for any $\\mathbf { v }$ . Such a mapping is called involution in geometry. For example, $\\phi \\colon \\mathbf { v } \\mapsto - \\mathbf { v }$ is one example of an involution. Note that the identity map itself, such as $\\phi \\colon \\mathbf { v } \\mapsto \\mathbf { v }$ , is excluded from the involution. ",
+        "bbox": [
+            174,
+            459,
+            825,
+            515
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.2 REFLECTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            531,
+            303,
+            546
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A reflection is an involution that reverses the location between two vectors in a Euclidean space through an affine hyperplane (mirror). Reflection is an idealized function because every point returns to its original location when reflection is applied twice: ",
+        "bbox": [
+            173,
+            558,
+            825,
+            599
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c1420a90afd4d9c4d80e3ad03975bdeaa973bd33265f239fa5c32d6fb4026a94.jpg",
+        "text": "$$\n\\forall \\mathbf { v } \\in \\mathbb { R } ^ { n } , \\qquad \\mathbf { v } = R e f _ { \\mathbf { a } , \\mathbf { c } } ( R e f _ { \\mathbf { a } , \\mathbf { c } } ( \\mathbf { v } ) { \\bf \\phi } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            356,
+            604,
+            638,
+            622
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given vector $\\mathbf { v }$ in Euclidean space $\\mathbb { R } ^ { n }$ , the formula for the reflection in the mirror is given: ",
+        "bbox": [
+            174,
+            626,
+            767,
+            642
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f5e6daab1290adc2cc6687450825ff05353bd37fe14ba39f2bfc0816ec8dd5fb.jpg",
+        "text": "$$\nR e f _ { \\mathbf { a } , \\mathbf { c } } ( \\mathbf { v } ) = \\mathbf { v } - 2 { \\frac { ( \\mathbf { v } - \\mathbf { c } ) \\cdot \\mathbf { a } } { \\mathbf { a } \\cdot \\mathbf { a } } } \\mathbf { a } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            646,
+            594,
+            679
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { a } \\in \\mathbb { R } ^ { n }$ is a vector orthogonal to the mirror (normal vector) and $\\mathbf { c } \\in \\mathbb { R } ^ { n }$ is a point through which the mirror passes. a and c are parameters that determine the mirror. ",
+        "bbox": [
+            174,
+            681,
+            823,
+            710
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.3 REFLECTION-BASED WORD ATTRIBUTE TRANSFER ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            727,
+            570,
+            741
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We apply reflection to the word attribute transfer to invert a specific attribute of an input word without its explicit attribute knowledge. We learn a mirror (hyperplane) in a pre-trained embedding space using training word pairs with a common (binary) attribute $\\mathbf { z }$ (Fig. 2). Here since the mirror is uniquely determined by two parameter vectors, a and $\\mathbf { c }$ , we estimate a and c from target attribute $\\mathbf { z }$ using fully connected multi-layer perceptrons: ",
+        "bbox": [
+            173,
+            752,
+            825,
+            823
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/71edb7617d9bee7c4e41ca6707f965422caac7a6d10139e89a087553bd7fe411.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { a } = M L P ( \\mathbf { z } ) , } \\\\ { \\mathbf { c } = M L P ( \\mathbf { z } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            429,
+            827,
+            534,
+            873
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Transferred vector $\\mathbf { v } _ { y }$ is obtained by inverting attribute $\\mathbf { z }$ of ${ \\bf v } _ { x }$ by reflection: ",
+        "bbox": [
+            171,
+            877,
+            678,
+            892
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8f5a84934ac0f226a4528f86b4cddcb11e7cc8378f69bf2f7ea1ebc5f0e5f5a5.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { v } _ { y } = R e f _ { \\mathbf { a } , \\mathbf { c } } ( \\mathbf { v } _ { x } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            418,
+            897,
+            547,
+            915
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/4b41c91e239af6f7ce1b9bc1201af457426e51e5f181fbb02278f0acf591f13a.jpg",
+        "image_caption": [
+            "Figure 2: Reflection-based word attribute transfer examples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            236,
+            99,
+            821,
+            253
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/5dec7c1bb005195aa8edc93418d25df229d78394ea104c2d37ea207d3d975c87.jpg",
+        "image_caption": [
+            "Figure 3: Mirror estimation methods "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            194,
+            291,
+            826,
+            445
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.4 PARAMETERIZED MIRRORS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            497,
+            405,
+            511
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Reflection with a mirror by Eqs. 11 and 12 assumes a single mirror depending only on z. Previous discussion assumed that there will be pairs sharing a stable attribute such king and queen. However, often gendered words don’t come in pairs, and gender is far from a stable attribute. For example, actress may be feminine, but actor is clearly neutral in many cases (Fig. 3). Thus, actor isn’t as obvious a masculine counterpart as king. In fact, it is known that there are biases in gender words in the embedding space (Zhao et al., 2018; Kaneko & Bollegala, 2019). This phenomenon can occur not only with the gender attribute, but also with other attributes. With this assumption of a single mirror, the mirror must be a hyperplane that goes through the midpoints for all word vector pairs. However, the vector pairs shown on the left of Fig. 3 cannot be transferred well since the single mirror does not satisfy this constraint due to the bias of the embedding space. To solve this problem, we introduce different mirrors for different words. We propose parameterized mirrors determined by input vector ${ \\bf v } _ { x }$ in addition to attribute $\\mathbf { z }$ . The following are the definitions of the mirror parameters: ",
+        "bbox": [
+            173,
+            523,
+            825,
+            690
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/929f4392610c2ee018118a401dd5871005bfdc6e6f559ca1b93f3dba7a426d83.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { a } = M L P ( [ \\mathbf { z } ; \\mathbf { v } _ { x } ] ) , } \\\\ { \\mathbf { \\ c } = M L P ( [ \\mathbf { z } ; \\mathbf { v } _ { x } ] ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            696,
+            552,
+            750
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $[ \\cdot ; \\cdot ]$ indicates the vector concatenation in the column. The parameterized mirrors are expected to work flexibly on different words. For instance, as shown in Fig. 3, suppose we learned the mirror (the blue line) that transfers $\\mathbf { v } _ { h e r o }$ to $\\mathbf { v } _ { h e r o i n e }$ in advance. If input word vector $\\mathbf { v } _ { a c t o r }$ resembles $\\mathbf { v } _ { h e r o }$ , a mirror that is similar to the one for $\\mathbf { v } _ { h e r o }$ should be derived and used for the attribute transfer. ",
+        "bbox": [
+            174,
+            755,
+            825,
+            825
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.5 LOSS FUNCTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            842,
+            331,
+            857
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Loss function $\\mathcal { L }$ is defined: ",
+        "bbox": [
+            173,
+            868,
+            352,
+            883
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9b751774a4916f96bde15e8a16015e267bbdc9ed5b02ef1596065d74333e1a24.jpg",
+        "text": "$$\n\\mathcal { L } ( \\boldsymbol { \\Theta } ) = \\frac { 1 } { | \\mathcal { A } | } \\sum _ { ( x _ { i } , t _ { i } , \\mathbf { Z } _ { i } ) \\in \\mathcal { A } } ( \\mathbf { v } _ { y _ { i } } - \\mathbf { v } _ { t _ { i } } ) ^ { 2 } + \\frac { 1 } { | \\mathcal { N } | } \\sum _ { x _ { j } \\in \\mathcal { N } } ( \\mathbf { v } _ { y _ { j } } - \\mathbf { v } _ { x _ { j } } ) ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            887,
+            694,
+            928
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\frac { 1 } { | \\mathcal { A } | } \\sum _ { ( x _ { i } , t _ { i } , \\mathbf { Z } _ { i } ) \\in \\mathcal { A } } ( \\mathbf { v } _ { y _ { i } } - \\mathbf { v } _ { t _ { i } } ) ^ { 2 } } \\end{array}$ is a term that draws target word vector $\\mathbf { v } _ { t _ { i } }$ closer to corresponding transferred vector $\\mathbf { v } _ { y _ { i } }$ and $\\begin{array} { r } { \\frac { 1 } { | \\mathcal { N } | } \\sum _ { x _ { j } \\in \\mathcal { N } } ( \\mathbf { v } _ { y _ { j } } - \\mathbf { v } _ { x _ { j } } ) ^ { 2 } } \\end{array}$ is a term that prevents words without a target attribute from being moved by transfer function $f _ { \\mathbf { Z } } . \\Theta$ represents the set of all the trainable parameters. The parameters in the proposed model are the MLP weights used to determine mirror hyperplanes via a and $\\mathbf { c }$ . We iteratively update $\\Theta$ to minimize $\\mathcal { L }$ : ",
+        "bbox": [
+            173,
+            102,
+            825,
+            181
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5f95d2f058b21889153b73f90232368949486244dc5ba9400070a42dcc3c3dd6.jpg",
+        "text": "$$\n\\begin{array} { r } { \\Theta _ { t + 1 } \\gets \\underset { \\Theta _ { t } } { \\operatorname { a r g m i n } } \\mathcal { L } ( \\Theta _ { t } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            185,
+            565,
+            212
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $t$ is the number of parameter updates at that time. ",
+        "bbox": [
+            173,
+            215,
+            542,
+            231
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            251,
+            318,
+            267
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We evaluated the performance of the proposed reflection-based word attribute transfer using data with some different attributes. ",
+        "bbox": [
+            176,
+            281,
+            823,
+            310
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            327,
+            375,
+            340
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We used three different datasets of word pairs with three binary attributes: Male-Female 1, SingularPlural and Capital-Country, shown in Table 1.These word pairs were collected from analogy test sets (Mikolov et al., 2013a; Gladkova et al., 2016), the Internet and Nguyen et al. (2017) for antonyms of noun2. Since these datasets are very small, we added Gaussian noise $\\sigma = 0 . 1$ ) to every input vector ${ \\bf v } _ { x }$ during training to avoid overfit. Random noise was applied independently to every sample in every iteration. For a non-attribute dataset $\\mathcal { N }$ , we sampled words from the three-million-word vocabulary of the word embedding model. We sampled from 4 to 50 words for training $0 ~ \\leq$ $| \\mathcal { N } _ { \\mathrm { t r a i n } } | \\leq 5 0 )$ and 1000 words for the test $\\lvert N _ { \\mathrm { t e s t } } \\rvert = \\bar { 1 } 0 0 0 )$ . We used a mixed dataset that included both $| \\mathcal { N } _ { \\mathrm { t r a i n } } |$ and $| \\mathcal { A } _ { \\mathrm { t r a i n } } |$ . Note that $\\mathcal { N } _ { \\mathrm { t e s t } }$ was sampled and excluded words from $\\mathcal { N } _ { \\mathrm { t r a i n } }$ and $\\mathcal { A } _ { \\mathrm { t r a i n } }$ . We had no $\\mathcal { N } _ { \\mathrm { v a l } }$ because the tuning was conducted with only $| \\mathcal { A } _ { \\mathrm { v a l } } |$ . We used word2vec (Mikolov et al., 2013b) 3 and GloVe (Pennington et al., 2014) 4 as the pre-trained embedding model. The embedded vector dimension is $n = 3 0 0$ . ",
+        "bbox": [
+            173,
+            352,
+            825,
+            518
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/14ad4493483190c3814551ef38028e23fe7d69aa13f6a41358f1c8878310a92e.jpg",
+        "table_caption": [
+            "Table 1: Statistics of binary attribute word pair datasets (in the number of word pairs) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset A</td><td>Train</td><td>Val</td><td>Test</td><td>Total</td></tr><tr><td>Male-Female (MF)</td><td>29</td><td>12</td><td>12</td><td>53</td></tr><tr><td>Singular-Plural (SP)</td><td>90</td><td>25</td><td>25</td><td>140</td></tr><tr><td>Capital-Country (CC)</td><td>59</td><td>25</td><td>25</td><td>109</td></tr><tr><td>Antonym (AN)</td><td>1354</td><td>290</td><td>290</td><td>1934</td></tr></table>",
+        "bbox": [
+            313,
+            554,
+            684,
+            640
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5.2 EVALUATION METRICS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            672,
+            375,
+            688
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We measured the accuracy and stability performances of the word attribute transfer. The accuracy measures how many input words in $\\mathcal { A } _ { \\mathrm { t e s t } }$ were transferred correctly to the corresponding target words. The stability score measures how many words in $\\mathcal { N } _ { \\mathrm { t e s t } }$ are not mapped to other words. For example, in a gender transfer, given man, the transfer is regarded as correct if woman is the closest word to the transferred vector; otherwise it is incorrect. Given apple, the transfer is regarded as correct if apple is the closest word to the transferred vector; otherwise its stability is incorrect. Here we used cosine similarity to measure the similarity of output vector $\\mathbf { v } _ { y }$ and target vector $\\mathbf { v } _ { t }$ . The accuracy and stability scores are calculated by the following formula: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/398d93badd0268cafe357f69a932f8a83c17fe80baa5d420877029308cfe289f.jpg",
+        "text": "$$\n\\delta ( { \\bf v } _ { y } , t ) = \\left\\{ \\begin{array} { l l } { 1 } & { \\mathrm { ~ i f ~ } \\quad \\underset { k \\in \\mathcal { V } } { \\arg \\operatorname* { m a x } } \\frac { { \\bf v } _ { y } \\cdot { \\bf v } _ { k } } { \\| { \\bf v } _ { y } \\| \\| { \\bf V } _ { k } \\| } = t , } \\\\ { 0 } & { \\mathrm { ~ o t h e r w i s e , } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/624cd63e8e1c191833731c942651b335d67dfa89b7067329bd1ad5abf1f3b707.jpg",
+        "text": "$$\n\\mathrm { A c c u r a c y } = \\frac { 1 } { | \\mathcal { A } _ { \\mathrm { t e s t } } | } \\sum _ { ( x _ { i } , t _ { i } , \\mathbf { Z } _ { i } ) \\in \\mathcal { A } _ { \\mathrm { t e s t } } } \\delta ( \\mathbf { v } _ { y _ { i } } , t _ { i } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            98,
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+            138
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fc02a56ff9df59bdc47b64cafec0fb1673ce782232d93548a1ac460583e8fc64.jpg",
+        "text": "$$\n\\mathrm { S t a b i l i t y } = \\frac { 1 } { | \\mathcal N _ { \\mathrm { t e s t } } | } \\sum _ { \\substack { x _ { i } \\in \\mathcal N _ { \\mathrm { t e s t } } } } \\delta ( \\mathbf v _ { y _ { i } } , x _ { i } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            154,
+            612,
+            193
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\nu$ is the vocabulary of the word embedding model. ",
+        "bbox": [
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+            199,
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+            214
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.3 METHODS AND CONFIGURATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            229,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In our experiment, we compared our proposed method with the following baseline methods: ",
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "REF Reflection-based word attribute transfer with a single mirror. We used a fully connected 2-layer MLP with 300 hidden units and ReLU activations to estimate a and c. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "REF+PM Reflection-based word attribute transfer with parameterized mirrors. We used the same MLP as the REF. ",
+        "bbox": [
+            173,
+            315,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "MLP Fully connected MLP: $\\mathbf v _ { y } = M L P ( [ \\mathbf v _ { x } ; \\mathbf z ] )$ . The highest accuracy models are a 2-layer MLP for Capital-Country and 3-layer MLP for the other datasets. ",
+        "bbox": [
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+            348,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "DIFF Analogy-based word attribute transfer with a difference vector: $\\mathbf { d } = \\mathbf { v } _ { m } - \\mathbf { v } _ { w }$ , where $m$ and $w$ are in the training data of $\\mathcal { A }$ . We chose d because it achieved the best accuracy in the validation data of $\\mathcal { A }$ . We determined whether to add or subtract $\\mathbf { d }$ to ${ \\bf v } _ { x }$ based on attribute knowledge (Eq. 4). ",
+        "bbox": [
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+            381,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "DIFF $^ +$ Analogy-based word attribute transfer with a difference vector regardless of the attribute knowledge. d was obtained in the same way as the DIFF. We added $\\mathbf { d }$ to ${ \\bf v } _ { x }$ for any input $x$ : $f _ { \\mathbf { Z } } ( \\mathbf { v } _ { x } ) = \\mathbf { v } _ { x } + \\mathbf { d }$ , $\\forall \\mathbf { v } _ { x } \\in \\mathbb { R } ^ { n }$ . ",
+        "bbox": [
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+            443,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "DIFF − Analogy-based word attribute transfer with a difference vector regardless of the attribute knowledge. d was obtained in the same way as the DIFF. We subtracted $\\mathbf { d }$ from ${ \\bf v } _ { x }$ for any input $x$ : ${ \\bf { \\bar { f } } } _ { \\mathbf { Z } } ( { \\bf v } _ { x } ) = { \\bf v } _ { x } - \\mathbf { d } , \\quad \\forall { \\bf v } _ { x } \\in \\mathbb { R } ^ { n } .$ . ",
+        "bbox": [
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+            488,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "MEANDIFF Analogy-based word attribute transfer with a mean difference vector $\\bar { \\bf d }$ : $\\bar { \\textbf { d } } =$ 1|Atrain| P(mi,wi)∈Atrain (vmi − vwi ). We determined whether to add or subtract d¯ to vx based on the attribute knowledge (Eq.4). ",
+        "bbox": [
+            173,
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+            580
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "MEANDIFF $^ +$ Analogy-based word attribute transfer with a mean difference vector regardless of the attribute knowledge: $f _ { \\mathbf { Z } } ( \\mathbf { v } _ { x } ) = \\mathbf { v } _ { x } + \\bar { \\mathbf { d } }$ , $\\forall \\mathbf { v } _ { x } \\in \\mathbb { R } ^ { n }$ . $\\bar { \\bf d }$ was obtained in the same way as the MEANDIFF. ",
+        "bbox": [
+            174,
+            584,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "MEANDIFF − Analogy-based word attribute transfer with a mean difference vector regardless of the attribute knowledge: $f _ { \\mathbf { Z } } ( \\mathbf { v } _ { x } ) = \\mathbf { v } _ { x } - \\bar { \\mathbf { d } }$ , $\\forall \\mathbf { v } _ { x } \\in \\mathbb { R } ^ { n }$ . d¯ was obtained in the same way as the MEANDIFF. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Based on the tuning, we used the Adam optimizer (Kingma & Ba, 2015) with a learning rate of $\\alpha = 1 0 ^ { - 4 }$ (the other hyperparameters were the same as the original one (Kingma & Ba, 2015)), and a batchsize of 62 for male-female, and 32 for the others. These hyperparameters were identical for the learning-based methods: REF, $\\boldsymbol { \\mathrm { R E F + P M } }$ , or MLP. We did not use such regularization methods as dropout (Srivastava et al., 2014) or batch normalization (Ioffe & Szegedy, 2015) because they did not show any improvement in our pilot test. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.4 ACCURACY AND STABILITY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            786,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 2 and 3 shows the transfer accuracy and stability score results. Both experiments using GloVe or word2vec obtained similar results. $\\boldsymbol { \\mathrm { R E F + P M } }$ achieved the best accuracy among the methods that did not use explicit attribute knowledge. This means that reflection can be used for word attribute transfers even without attribute knowledge. In the stability evaluation, reflection-based methods (REF, $\\mathsf { R E F } + \\mathsf { P M } )$ and the analogy-based methods with a mean difference vector (MEANDIFF $-$ , MEANDIFF $^ +$ ) achieved high stability. In particular, reflection-based transfers achieved outstanding stability scores exceeding $9 9 \\%$ . The stability of DIFF $^ +$ and DIFF $-$ was much lower than the other methods. Although MEANDIFF − and MEANDIFF + achieved high stability, their accuracy results were very low. Interestingly, reflection-based transfer with parameterized mirrors $( \\mathrm { R E F } + \\mathrm { P M } )$ achieved high performance in both accuracy and stability. For example, the accuracy of RE $\\boldsymbol { \\mathbf { \\ell } } + \\mathbf { P M }$ was $4 1 . 6 7 \\%$ , and the stability was $9 9 . 9 \\%$ in Male-Female (MF), and the accuracy was $58 \\%$ and the stability was $9 9 . 4 0 ~ \\%$ in Capital-Country (CC). These results show that the proposed method transfers an input word if it has a target attribute and does not transfer an input word even though it does not use explicit attribute knowledge on the input words. MLP worked poorly both in accuracy and stability. In the antonym (AN), while the transfer accuracy by the proposed method was a bit lower than that by MLP, the stability of the proposed method was $100 \\%$ and that of MLP was really poor (almost $0 \\%$ ). ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/0f16f08f1cfb26a955ee4b00b6e6a413e2be9fee4306051faeabfe12e96eadd5.jpg",
+        "table_caption": [
+            "Table 2: Results in accuracy and stability scores (word2vec). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Knowledge</td><td colspan=\"4\">Accuracy (%)</td><td colspan=\"4\">Stability (%)</td></tr><tr><td>MF</td><td>SP</td><td>CC</td><td>AN</td><td>MF</td><td>SP</td><td>CC</td><td>AN</td></tr><tr><td>REF</td><td></td><td>20.83</td><td>0.00</td><td>36.00</td><td>0.00</td><td>99.80</td><td>100.00</td><td>99.80</td><td>100.00</td></tr><tr><td>REF+PM</td><td></td><td>41.67</td><td>22.00</td><td>58.00</td><td>28.79</td><td>99.90</td><td>99.40</td><td>99.40</td><td>100.00</td></tr><tr><td>MLP</td><td></td><td>8.33</td><td>4.00</td><td>12.00</td><td>35.86</td><td>2.20</td><td>0.00</td><td>2.70</td><td>1.90</td></tr><tr><td>DIFF+</td><td></td><td>25.00</td><td>2.00</td><td>32.00</td><td>-</td><td>72.10</td><td>77.90</td><td>53.90</td><td>-</td></tr><tr><td>DIFF-</td><td></td><td>25.00</td><td>2.00</td><td>30.00</td><td>=</td><td>49.60</td><td>78.20</td><td>56.30</td><td>1</td></tr><tr><td>MEANDIFF +</td><td></td><td>4.17</td><td>0.00</td><td>22.00</td><td>=</td><td>98.60</td><td>99.40</td><td>87.60</td><td></td></tr><tr><td>MEANDIFF</td><td></td><td>8.33</td><td>0.00</td><td>14.00</td><td>=</td><td>97.20</td><td>99.30</td><td>92.40</td><td></td></tr><tr><td>DIFF</td><td></td><td>62.50</td><td>4.00</td><td>64.00</td><td></td><td>1</td><td>=</td><td>=</td><td></td></tr><tr><td>MEANDIFF</td><td></td><td>12.50</td><td>0.00</td><td>36.00</td><td></td><td></td><td></td><td></td><td></td></tr></table>",
+        "bbox": [
+            173,
+            125,
+            826,
+            296
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/7d9f1402357ae6ccceaee695adca5cfbc4a413c424c0789053a09d2b01081b82.jpg",
+        "table_caption": [
+            "Table 3: Results in accuracy and stability scores (GloVe). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Knowledge</td><td colspan=\"4\">Accuracy (%)</td><td colspan=\"4\">Stability (%)</td></tr><tr><td>MF</td><td>SP</td><td>CC</td><td>AN</td><td>MF</td><td>SP</td><td>CC</td><td>AN</td></tr><tr><td>REF</td><td></td><td>12.50</td><td>2.00</td><td>26.00</td><td>0.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td></tr><tr><td>REF+PM</td><td></td><td>45.83</td><td>50.00</td><td>76.00</td><td>33.54</td><td>99.70</td><td>99.10</td><td>99.20</td><td>100.00</td></tr><tr><td>MLP</td><td></td><td>4.17</td><td>10.00</td><td>18.00</td><td>36.72</td><td>5.10</td><td>7.00</td><td>5.20</td><td>1.20</td></tr><tr><td>DIFF+</td><td></td><td>25.00</td><td>2.00</td><td>26.00</td><td>-</td><td>99.30</td><td>94.20</td><td>99.30</td><td>1</td></tr><tr><td>DIFF-</td><td></td><td>25.00</td><td>2.00</td><td>24.00</td><td>=</td><td>100.60</td><td>99.90</td><td>99.50</td><td>=</td></tr><tr><td>MEANDIFF +</td><td></td><td>0.00</td><td>0.00</td><td>22.00</td><td></td><td>100.00</td><td>100.00</td><td>100.00</td><td>一</td></tr><tr><td>MEANDIFF</td><td></td><td>0.00</td><td>0.00</td><td>0.00</td><td></td><td>100.00</td><td>100.00</td><td>100.00</td><td></td></tr><tr><td>DIFF</td><td></td><td>50.00</td><td>4.00</td><td>44.00</td><td></td><td>=</td><td>=</td><td></td><td></td></tr><tr><td>MEANDIFF</td><td></td><td>0.00</td><td>0.00</td><td>0.00</td><td></td><td></td><td>-</td><td></td><td></td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We investigated the relation between the size of $| \\mathcal { N } _ { \\mathrm { t r a i n } } |$ and the stability of learning-based methods by conducting an additional experiment by varying $| \\mathcal { N } _ { \\mathrm { t r a i n } } |$ from 0 to 50. The stability scores by MLP did not improve (Table 4). On the other hand, REF and $\\boldsymbol { \\mathrm { R E F } } + \\boldsymbol { \\mathrm { P M } }$ achieved high stability scores with just $| \\bar { \\mathcal { N } } _ { \\mathrm { t r a i n } } | = 4$ and maintained the accuracy. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.5 TRANSFER EXAMPLE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            743,
+            362,
+            757
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 5 shows examples of a gender transfer at the sentence level, where the attribute transfer was applied to words in sentence $\\bar { X } = \\{ x _ { 1 } , x _ { 2 } , \\ldots \\}$ . Here since such words as $a$ and . are not in the vocabulary of the original word embedding model, we omitted them from the inputs. MLP made many wrong transfers on words without gender attributes, e.g., the became By Katie Klingsporn, was became she, when became Doughty Evening Chronicle, and woman became girlfriend. $\\mathrm { D I F F ^ { + } }$ can transfer if $x$ is female, e.g., it transferred from woman to man, but it could not transfer grandfather and boy. Similarly, DIFF − failed to transfer from female to male. In addition, since the stability of these methods was low, they erroneously transferred. For example, in DIFF $^ -$ , the and when became she. REF + PM can selectively transfer words with a gender attribute without using explicit gender information. For example, when woman was given, $R e f ( \\mathbf { v } _ { w o m a n } )$ became man without knowledge that woman is a female word, and when man was given, it became woman. When non-attribute word married was given, $R e f ( \\mathbf { v } _ { m a r r i e d } )$ became married without knowledge that married has no gender attribute. When we applied the reflection-based transfer twice, the transferred word returned to its original word, e.g., $R e f ( R e f ( \\mathbf { v } _ { w o m a n } ) )$ gives woman. ",
+        "bbox": [
+            173,
+            770,
+            825,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/b2e123d72b0df2dac4f5b0e1da33809deaceb6862e5f622c69cc946f381b3eab.jpg",
+        "table_caption": [
+            "Table 4: Relation among size of $| \\mathcal { N } _ { \\mathrm { t r a i n } } |$ and stability of learning-based methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"2\"></td><td colspan=\"4\">Accuracy (%)</td><td colspan=\"4\">Stability (%)</td></tr><tr><td colspan=\"4\">Wtrainl</td><td colspan=\"4\">|Wtrainl</td></tr><tr><td></td><td></td><td>0</td><td>4</td><td>10</td><td>50</td><td>0</td><td>4</td><td>10</td><td>50</td></tr><tr><td rowspan=\"3\">MF</td><td>REF</td><td>16.67</td><td>20.83</td><td>20.83</td><td>20.83</td><td>98.30</td><td>98.80</td><td>99.40</td><td>99.80</td></tr><tr><td>REF+PM</td><td>41.67</td><td>45.83</td><td>20.83</td><td>41.67</td><td>38.30</td><td>98.40</td><td>100.00</td><td>99.90</td></tr><tr><td>MLP</td><td>4.17</td><td>8.33</td><td>8.33</td><td>8.33</td><td>0.00</td><td>0.30</td><td>0.30</td><td>2.20</td></tr><tr><td rowspan=\"3\">SP</td><td>REF</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>99.90</td><td>99.90</td><td>99.90</td><td>100.90</td></tr><tr><td>REF+PM</td><td>12.00</td><td>22.00</td><td>18.00</td><td>18.00</td><td>98.40</td><td>99.40</td><td>99.30</td><td>99.80</td></tr><tr><td>MLP</td><td>4.00</td><td>4.00</td><td>2.00</td><td>2.00</td><td>0.00</td><td>0.00</td><td>0.10</td><td>3.40</td></tr><tr><td rowspan=\"3\">CC</td><td>REF</td><td>36.00</td><td>36.00</td><td>36.00</td><td>34.00</td><td>99.80</td><td>99.80</td><td>99.80</td><td>100.00</td></tr><tr><td>REF+PM</td><td>58.00</td><td>56.00</td><td>58.00</td><td>54.00</td><td>73.80</td><td>99.70</td><td>99.40</td><td>99.40</td></tr><tr><td>MLP</td><td>6.00</td><td>6.00</td><td>8.00</td><td>12.00</td><td>0.00</td><td>0.30</td><td>0.50</td><td>2.70</td></tr><tr><td rowspan=\"3\">AN</td><td>REF</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>99.90</td><td>99.90</td><td>99.90</td><td>99.80</td></tr><tr><td>REF+PM</td><td>21.72</td><td>27.24</td><td>28.62</td><td>28.79</td><td>95.30</td><td>99.20</td><td>99.50</td><td>99.80</td></tr><tr><td>MLP</td><td>34.14</td><td>35.00</td><td>34.31</td><td>35.86</td><td>0.00</td><td>0.01</td><td>0.02</td><td>1.90</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            381,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/fe0874db2874fadee852b2743932f7450fffd94af80d5e2234e3cc86521453fc.jpg",
+        "table_caption": [
+            "Table 5: Transfer results when sentence $X = \\{ \\mathrm { t h e , . . . , b o y } \\}$ was given. Out-of-vocabulary words $a$ and . were not given as input. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>X</td><td>the woman was married when your grandfather was (a) boy (.)</td></tr><tr><td>Ref(x)</td><td>the man was married when your grandmother was (a) girl (.)</td></tr><tr><td>Ref(Ref(x))</td><td>the woman was married when your grandfather was (a) boy (.)</td></tr><tr><td>MLP</td><td>By_Katie_Klingsporn girlfriend she fiancee Doughty_Evening_Chronicle ma&#x27;am daughter she (a) mother (.)</td></tr><tr><td>DIFF+</td><td>the man was married when your grandfather was (a) boy (.)</td></tr><tr><td>DIFF-</td><td>she woman was married she your grandmother was (a) girl (.)</td></tr></table>",
+        "bbox": [
+            230,
+            473,
+            766,
+            595
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We can transfer some different attributes of words with reflection-based transfer one-by-one. Table 6 shows that the words having different target attributes were transferred by each reflection-based transfer in the order of Male-Female, Singular-Plural, and Country-Capital. Given actress for a Male-Female transfer, it was transferred to actor and to actors for Singular-Plural. Given Tokyo for Male-Female, Singular-Plural, and Antonym, it was not transferred, but it was transferred to Japan for Country-Capital. Given rich for Male-Female, Singular-Plural, and Capital-Country, it was not transferred, but it was transferred to poor for Antonym. ",
+        "bbox": [
+            173,
+            616,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/df8cc0d4b43b980d0e7ac412654950fbe34e7e0d88681e74378176b8b226faaa.jpg",
+        "table_caption": [
+            "Table 6: Transfer of different attributes with reflection-based word attribute transfer with parameterized mirrors. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>X</td><td>the rich actress and the poor actor want to stay the beautiful city in Tokyo.</td></tr><tr><td>+Male-Female</td><td>the rich actor and the poor actress want to stay the beautiful city in Tokyo.</td></tr><tr><td>+ Singular-Plural</td><td></td></tr><tr><td></td><td>the rich actors and the poor actresses want to stay the beautiful cities in Tokyo.</td></tr><tr><td>+Capital-Country</td><td>the rich actors and the poor actresses want to stay the beautiful citie in Japan.</td></tr><tr><td>+ Antonym</td><td>the poor actors and the rich actresses want to stay the beautiful cities in Japan.</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "6 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The embedded vectors obtained by SGNS (Mikolov et al., 2013a;b) and GloVe (Pennington et al., 2014) have analogic relations. The theory of analogic relations in word embeddings has been widely discussed: Levy & Goldberg (2014b); Arora et al. (2016); Gittens et al. (2017); Ethayarajh et al. (2019); Allen & Hospedales (2019); Linzen (2016). Levy & Goldberg (2014b) offer the explanation that SGNS factorizes a shifted PMI matrix. Allen & Hospedales (2019) and Ethayarajh et al. (2019) argued that they proved the existence of such analogic relations without strong assumptions. In our work, we focus on the analogic relations in a word embedding space and propose a novel framework to obtain a word vector with inverted attributes. Style transfers (Niu et al., 2018; Prabhumoye et al., 2018; Jain et al., 2019; Logeswaran et al., 2018; Dai et al., 2019; Zhang et al., 2018) resemble our task. In a style transfer, the text style of the input sentences is changed. For instance, Jain et al. (2019) transferred from formal to informal sentences. Logeswaran et al. (2018) transferred sentences by controlling such attributes as mood and tense. These style transfer tasks use sentence pairs; our word attribute transfer task uses word pairs. Style transfer changes sentence styles, but our task changes the word attributes (contents). Soricut & Och (2015) studied the problem of morphological transformation based on character information. Our work aims more general attribute transfer such as gender transfer and country-capital and is not limited to the morphological transformation. ",
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+    {
+        "type": "text",
+        "text": "7 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We proposed a novel representation learning framework based on reflection to invert a certain attribute of a word vector. We proposed a reflection-based method for word attribute transfers without relying on the explicit attribute knowledge of an input word, which is necessary for a simple analogy-based transfer. Experimental results showed that our proposed method can transfer the word attributes if the input word has a target attribute. If not, reflection does not transfer the word. ",
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+    {
+        "type": "text",
+        "text": "Future work includes applications to other transfer tasks: sentence by sentence transfer, such Niu et al. (2018); Prabhumoye et al. (2018); Jain et al. (2019), and entity prediction on an analogic graph embedding space (Liu et al., 2017), in the field of computer vision, visual analogy (Reed et al., 2015), or style transfer (Zhu et al., 2017; Liao et al., 2017) with GANs (Radford et al., 2016; Goodfellow et al., 2014) because their latent space holds analogic relations (Radford et al., 2016). ",
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+        "type": "text",
+        "text": "A TOP THREE ACCURACY AND STABILITY ",
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+        "table_caption": [
+            "Table 7: Male-Female "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Knowledge</td><td colspan=\"4\">Accuracy (%)</td><td colspan=\"4\">Stability (%)</td></tr><tr><td>Mean@3</td><td>@1</td><td>@2</td><td>@3</td><td>Mean@3</td><td>@1</td><td>@2</td><td>@3</td></tr><tr><td>REF</td><td></td><td>50.00</td><td>20.83</td><td>62.50</td><td>66.67</td><td>99.80</td><td>99.80</td><td>99.80</td><td>99.80</td></tr><tr><td>REF+PM</td><td></td><td>55.56</td><td>41.67</td><td>58.33</td><td>66.67</td><td>99.13</td><td>99.00</td><td>99.20</td><td>99.20</td></tr><tr><td>MLP</td><td></td><td>20.83</td><td>8.33</td><td>20.83</td><td>33.33</td><td>2.20</td><td>2.20</td><td>2.20</td><td>2.20</td></tr><tr><td>DIFF+</td><td></td><td>31.94</td><td>25.00</td><td>33.33</td><td>37.50</td><td>75.43</td><td>72.10</td><td>75.80</td><td>78.40</td></tr><tr><td>DIFF-</td><td></td><td>31.94</td><td>25.00</td><td>33.33</td><td>37.50</td><td>55.80</td><td>49.60</td><td>57.30</td><td>60.50</td></tr><tr><td>MEANDIFF +</td><td></td><td>23.61</td><td>4.17</td><td>33.33</td><td>33.33</td><td>98.93</td><td>98.60</td><td>99.10</td><td>99.10</td></tr><tr><td>MEANDIFF</td><td></td><td>23.61</td><td>8.33</td><td>29.17</td><td>33.33</td><td>97.63</td><td>97.20</td><td>97.80</td><td>97.90</td></tr><tr><td>DIFF</td><td>√</td><td>68.05</td><td>62.50</td><td>66.66</td><td>75.00</td><td>-</td><td>1</td><td>-</td><td>1</td></tr><tr><td>MEANDIFF</td><td>1</td><td>47.22</td><td>12.50</td><td>62.50</td><td>66.67</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></table>",
+        "bbox": [
+            173,
+            161,
+            852,
+            333
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/741e0783888d0e58705ba45537e365b8096c9e2381cdc8ba0c2507cd4ce50193.jpg",
+        "table_caption": [
+            "Table 8: Singular-Plural "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Knowledge</td><td colspan=\"4\">Accuracy (%)</td><td colspan=\"4\">Stability (%)</td></tr><tr><td>Mean@3</td><td>@1</td><td>@2</td><td>@3</td><td>Mean@3</td><td>@1</td><td>@2</td><td>@3</td></tr><tr><td>REF</td><td></td><td>50.00</td><td>0.00</td><td>72.00</td><td>78.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td></tr><tr><td>REF+PM</td><td></td><td>56.67</td><td>22.00</td><td>72.00</td><td>76.00</td><td>99.70</td><td>99.40</td><td>99.80</td><td>99.90</td></tr><tr><td>MLP</td><td></td><td>7.33</td><td>4.00</td><td>8.00</td><td>10.00</td><td>0.13</td><td>0.00</td><td>0.20</td><td>0.20</td></tr><tr><td>DIFF+</td><td></td><td>36.67</td><td>2.00</td><td>50.00</td><td>58.00</td><td>80.27</td><td>77.90</td><td>79.70</td><td>83.20</td></tr><tr><td>DIFF-</td><td></td><td>36.67</td><td>2.00</td><td>52.00</td><td>56.00</td><td>80.27</td><td>78.20</td><td>80.70</td><td>81.90</td></tr><tr><td>MEANDIFF +</td><td></td><td>42.00</td><td>0.00</td><td>56.00</td><td>70.00</td><td>99.47</td><td>99.40</td><td>99.40</td><td>99.60</td></tr><tr><td>MEANDIFF -</td><td></td><td>39.33</td><td>0.00</td><td>56.00</td><td>62.00</td><td>99.50</td><td>99.30</td><td>99.60</td><td>99.60</td></tr><tr><td>DIFF</td><td>√</td><td>49.33</td><td>4.00</td><td>70.00</td><td>74.00</td><td>1</td><td>-</td><td>1</td><td>1</td></tr><tr><td>MEANDIFF</td><td>√</td><td>52.00</td><td>0.00</td><td>76.00</td><td>80.00</td><td>1</td><td>=</td><td>-</td><td>-</td></tr></table>",
+        "bbox": [
+            173,
+            378,
+            872,
+            547
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/5a4299c1e12fe95ceebcc2c62c0707cdcc47295b1775497065c002a59b6edded.jpg",
+        "table_caption": [
+            "Table 9: Capital-Country "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Knowledge</td><td colspan=\"4\">Accuracy (%)</td><td colspan=\"4\">Stability (%)</td></tr><tr><td>Mean@3</td><td>@1</td><td>@2</td><td>@3</td><td>Mean@3</td><td>@1</td><td>@2</td><td>@3</td></tr><tr><td>REF</td><td></td><td>66.67</td><td>36.00</td><td>78.00</td><td>86.00</td><td>99.80</td><td>99.80</td><td>99.80</td><td>99.80</td></tr><tr><td>REF+PM</td><td></td><td>72.67</td><td>58.00</td><td>74.00</td><td>86.00</td><td>99.53</td><td>99.40</td><td>99.60</td><td>99.60</td></tr><tr><td>MLP</td><td></td><td>35.33</td><td>12.00</td><td>40.00</td><td>54.00</td><td>2.80</td><td>2.70</td><td>2.80</td><td>2.90</td></tr><tr><td>DIFF+</td><td></td><td>39.33</td><td>32.00</td><td>42.00</td><td>44.00</td><td>64.87</td><td>53.90</td><td>69.60</td><td>71.10</td></tr><tr><td>DIFF</td><td></td><td>34.67</td><td>30.00</td><td>36.00</td><td>38.00</td><td>58.63</td><td>56.30</td><td>58.90</td><td>60.70</td></tr><tr><td>MEANDIFF +</td><td></td><td>36.00</td><td>22.00</td><td>42.00</td><td>44.00</td><td>89.33</td><td>87.60</td><td>89.70</td><td>90.70</td></tr><tr><td>MEANDIFF-</td><td></td><td>34.67</td><td>14.00</td><td>44.00</td><td>46.00</td><td>93.07</td><td>92.40</td><td>93.10</td><td>93.70</td></tr><tr><td>DIFF</td><td>√</td><td>80.00</td><td>64.00</td><td>86.00</td><td>90.00</td><td>-</td><td>-</td><td>-</td><td>1</td></tr><tr><td>MEANDIFF</td><td>1</td><td>70.67</td><td>36.00</td><td>86.00</td><td>90.00</td><td>=</td><td>-</td><td>-</td><td>-</td></tr></table>",
+        "bbox": [
+            173,
+            593,
+            852,
+            765
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/8a6a764945a8d934c5cb0b81fd67ce31d79f09680f10a17c748d0c97d549e421.jpg",
+        "table_caption": [
+            "Table 10: Antonym "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Knowledge</td><td colspan=\"4\">Accuracy (%)</td><td colspan=\"4\">Stability (%)</td></tr><tr><td>Mean@3</td><td>@1</td><td>@2</td><td>@3</td><td>Mean@3</td><td>@1</td><td>@2</td><td>@3</td></tr><tr><td>REF</td><td></td><td>0.94</td><td>0.00</td><td>1.19</td><td>16.21</td><td>99.97</td><td>99.90</td><td>100.00</td><td>100.00</td></tr><tr><td>REF+PM</td><td></td><td>38.56</td><td>28.79</td><td>41.38</td><td>45.52</td><td>99.93</td><td>99.80</td><td>100.00</td><td>100.00</td></tr><tr><td>MLP</td><td></td><td>41.38</td><td>35.86</td><td>42.59</td><td>45.69</td><td>1..97</td><td>1.90</td><td>2.00</td><td>2.00</td></tr></table>",
+        "bbox": [
+            173,
+            126,
+            848,
+            212
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B VISUALIZATION OF PARAMETERIZED MIRRORS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            236,
+            598,
+            251
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Figures below visualize PCA results of a obtained for the test words. We normalized the L2 norm of a to 1 $( \\frac { \\mathbf { a } } { \\lVert \\mathbf { a } \\rVert } )$ . Corresponding word pairs are connected by solid lines. Figs. 4 and 5 suggest not only the mirror parameters of paired words are similar to each other but also the parameters with the attribute form a cluster — words with the same attribute has similar mirror parameter a. The mirrors of paired words are close to each other in the same attribute (Fig. 4 and 5). Some MF pairs in Fig. 5 are placed away from the cluster of the MF words. This may come from missing principal components due to the small data size used for PCA. Figs. 6, 7, 8, 9, 10, 11, 12, and 13 are the detailed PCA results for four different attributes: MF, SP, CC, and AN. These results show that a reflection transfers a paired word each other by using a similar mirror. For example, rich and poor use almost the same mirror (Fig. 9). On the other hand, different mirrors are used for different word pairs since the mirrors are parameterized. These results shows the effect of the mirror as described in section 4.4. ",
+        "bbox": [
+            173,
+            266,
+            826,
+            435
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/786866719e752120c080bcbcca2472a8fa6198330a10cbf7cb7107e46f7e6c45.jpg",
+        "image_caption": [
+            "Figure 4: A PCA result of a (word2vec). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            245,
+            449,
+            750,
+            638
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/87bbae84df859c8cedc508264a9f16e82a8e52bf4ef2c7e2881954da13947ebc.jpg",
+        "image_caption": [
+            "Figure 5: A PCA result of a (GloVe). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            245,
+            685,
+            751,
+            875
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/3a16a3d5f9aeae75fe79399dd9a0de03e620603e20747108fc9ecaf439fdca14.jpg",
+        "image_caption": [
+            "Figure 6: Male-Female (word2vec) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            111,
+            795,
+            338
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/028b7a5a4d0e9d42e17b9c8f8b49f949de875485693c188f88a69690d71dddc1.jpg",
+        "image_caption": [
+            "Figure 7: Singular-Plural (word2vec) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            200,
+            387,
+            795,
+            613
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/2e61fbd48df681210a7065e038778434d49bc23d8c225b79e76148c229bc3800.jpg",
+        "image_caption": [
+            "Figure 8: Capital-Country (word2vec) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            662,
+            797,
+            890
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/5f6a179ee60381eba5262a3ceaa8a9fe7c05976f4050571f9bb44f3f008913da.jpg",
+        "image_caption": [
+            "Figure 9: Antonym (word2vec) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            112,
+            797,
+            337
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/e48e2dec7a093b8d622deaf6b992ade918e682e9aa59d2dccf89851b162157b1.jpg",
+        "image_caption": [
+            "Figure 10: Male-Female (GloVe) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            388,
+            795,
+            611
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/56e0d4c4e8fc38ff738ff48ea9b51fe47a995ddebc53f3b37269a7e651b3581d.jpg",
+        "image_caption": [
+            "Figure 11: Singular-Plural (GloVe) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            660,
+            795,
+            887
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/ca7242aaa1ea2d5e327de2696960c27280d651ea2e1fea3fabdadc51337cdabf.jpg",
+        "image_caption": [
+            "Figure 12: Capital-Country (GloVe) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            181,
+            795,
+            402
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/f2d0836f02185abbc6a71731fec5a2233e99faeaf783994e832748d9de2896c2.jpg",
+        "image_caption": [
+            "Figure 13: Antonym (GloVe) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            593,
+            799,
+            819
+        ],
+        "page_idx": 16
+    }
+]

parse/train/OdklztJBBYH/OdklztJBBYH.md

@@ -0,0 +1,316 @@
+# Exploring Architectural Ingredients of Adversarially Robust Deep Neural Networks
+
+Hanxun Huang1 Yisen Wang2,3 Sarah Erfani1 Quanquan $\mathbf { G u ^ { 4 } }$ James Bailey1 Xingjun $\mathbf { M } \mathbf { a } ^ { \mathsf { \pm } }$
+
+1School of Computing and Information Systems, The University of Melbourne, Victoria, Australia   
+2Key Lab. of Machine Perception, School of Artificial Intelligence, Peking University, Beijing, China 3Institute for Artificial Intelligence, Peking University, Beijing, China 4University of California, Los Angeles, USA 5School of Computer Science, Fudan University, Shanghai, China
+
+# Abstract
+
+Deep neural networks (DNNs) are known to be vulnerable to adversarial attacks. A range of defense methods have been proposed to train adversarially robust DNNs, among which adversarial training has demonstrated promising results. However, despite preliminary understandings developed for adversarial training, it is still not clear, from the architectural perspective, what configurations can lead to more robust DNNs. In this paper, we address this gap via a comprehensive investigation on the impact of network width and depth on the robustness of adversarially trained DNNs. Specifically, we make the following key observations: 1) more parameters (higher model capacity) does not necessarily help adversarial robustness; 2) reducing capacity at the last stage (the last group of blocks) of the network can actually improve adversarial robustness; and 3) under the same parameter budget, there exists an optimal architectural configuration for adversarial robustness. We also provide a theoretical analysis explaning why such network configuration can help robustness. These architectural insights can help design adversarially robust DNNs. Code is available at https://github.com/HanxunH/RobustWRN.
+
+# 1 Introduction
+
+Deep neural networks (DNNs) are becoming standard models for many real-world applications such as image classification [1], object detection [2] and natural language processing [3]. However, a line of research has shown that DNNs are vulnerable to adversarial examples (attacks), which can be easily crafted by slightly perturbing the input instance to maximize the model’s prediction error [4–6]. This vulnerability of DNNs has become a major concern for their deployment in security-critical applications such as autonomous driving [7, 8] and medical diagnosis [9, 10].
+
+A number of defense methods have been proposed to train adversarially robust DNNs [11–14], among which adversarial training has demonstrated the most promising results [15–17]. Adversarial training can be viewed as a type of data augmentation that trains DNNs on adversarial (instead of natural) examples [15–19]. Based on adversarial training, a set of works have been proposed to understand its learning and convergence behaviors, and the key factors for training adversarially robust DNNs. For example, it has been found that adversarial training encourages the model to learn more robust or compact features [20, 21], and it requires more data [22–25] or higher capacity models to gain more robustness [15, 26]. While these understandings have motivated several improved defense methods, it is still not clear, from an architectural perspective, what makes an adversarially robust DNN.
+
+In this paper, we present the first comprehensive investigation on the architectural ingredients of adversarially robust DNNs. Our investigation is based on adversarial training and WideResNet-34-10 (WRN-34-10) [27], one extensively tested architecture in the defense literature. Based on the base architectural configuration of WRN-34-10, we apply a finely-controlled grid search to explore the impact of network width and depth configurations on the robustness of adversarial trained DNNs.
+
+The standard WRN-34-10 consists of 3 stages with each stage being a group of 5 (i.e., depth) residual blocks and each residual block having 2 convolutional layers. We denote the three stages as Stage-1, Stage-2 and Stage-3 following the direction from the input to the output. Each stage is configured by a depth (number of residual blocks) and a width (number of filters) factor. The hyper-parameters for width and depth of each stage control the scale of learnable parameters (capacity). In this paper, we explore different configurations of width and depth for each of the three stages. Based on our explorations, we make the following key observations:
+
+• Simply increasing the number of parameters (model capacity) by upscaling width or depth does not necessarily lead to improved robustness. This contrasts with current beliefs that, under the same type of architecture, more parameters (higher model capacity) can improve adversarial robustness [15, 26, 28]. Adversarial training does require larger capacity models, but there exists a trade-off. We provide both theoretical and empirical evidences that wider/deeper models increase Lipschitzness (larger Lipschitz constant). • For a larger model used in adversarial training, reducing capacity at the last stage (Stage-3) of WRNs can achieve a better trade-off between capacity and Lipschitzness, thus improving adversarial robustness. This can be achieved by reducing either depth or width, with width reduction being slightly more effective. This highlights that smaller DNNs can also have better robustness if the parameter reduction is applied at the right place (i.e., the last stage). • Under the same type of architectures (i.e., WRNs) and parameter budget, there may exist an optimal architectural configuration that can produce the most robust DNN. We show that the same configuration rule can also be applied to improve the robustness of VGGs, DenseNets (DNs), as well as networks found by Differentiable Architecture Search (DARTS).
+
+Furthermore, we provide a series of understandings for the above findings, which can not only provide useful insights for training more robust models with adversarial training, but also shed new light on the architectural ingredients of adversarially robust DNNs.
+
+# 2 Related Work
+
+# 2.1 Adversarial Training
+
+Adversarial training has been demonstrated to be the most reliable training method for obtaining adversarially robust DNNs [29, 30]. The standard adversarial training (SAT) can be formulated as a min-max optimization framework as follows:
+
+$$
+\underset { \pmb { \theta } } { \arg \operatorname* { m i n } } \mathbb { E } _ { ( \pmb { x } , \pmb { y } ) \sim \mathbb { D } } \left[ \underset { \pmb { x } ^ { \prime } } { \operatorname* { m a x } } \mathcal { L } ( f _ { \pmb { \theta } } , \pmb { x } ^ { \prime } , \pmb { y } ) \right] ,
+$$
+
+where the inner maximization generates adversarial examples $\mathbf { x } ^ { \prime }$ , the outer minimization trains the model on $\mathbf { x } ^ { \prime }$ , $f _ { \theta }$ denote the neural network and $\mathcal { L } ( \cdot )$ is the cross entropy (CE) loss. During the inner maximization process, SAT uses PGD to generate adversarial examples [15]:
+
+$$
+\pmb { x } _ { k } ^ { \prime } = \Pi _ { \epsilon } ( \pmb { x } _ { k - 1 } ^ { \prime } + \alpha \cdot \mathrm { s i g n } ( \nabla _ { \pmb { x } } \mathcal { L } ( f _ { \pmb { \theta } } , \pmb { x } _ { k - 1 } ^ { \prime } , y ) ) ) ,
+$$
+
+where $\mathrm { s i g n } ( \cdot )$ is the sign function, $ { \boldsymbol { { x } } } _ { k } ^ { \prime }$ is the adversarial example obtained at the $k$ -th (for overall $K$ steps) perturbation step, $\alpha$ is the step size, and $\Pi _ { \epsilon }$ is a projection (clipping) operation that projects the perturbation back onto the $\epsilon$ -ball centered around $_ { \textbf { \em x } }$ if it goes beyond.
+
+Improved variants of SAT have also been proposed, such as the trade-off between adversarial robustness and natural accuracy (TRADES) [16], Dynamic AdveRsarial Training (DART) [17], Friendly Adversarial Training (FAT) [31], Misclassification Aware adveRsarial Training (MART) [18], Robust Self-Training (RST) [23], Unsupervised Adversarial Training (UAT) [32], Guided Adversarial Training (GAT) [33], Max-Margin AT [34], using Max-Mahalanobis Center (MMC) loss [35], accelerated AT [36–38], using pre-training [39], incorporating hypersphere embedding [40], self-progressing robust training [41], Adversarial Weight Perturbation (AWP) [19], Adversarial Distributional Training (ADT) [42], Channel-wise Activation Suppressing (CAS) [21], GeometryAware Instance-Reweighted Adversarial Training (GAIRAT) [43] and robustness distillation [44, 45]. Adversarial Training has also been found to cause robust overfitting [46], but it can be mitigated by smoothing techniques [47].
+
+# 2.2 Understanding Adversarially Trained DNNs
+
+Understanding the working mechanism of adversarial training has been a hot research area. For example, it has been found that adversarial training encourages the model to learn more robust features [20, 48], have good generative ability [49, 50], improve the model’s transferability to downstream tasks [51, 52] and improve performance on clean data [53]. It has also been found that using auxiliary training data with adversarial training can further improve adversarial robustness [22, 23], and that weight decay plays an important role in adversarial training [54]. Another important observation is that using WRNs instead of ResNets (RNs) can bring $\sim 3 \% { - } 5 \%$ more robustness [16–18]. Other works also suggest that adversarial training requires deeper and wider models [15, 26, 55]. Also, the skip-connection operation used in WRN has been found can improve robustness for deeper architectures [56] and there exists a trade-off between depth and width for approximating natural functions [57]. On the other hand, there are also works showing that increasing the number of parameters for the same type of DNN architectures can only lead to limited robustness improvement [28, 58]; and wider networks may cause more perturbation instability [59].
+
+Several recent works have applied neural architecture search (NAS) to search for more robust DNN architectures [60]. They found that, 1) densely connected cells result in improved robustness; and 2) under certain computational budget, adding convolution operations to direct connection edge is effective. Other works improve the NAS search strategy by searching on targeted capacity [61], maximizing certified lower bound [62], using the log-normal distribution to approximate the Lipschitz constant [63], using lower and upper confidence bounds in Bandit [64], or using perturbation-based regularization [65]. Another study on hand-crafted versus NAS-based architectures shows that, without adversarial training, NAS-based architectures are more robust for small-scale datasets and simple tasks than hand-crafted architectures, however, hand-crafted architectures are more robust than NAS-based architectures as the dataset size or the task complexity increases [66]. Note that NAS is extremely time-consuming, especially when applied with adversarial training. Previous works using NAS find optimal topological connections within the cell structure [60], but did not investigate depth/width configurations, which arguably has more impact on robustness (e.g., RNs vs. WRNs). In this work, we focus on fine-grained configuration exploration rather than blind search, which can produce more precise understandings of how depth and width affect robustness.
+
+# 3 Wider and Deeper Models Increase Lipschitz Upper Bound
+
+It has been theoretically shown that high Lipschitzness (larger Lipschitz constant) corresponds to low stability of the model’s output to input perturbations [59]. However, adversarial training does require a larger capacity model (e.g., RN vs. WRN) [15], an empirical finding that goes against the theoretical expectation. In this section, we first theoretically show a trade-off between network capacity (width/depth) and the Lipschitz upper bound. In Section 4, we will empirically examine this trade-off and its relation to the improved adversarial robustness for larger capacity models.
+
+The Lipschitz constant $L$ of a DNN measures the maximum rate of change in the output with the change in the input, and is closely related to adversarial robustness [4]. Formally, it is $\| f _ { \pmb \theta } ( \pmb x ) - \bar { f } _ { \pmb \theta } ( \pmb x ^ { \prime } ) \| \le \bar { L } \| \pmb x - \pmb x ^ { \prime } \|$ .
+
+Theorem 1 (Lipschitz Constant Upper Bound of a Neural Network with Gaussian Distributed Weights). Consider an $n$ layer DNN $f$ , where the weight parameters $\pmb \theta$ are independent Gaussian random variables distributed as $\mathcal { N } ( 0 , \sigma _ { \theta } ^ { 2 } )$ with $\boldsymbol { \sigma } _ { \theta } ^ { 2 }$ denoting the variance of the Gaussian distribution, and where the activation functions are $I$ -Lipschitz. The expected Lipschitz constant of a DNN with hidden layer size $h$ is upper bounded by:
+
+$$
+L ( f _ { \pmb \theta } ) \leq \prod _ { j = 1 } ^ { n } \left( \sqrt { h _ { j - 1 } } + \sqrt { h _ { j } } \right) \cdot \sigma _ { \pmb \theta _ { j } }
+$$
+
+Theorem 2. For a convolutional neural network $f$ , each layer’s convolution operation with feature map size $W \times m \times m$ and kernel size $k \times k$ , where the weight parameters $\pmb { \theta }$ are independent Gaussian
+
+random variables distributed as $\mathcal { N } ( 0 , \sigma _ { \theta } ^ { 2 } )$ , the expected Lipschitz constant is upper bounded by:
+
+$$
+L ( f _ { \pmb \theta } ) \leq \prod _ { j = 1 } ^ { n } ( m _ { j } \sqrt { W _ { j - 1 } } + ( m _ { j } - k _ { j } + 1 ) \sqrt { W _ { j } } ) \cdot \sigma _ { { \pmb \theta } _ { j } }
+$$
+
+The proof for Theorem 1 and 2 is inspired by [67–69] and can be found in Appendix A.1. This establishes a connection between the upper bound on the Lipschitz constant of a feed-forward DNN with $n$ layers and width of $h _ { j }$ for each layer. For the $j$ -th layer of convolution operations, the Lipschitz constant upper bound increases with its input dimension $( W _ { j - 1 } \times m _ { j } \times m _ { j } )$ and the number of output channels $W _ { j }$ . More simply, it is upper-bounded by the variance of the weight matrix and the input representation’s dimension plus the output representation’s dimension. For the entire network, the Lipschitz constant upper bound grows exponentially with the depth. This suggests that wider and deeper models have a relatively larger change of the output due to the changes in the input, i.e., lower adversarial robustness.
+
+Several works attempt to regularize the network’s Lipschitz constant by using Parseval tight frames on the weight matrixes [70], enforcing constraints on the singular values of the weight matrixes [71], or via a Lipschitz-margin training [72]. However, a follow-up work points out that there exist both experimental and theoretical limitations for the above approaches [73]. Whilst the Lipschitz constant may not be used as a regularization, it has been widely adopted for analyzing the stability and adversarially robustness of DNNs [4, 59, 74].
+
+# 4 Exploring Adversarially Robust Architectures
+
+Our exploration of the relationship between DNN architectural configuration, Lipschitzness (size of the Lipschitz constant) and adversarial robustnessare starts with a fine-controlled grid search on the width/depth of the WideResNet (WRN) [27]. In Sections 4.1 and 4.2, we show our exploration results with depth and width, respectively. Based on these results, a pattern of robust depth/width configuration is discovered. In Section 4.4, we examine a linear scaling effect with the discovered robust configuration. In Section 4.5, we provide an analysis on the trade-off between model capacity and Lipschitzness, and the key factors contributing to improved adversarial robustness.
+
+![](images/2f9baf00f136e5769fb34335b7fe2d63f3c30fcb5a6832f987221d1bd62d1435.jpg)  
+Figure 1: (a): Illustration of WRN-34-10 denoted as $^ { \mathrm { d } } 5  – 5 – 5$ . (b): Grid search results on different depth configurations. The three-digit numbers highlight the depth configurations of only those networks that have either a low $\mathrm { < = 5 0 . 0 \% }$ ) or a high $> = 5 2 . 5 \%$ ) adversarial robustness (against $\mathrm { P G D ^ { 2 0 } }$ ).
+
+Base architecture. We take the standard WRN-34-10 designed for CIFAR-10 as our base architecture. Figure 1a provides an overview of the architecture and the detailed configurations are summarized in Appendix Table 3. The standard WRN architecture consists of 3 stages (groups) of residual blocks and 4 fixed convolutional layers. Here, we focus on the configuration of the 3 stages, which are the key components of the network. We denote the depth and width configuration for the $i$ -th $( i \in \{ 1 , 2 , 3 \} )$ ) stage as $D _ { i }$ and $W _ { i }$ , respectively. For standard WRN-34-10, $D _ { 1 / 2 / 3 } = 5$ (denoted as $^ { \mathrm { d } } 5  – 5 – 5 )$ and $W _ { 1 / 2 / 3 } = 1 0$ (denoted as $^ { \mathrm { w } } 1 0 \mathrm { - } 1 0 \mathrm { - } 1 0 $ ). For the rest of this paper, we use $^ { \mathrm { d } } D _ { 1 } { - } D _ { 2 } { - } D _ { 3 }$ and $^ { \mathrm { w } } W _ { 1 ^ { - } } W _ { 2 ^ { - } } W _ { 3 }$ to represent the exact width and depth configurations. We explore the stage-wise depth and width configurations while keeping other configurations unchanged.
+
+Experimental settings. We train all explored networks on CIFAR-10 dataset [1] using the standard adversarial training (SAT) with Projected Gradient Descent (PGD) [15] (see definition in equation (2)). Following the typical adversarial training setting, we constrain the $L _ { \infty }$ -norm of the maximum adversarial perturbation to $\epsilon = 8 / 2 5 5$ , and use 10-step PGD $( { \mathrm { P G D } } ^ { 1 0 } )$ with step size $\alpha = 2 / 2 5 5$ . After training, we test the robustness of the network on PGD adversarial examples crafted on the entire test set of CIFAR-10, under the same perturbation constraint $\epsilon = 8 / 2 5 5$ . For evaluation, we use the 20-step PGD $( \mathrm { P G D ^ { 2 0 } } )$ with step size $\bar { \alpha } = \epsilon / 1 0$ . The robustness is measured by the network’s accuracy on the $\mathrm { P G D ^ { 2 0 } }$ test adversarial examples. More details can be found in Appendix B.
+
+# 4.1 Exploring Different Depths
+
+We first explore different depth configurations based on the base WRN-34-10 architecture introduced above. For each of the three stages (e.g., Stage-1, Stage-2, Stage-3), we explore different depth $D _ { i } \in \{ 1 , 3 , 5 , 7 , 9 \}$ . Since each stage has 5 possible depth configurations, the total number of all possible depth configuration for all 3 stages are 125 (5x5x5, permutation with replacement). We first perform a grid search on all the 125 depth configurations, then take a closer look at the impact at each individual stage. The adversarial robustness of the 125 networks (adversarially trained using SAT) against $\mathrm { P G D ^ { 2 0 } }$ test adversarial examples is plotted in Figure 1b. Note that the depth configuration of standard WRN-34-10 is $^ { \mathrm { d } } 5  – 5 – 5$ . By investigating the robustness scores along the $\mathbf { X }$ -axis (number of parameters), we find that more parameters does not necessarily lead to improved robustness. For example, the networks with more than 80M (million) parameters are even less robust than some of those with only 20M parameters. Given the same level of parameters, for example ${ \sim } 2 0 \mathbf { M }$ , different depth configurations can lead to $\sim 6 \%$ difference in robustness. This implies that, under the same parameter budget, there may exist an optimal depth configuration for adversarial robustness.
+
+Next, we take a closer look at the above grid search result and investigate the common characteristics of the top-5 most robust networks, the details of which are reported in Figure 2d. Interestingly, we find that the top-5 networks all have a significantly reduced depth of 1 or 3 at the last stage (i.e., Stage-3). This trend indicates that reducing model capacity at the last (deepest) stage can actually improve robustness. The other observation is that, having more residual blocks (higher depth) at the two shallow stages (i.e., Stage-1/2) can also improve robustness. For example, the top-2 networks have 9 residual blocks at Stage-1, and all top-4 networks have 9 or at least 7 residual blocks at Stage-2. This suggests that capacity is more important for the shallow layers. We conjecture this is because the network still needs sufficient capacity to learn the augmented examples by adversarial training. Note that the best performing model $- 9 - 7 - 1$ only uses half of the parameters of the standard WRN-34-10 (d5-5-5), which is only ranked the 45-th out of all 125 models.
+
+![](images/591388548c726a1145488d9a8b95e16b1587118fa8538106e5fb4c9b6f0b9ecc.jpg)  
+Figure 2: (a-c) The impact of depth on adversarial robustness at different stages. When studying one stage, the depths of other two stages are fixed to 5. (d) Clean accuracy and adversarial robustness of the top-5 most robust depth configurations discovered in the grid search. All networks are trained using SAT [15] on CIFAR-10. Robustness evaluated using $\mathrm { P G D ^ { 2 0 } }$ . d5-5-5 is the depth configuration of the baseline WRN-34-10 model.
+
+We further explore the distinctive impacts of depth on adversarial robustness at different stages via a control study. Specifically, we add or remove residual blocks from each individual stage of WRN-34-10 (d5-5-5) while keeping the other two stages fixed to depth 5. The robustness results are shown in Figure 2a-2c. As can be observed, reducing depth at the first two stages constantly degrades the robustness, however, it is the other way around at the last stage (i.e., Stage-3). In relation to previous understanding that higher model capacity can lead to more robust models [28, 58], our finding indicates that it is true for the shallow layers but quite the opposite for the deeper layers. In other words, more parameters can improve adversarial robustness only when added to the shallow layers (e.g., layers in Stage-1 and Stage-2).
+
+# 4.2 Exploring Different Widths
+
+We further explore whether width also has a similar effect as depth. The standard WRN-34-10 has a width upscaling factor 10 applied to each stage, that is, $^ { \mathrm { w } } 1 0 \mathrm { - } 1 0 \mathrm { - } 1 0$ . Based on our above findings with the depth in Section 4.1, here we skip the grid search and directly investigate the impact of width at different stages. At each stage, we investigate different width configurations $W _ { i } \in \{ 2 , 4 , 6 , 8 , 1 0 \}$ for $i = { 1 , 2 , 3 }$ .
+
+![](images/a6e0d6bd4eebb11e4655603cc36f37e7e3881fc56090e66da6c3fba5eb7cf113.jpg)  
+Figure 3: (a-c): The impact of width on adversarial robustness at different stages. When studying one stage, the widths of other two stages are fixed to 10. (d): Clean accuracy and adversarial robustness of the networks obtained by reducing width in the last stage (i.e., Stage-3). All networks are trained using SAT [15] on CIFAR-10. Robustness is evaluated using $\mathrm { P G D ^ { 2 0 } }$ . w10-10-10 is the depth configuration of the baseline WRN-34-10 model.
+
+The robustness results are illustrated in Figure 3. We find that width reduction generally has a similar effect as depth reduction: reducing width at the first two stages harms robustness until $W _ { \frac { 1 } { 2 } / 2 } = 4$ , however, the same operation can improve robustness when applied to the last stage. This confirms the importance of high capacity at the shallow layers and low capacity at the deeper layers. Compared to depth reduction, we find that, with the same amount of robustness improvement, width reduction (at the last stage) can lead to smaller models. For example, $^ { \mathrm { w } } 1 0 ^ { . } 1 0 ^ { . } 4$ (the second row in Figure 3d) achieves a similar robustness $( \sim 5 4 \% )$ as $^ { \mathrm { d } } 9 . 7 . 1$ (the first row in Figure 2d). However, the number of parameters of the $^ { \mathrm { w } } 1 0 ^ { \phantom { - } } 1 0 ^ { \phantom { - } } $ configuration is only 17.05M, which is much less than the 22.19M of the d9-7-1 configuration.
+
+Another interesting observation is that adversarial robustness does not change much if we reduce the width from $W _ { 1 / 2 } = 4$ to $W _ { 1 / 2 } = 2$ at Stage-1 or Stage-2, whereas the same reduction at Stage-3 hurts robustness. This is somewhat expected since, on one hand, the robustness might not be affected much unless a sufficient number of filters (channels) are removed, which is different to depth that configures the entire residual block. On the other hand, if too many filters are removed at the last stage, the network may lose the capacity required for proper learning, while in our depth exploration, there exists at least one residual block $D _ { 3 } \geq 1 \underline { { \cdot } }$ ) at the last stage.
+
+# 4.3 Exploring Depth-Width Combinations
+
+Although reducing capacity at the last stage via either depth or width can improve robustness, there exists a limit. For example, if we reduce depth and width at the same time or too much of the width, the network may end up with insufficient capacity for proper learning. We first explore an extreme case that removes the entire Stage-3 as $\mathrm { d } 5 . { \dot { 5 } } { - } 0$ . This ends up with $2 \%$ less robustness than baseline WRN-34-10. This result verifies the necessity of Stage-3. We then reduce depth and width simultaneously by setting the depth to $^ { \mathrm { d } } 5  – 5 – 1$ and width to $^ { \mathrm { w } } 1 0 – 1 0 – 2$ . This produces a new network with a similar robustness $( \sim 5 2 \% )$ ) to $\mathrm { P G D ^ { 2 0 } }$ as WRN-34-10. Note that, in this case, comparing to WRN-34-10, the number of parameters has been reduced by $70 \%$ . We then explore all the 25 possible depth-width combinations between the top-5 depth and width configurations in Figure 2d and 3d, respectively. Surprisingly, we find that none of these models can achieve better robustness than simply reducing the width to $^ { \mathrm { w } } 1 0 ^ { . } 1 0 ^ { . } 4$ . These models achieved the same level of robustness $( \sim 5 4 \% )$ ), but require more computations (FLOPS). For instance, the network with depth $^ { \mathrm { d } } 7 . 9 . 3$ and width $^ { \mathrm { w } } 1 0 { - } 1 0 { - } 4$ requires 2 times more FLOPS than WRN-34-10. This does not benefit adversarial training since it is known to be time-consuming. Although a more fine-grained (with decimals) exploration of the width configuration may lead to even more robust WRN models, here we simply take the $^ { \mathrm { w } } 1 0 { - } 1 0 { - } 4$ configuration as our choice of the optimally-reduced WRN.
+
+# 4.4 Scaling with the Discovered Configuration
+
+Previous works [15, 16] have shown that a wider network like WRN-34-10 can be trained to be more robust than a standard ResNet like RN-34. Here, we investigate if we can obtain more robust models by scaling the discovered width configuration $^ { \mathrm { w } } 1 0 ^ { \phantom { - } } 1 0 ^ { \phantom { - } } 4$ . We test different scaling ratios $\gamma \in [ 0 . 2 5 , 2 . 0 ]$ , and show the robustness results in Table 1. Compared to $^ { \mathrm { w } } 1 0 ^ { \phantom { - } } 1 0 ^ { \phantom { - } } $ $( \gamma = 1 . 0 )$ ), scaling down $\gamma$ to 0.5 or 0.25 decreases the robustness while scaling up $\gamma$ can further improve the robustness, although the improvement become less significant when $\gamma$ goes above 1.5. Note that the network with $\gamma = 0 . 5$ has 10 times fewer parameters than the baseline WRN-34-10, but can already achieve a better robustness against $\mathrm { P G D ^ { 2 0 } }$ . The best robustness is achieved at $\gamma = 2 . 0$ , i.e., $^ { \mathrm { w } } 2 0 – 2 0 – 8$ . We denote the corresponding WRN-34 network as WRN-34-R, more details can be found in Appendix Table 3. In Section 5, we will apply the $^ { \mathrm { w } } 1 0 ^ { \phantom { - } } 1 0 ^ { \phantom { - } } 4$ configuration rule to more network architectures and evaluate their (along with WRN-34-R) adversarial robustness more systematically.
+
+Table 1: Clean accuracy and adversarial robustness for scaled $^ { \mathrm { w } } 1 0 – 1 0 – 4$ configurations. All networks are trained using SAT [15] on CIFAR-10 and evaluated using $\mathrm { P G D ^ { 2 0 } }$ .   
+
+<table><tr><td>Scaling Ratio</td><td>Params (M)</td><td>Clean (%)</td><td>PGD20 (%)</td></tr><tr><td>γ= 0.25</td><td>1.07</td><td>80.61</td><td>50.90</td></tr><tr><td>γ= 0.5</td><td>4.27</td><td>84.31</td><td>54.07</td></tr><tr><td>γ=1.0</td><td>17.05</td><td>87.00</td><td>54.99</td></tr><tr><td>γ= 1.5</td><td>38.33</td><td>87.68</td><td>55.33</td></tr><tr><td>Y=2.0</td><td>68.12</td><td>88.12</td><td>55.35</td></tr></table>
+
+# 4.5 Empirical Understanding
+
+We apply two closely related metrics, including Perturbation Stability [59] and Empirical Lipschitz [74] to explore the distinctive impact of the deeper layers to adversarial robustness. They measure the output stability of the neural network.
+
+Perturbation Stability. Adversarial robustness is typically measured by the percentage of correctly classified adversarial examples, which can be further decomposed into the set of correct clean examples intersect with stable examples [59]. Correct clean examples refer to clean examples that can be correctly classified by the model $\{ ( \pmb { x } , y ) \sim \mathbb { D } , f _ { \pmb { \theta } } ( \pmb { x } ) \ \stackrel { \ } { = } \ y \}$ . Stable examples are defined as, $\{ \pmb { x } : \forall \pmb { x } ^ { \prime } \in \mathcal { X } , f _ { \pmb { \theta } } ( \pmb { x } ) = f _ { \pmb { \theta } } ( \pmb { x } ^ { \prime } ) \}$ , where $\mathcal { X }$ is the domain of the $\epsilon$ -ball around $_ { \textbf { \em x } }$ . This perturbation stability measures the fraction of examples whose outputs cannot be adversarially perturbed. While many factors can affect the model’s performance on the correct clean examples such as the generalization capability of the model, the perturbation stability only measures if adversarial perturbations can change the prediction of the output. We apply this metric to understand the role of neural network architecture in adversarial robustness. More specifically we are interested to find out whether the improved robustness is a result of improved generalization, stability or both.
+
+Empirical Lipschitz constant. The empirical Lipschitz constant is defined as [74]:
+
+$$
+\frac { 1 } { n } \sum _ { i = 1 } ^ { n } \operatorname* { m a x } _ { \pmb { x } ^ { \prime } \in \mathcal { X } } \frac { \| f _ { \pmb { \theta } } ( \pmb { x } _ { i } ) - f _ { \pmb { \theta } } ( \pmb { x } _ { i } ^ { \prime } ) \| _ { 1 } } { \| \pmb { x } _ { i } - \pmb { x } _ { i } ^ { \prime } \| _ { \infty } } ,
+$$
+
+where $\mathcal { X }$ is the domain of the $\epsilon$ -ball around $_ { \textbf { \em x } }$ and $\mathbf { x } ^ { \prime }$ can be generated by an adversarial attack (i.e., $\mathrm { P G D ^ { 2 0 } }$ ). A lower value of the empirical Lipschitz constant implies a smoother and more adversarially robust classifier. For our analysis, we measure the empirical Lipschitz constant of the functions represented by the output layers of different residual blocks or the entire network (logits output). e.g., for block-5, we measure the maximum rate of change in its representation output between clean $( { \pmb x } )$ and adversarial $( { \pmb x } ^ { \prime } )$ inputs.
+
+![](images/6ac9f1800b40e21419d866a7203da0c6c55573dc35565fd9e790a026b8ad5f7c.jpg)  
+Figure 4: The change of perturbation stability and empirical Lipschitz constant when (a) depth of Stage-3 is reduced, (b) width of Stage-3 is reduced, or (c) linear scaling with $\gamma$ and $^ { \mathrm { w } } 1 0 ^ { \phantom { - } } 1 0 ^ { \phantom { - } } 4$ .
+
+The Trade-off Between Capacity and Lipschitzness. We compute the above two metrics on the test set of CIFAR-10 for models with different depth and width configurations explored in Section 4.1 and 4.2. The results are illustrated in Figure 4, where the empirical Lipschitz constant is computed for the entire network. We can observe that, when depth or width for the given network is reduced, the empirical Lipschitz constant is also reduced, and the perturbation stability improves. This is consistent with our theoretical analysis in the Lipschitz constant upper bound. As shown in Figure 4b and 4c, this observation is more obvious for the width reduction. There exists a trade-off between the network capacity and Lipschitzness. For example, with $\gamma = 0 . 2 5$ scaling, the network achieves a much lower Lipschitzness and better stability, however, it also significantly reduces the clean accuracy. This indicates that adversarial training does require larger capacity models and a better trade-off can be achieved by balancing model capacity and Lipschitzness using proper architectural reduction.
+
+![](images/8f0507b8575bf1878004dd9d1b5ab89ae137a31f57b4eb97d38fd773dc6beac1.jpg)
+
+![](images/1150a3d5e465a31af54c3acc12b01307d78309e86f0793115d7537c5460196a9.jpg)
+
+![](images/97c2d9d09aa79eece94769c1ede44826b43d79cf301db049130142b32066a6d7.jpg)  
+(a) Reducing width at Stage-1. (b) Reducing width at Stage-2. (c) WRN-34-12 and WRN-34-R Figure 5: Empirical Lipschitz constant of the output layers of different residual blocks (bins 1-15) or the entire network (bin 16). All experiments are run on CIFAR-10.
+
+In Figure 5, we plotted the empirical Lipschitz constant of the output layer of each residual block or the entire network. The $f _ { \pmb \theta } ( \pmb x )$ in equation (3) is replaced with the output of each residual block $f _ { \pmb { \theta } _ { j } } ( \pmb { x } )$ (from input to block output), and the last (16-th) bin is the empirical Lipschitz constant of the entire network (from input to logits). From Figure 5, we find that: 1) within each stage, the empirical Lipschitz increases with depth; 2) when transitioning from one stage to the next, the spatial dimension decreases while the empirical Lipschitz decreases; 3) comparing WRN-34-12 with WRN-34-R (Figure 5c), the empirical Lipschitz increase/decrease with the network width. This provides empirical results for our theoretical analysis in Section 3.
+
+Reducing Parameters at Deeper Layers Improves both Perturbation Stability and Lipschitzness. Based on our theoretical analysis, reducing the width at Stage-1 and Stage-2 should improve the Lipschitzness of the corresponding stage as well as the entire network. However, empirically, it is true for the corresponding stage but not necessarily for the entire network. This is because the theoretical analysis in Section 3 only considers the interplay between two adjacent layers (or blocks), not including that of the non-adjacent layers. The empirical results in Figure 5 can fill this gap.
+
+Specifically, Stage-1 width reduction (Figure 5a) lowers the Lipschitzness of Stage-1 blocks but not the overall Lipschitzness. Stage-2 width reduction (Figure 5b) can improve both Stage-2 and the overall Lipschitzness but fails to improve clean accuracy, perturbation stability nor adversarial robustness. WRN-34-R in Figure 5c marks our discovered reduction and scaling rule. Compared with standard WRNs, WRN-34-R not only reduces the width of Stage-3 (decreasing Lipschitzness) but also increases the widths of Stage-1 and Stage-2 (increasing Lipschitzness). Figure 5c shows that the increased Lipschitzness at Stage-1 and Stage-2 of WRN-34-R can be effectively mitigated at Stage-3, leading to decreased overall Lipschitzness and improved robustness (see Table 4.4). This also results in higher clean accuracy and perturbation stability (Figure $_ \mathrm { 4 c }$ ). We conjecture this is because Stage-3 (the last stage) is closer to the final output, thus has a more direct impact on the overall Lipschitzness. These empirical results provide a more in-depth understanding of the impact of width and depth configurations to the overall Lipschitzness, perturbation stability and adversarial robustness.
+
+# 5 Adversarial Robustness Evaluation
+
+In this section, we apply the discovered $^ { \mathrm { w } } 1 0 { - } 1 0 { - } 4$ width configuration rule to VGG, DenseNet (DN), DNNs discovered by NAS, WRN-34-R $^ { \mathrm { w } } 1 0 – 1 0 – 4$ scaled by $\gamma = 2 . 0$ ), and evaluate their robustness with various adversarial attacks and defence methods on CIFAR [1] in the white-box setting. Additional results for CIFAR-10 black-box and ImageNet [75] using FastAT [37] can be found in Appendix D.
+
+Experimental Settings. We consider VGG-11 [76], DenseNet-121 (DN-121) [77] and a network found by DARTS [78] with 11 cells. We denote the optimized VGG-11, DN-121 and DARTS networks as VGG-11-R, DN-121-R and DARTS-R (see Appendix B.2 for details), respectively. For a fair comparison between the discovered WRN-34-R (scaled by $\gamma = 2 . 0$ ) and the standard
+
+WRN, we upscale WRN-34-10 to WRN-34-12 to make sure the two models have a similar amount of parameters. We train all networks using 4 adversarial training methods: Standard Adversarial Training (SAT) [15], TRADES [16], Misclassification Aware adveRsarial Training (MART) [18] and Robust Self-Training (RST) with 500K additional data [23]. More details are in Appendix B.1.
+
+Table 2: White-box robustness results on CIFAR-10 and CIFAR-100. 500K: Additional data used as in [23]. Params: number of parameters. SAT: Adversarial Training [15]; TRADES [16]; MART [18]; GAMA100: 100-step GAMA attack [33]; AA: AutoAttack [30]; $\mathrm { C W } _ { \infty }$ : $L _ { \infty }$ version CW attack [79] optimized by PGD; -R: reconfigured networks following our discovered robust architectural configuration; Last: Results evaluated at the last checkpoint. Best: Results evaluated at the best checkpoint according to $\mathrm { P G D ^ { 2 0 } }$ . The best results are in bold.
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Model</td><td rowspan="2">Method</td><td rowspan="2">Params (M)</td><td colspan="2">Clean (%)</td><td colspan="2">FGSM (%）</td><td colspan="2">PGD20 (%)</td><td colspan="2">GAMA100 (%）</td><td colspan="2">CW (%）</td><td colspan="2">AA (%)</td></tr><tr><td>Last</td><td>Best</td><td>Last</td><td>Best</td><td>Last</td><td>Best</td><td>Last</td><td>Best</td><td>Last</td><td>Best</td><td>Last</td><td>Best</td></tr><tr><td rowspan="10">CIFAR-10</td><td>VGG-11</td><td>SAT</td><td>9.23</td><td>79.24</td><td>77.84</td><td>55.98 57.35</td><td>56.68</td><td>42.62</td><td>45.46</td><td>38.59</td><td>40.65</td><td>45.45</td><td>46.29</td><td>37.21 38.44</td><td>39.84</td></tr><tr><td>VGG-11-R</td><td>SAT</td><td>5.83</td><td>79.63</td><td>77.34</td><td></td><td>57.11</td><td>43.93</td><td>45.97</td><td>39.71</td><td>41.31</td><td>46.49</td><td>47.23</td><td></td><td>40.65</td></tr><tr><td>DN-121</td><td>SAT</td><td>6.96</td><td>86.87</td><td>86.07</td><td>65.56</td><td>66.58</td><td>51.67</td><td>54.79</td><td>48.60</td><td>51.00</td><td>52.03</td><td>54.00</td><td>47.16</td><td>50.34</td></tr><tr><td>DN-121-R</td><td>SAT</td><td>6.00</td><td>87.22</td><td>86.01</td><td>67.12</td><td>67.20</td><td>52.52</td><td>55.16</td><td>49.37</td><td>51.44</td><td>53.07</td><td>54.67</td><td>47.75</td><td>50.54</td></tr><tr><td>DARTS</td><td>SAT</td><td>6.58</td><td>86.76</td><td>86.55</td><td>64.48</td><td>67.10</td><td>49.44</td><td>54.23</td><td>46.52</td><td>50.74</td><td>52.03</td><td>54.00</td><td>45.16</td><td>49.98</td></tr><tr><td>DARTS-R</td><td>SAT</td><td>2.53</td><td>87.20</td><td>85.79</td><td>66.74</td><td>66.61</td><td>52.36</td><td>55.01</td><td>48.71</td><td>50.94</td><td>53.07</td><td>54.67</td><td>47.75</td><td>50.54</td></tr><tr><td>WRN-34-12</td><td>SAT</td><td>66.46</td><td>86.71</td><td>87.20</td><td>64.06</td><td>66.26</td><td>49.92</td><td>53.09</td><td>47.45</td><td>50.40</td><td>52.23</td><td>53.58</td><td>46.06</td><td>49.18</td></tr><tr><td>WRN-34-R</td><td>SAT</td><td>68.12</td><td>87.62</td><td>87.85</td><td>66.23</td><td>68.15</td><td>51.08</td><td>55.35</td><td>48.45</td><td>51.36</td><td>52.42</td><td>54.57</td><td>46.75</td><td>50.03</td></tr><tr><td>WRN-34-12</td><td>TRADES</td><td>66.46</td><td>85.84</td><td>84.59</td><td>65.70</td><td>66.85</td><td>53.02</td><td>56.01</td><td>49.60</td><td>52.35</td><td>53.35</td><td>54.72</td><td>48.48</td><td>51.83</td></tr><tr><td>WRN-34-R</td><td>TRADES</td><td>68.12</td><td>86.77</td><td>86.02</td><td>67.99</td><td>68.49</td><td>55.15</td><td>57.66 57.95</td><td>51.92</td><td>53.86</td><td>55.41</td><td>56.30 54.61</td><td>50.90</td><td>53.46</td></tr><tr><td></td><td>WRN-34-12</td><td>MART</td><td>66.46</td><td>85.98</td><td>82.62</td><td>66.85</td><td>67.00</td><td>54.30</td><td></td><td>49.58</td><td>52.20</td><td>52.29</td><td></td><td>47.68</td><td>51.21</td></tr><tr><td>CIFAR-10</td><td>WRN-34-R</td><td>MART</td><td>68.12</td><td>86.09</td><td>83.69</td><td>68.79</td><td>68.18</td><td>56.31</td><td>59.13</td><td>51.40</td><td>53.22</td><td>54.20</td><td>55.44</td><td>49.90</td><td>52.48</td></tr><tr><td>+500K</td><td>WRN-34-12</td><td>RST</td><td>66.46</td><td>90.52</td><td>90.36</td><td>76.01</td><td>76.02</td><td>65.52</td><td>65.56</td><td>61.67</td><td>61.70</td><td>64.30</td><td>64.26</td><td>60.90</td><td>60.96</td></tr><tr><td rowspan="2"></td><td>WRN-34-R</td><td>RST</td><td>68.12</td><td>90.73</td><td>90.56</td><td>76.51</td><td>76.44</td><td>66.46</td><td>66.51</td><td>62.38</td><td>62.49</td><td>65.12</td><td>65.10</td><td>61.49</td><td>61.56</td></tr><tr><td>WRN-34-12</td><td>SAT</td><td>66.53</td><td>59.63</td><td>60.64</td><td>33.67</td><td>37.28</td><td>24.50</td><td>27.61</td><td>23.38</td><td>24.95</td><td>43.78</td><td>42.02</td><td>22.27</td><td>24.42</td></tr><tr><td rowspan="6">CIFAR-100</td><td>WRN-34-R</td><td>SAT</td><td>68.16</td><td>61.17</td><td>61.33</td><td>35.00</td><td>38.72</td><td>25.03</td><td>29.02</td><td>23.38</td><td>25.70</td><td>43.52</td><td>41.46</td><td>22.72</td><td>25.20</td></tr><tr><td>WRN-34-12</td><td>TRADES</td><td>66.53</td><td>55.62</td><td>56.47</td><td>35.35</td><td>36.90</td><td>27.52</td><td>29.48</td><td>24.94</td><td>25.21</td><td>44.75</td><td>46.19</td><td>24.58</td><td>24.85</td></tr><tr><td>WRN-34-R</td><td>TRADES</td><td>68.16</td><td>56.83</td><td>56.75</td><td>36.95</td><td>37.68</td><td>29.17</td><td>29.92</td><td>25.48</td><td>25.48</td><td>45.04</td><td>45.52</td><td>25.13</td><td>25.23</td></tr><tr><td>WRN-34-12</td><td>MART</td><td>66.53</td><td>58.51</td><td>57.29</td><td>36.06</td><td>39.48</td><td>26.50</td><td>32.43</td><td>23.85</td><td>27.64</td><td>41.53</td><td>38.73</td><td>23.33</td><td>26.92</td></tr><tr><td>WRN-34-R</td><td>MART</td><td>68.16</td><td>61.72</td><td>58.27</td><td>39.68</td><td>41.24</td><td>29.94</td><td>34.12</td><td>26.27</td><td>29.33</td><td>39.20</td><td>38.45</td><td>25.60</td><td>28.63</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+White-box Robustness. We evaluate the robustness of the networks to 5 adversarial attacks including Fast Gradient Sign Method (FGSM) [5], Projected Gradient Descent (PGD) [15], Carlini and Wagner (CW) [79], Guided Adversarial Margin Attack (GAMA) [33] and AutoAttack (AA) [30]. We apply these attacks on the test sets of CIFAR-10 and CIFAR-100 with the same maximum adversarial perturbation $\epsilon = 8 / 2 5 5$ as adopted for model training. For PGD, we use the 20-step PGD $( \mathrm { P G D ^ { 2 0 } } )$ with step size $\alpha = \epsilon / 1 0$ . For GAMA attack, we set its perturbation steps to 100 following the original paper. We report the model’s accuracy on the test adversarial examples crafted by these evaluation attacks for models obtained at both the best and the last checkpoints, following [16, 46, 18].
+
+The white-box evaluation results including both clean accuracy and adversarial robustness are reported in Table 2. As can be inferred, a robustness gain of $1 \sim 3 \%$ can be consistently achieved when the networks are reconfigured following our discovered configuration rule. And the improvements are not restricted to a particular architecture nor adversarial training method, except there is slight decrease for $\mathrm { C W } _ { \infty }$ on CIFAR-100 for SAT and MART. This wide range of robustness improvements by a simple architectural reconfiguration confirms that our findings are very general and can be immediately applied to commonly used DNNs to obtain more adversarial robustness. For VGG-11, DN-121 and DARTS, our robust reconfiguration can reduce the parameters by a considerable amount.
+
+# 6 Relation to Existing Understandings
+
+One recent work [59] shows that wider networks tend to increase the Lipschitzness, a finding that is consistent with ours. In [59], the theoretical analysis is based on Neural Tangent Kernel (NTK) under the assumption that all layers share the same width. By contrast, our analysis provides a more in-depth understanding related to the width and depth of each individual layer. Whilst in [59], the wider network utilizes a stronger regularization to mitigate the vulnerability (instability) caused by increased Lipschitzness, our work shows that this can be achieved alternatively by a simple reconfiguration of the architecture.
+
+It has also been found that weight decay plays an important role in adversarial training [54, 80]. This can be explained by our theoretical analysis in Theorem 1 and 2. Considering the weight matrix is normally distributed $\mathcal { N } ( 0 , \sigma _ { \theta } ^ { 2 } )$ , weight decay encourages the model to learn weights of smaller magnitudes, thus reducing the variance $\sigma _ { \theta } ^ { 2 }$ of the weight matrix. This will lead to reduced upper bound of the Lipschitz constant and improved robustness. See Appendix E for more discussions.
+
+# 7 Conclusion
+
+In this paper, we explored the architectural ingredients of adversarially robust DNNs via extensive fine-controlled experiments and theoretical analysis. Our findings are: 1) more parameters does not necessarily lead to more adversarially robust models; 2) reducing capacity (up to a limit) via either depth or width at the deeper layers improves adversarial robustness; and 3) under the same type of architectures and parameter budget, there may exist an architectural configuration that can exploit the full robustness potential of the network. We also showed that depth and width offer different levels of flexibility for capacity reduction and robustness improvement. Following a width reduction and scaling rule, we showed that our findings are generic, not restricted to a particular adversarial training method, and can be immediately applied to improve both manually-designed or NAS-discovered DNNs. We also provide a series of empirical understandings on the distinctive impacts of the deeper layers on adversarial robustness. Our work can provide useful insights into the architectural perspective of adversarial robustness, and help design more adversarially robust DNNs.
+
+# Border Impacts
+
+Adversarial training is currently the most effective defense against adversarial attacks, although its performance is yet to be improved. In this work, we extensively studied the impact of network architecture to adversarial robustness. Our findings suggest that models can be made more robust by even reducing capacity at the deep layers. Such reduction can also help save the training cost of adversarial training which is known to be extremely time-consuming. We will open source the discovered architectural configurations to help future research design more robust architectures.
+
+# Acknowledgment
+
+Yisen Wang is partially supported by the National Natural Science Foundation of China under Grant 62006153, and Project 2020BD006 supported by PKU-Baidu Fund. This research was undertaken using the LIEF HPC-GPGPU Facility hosted at The University of Melbourne. This Facility was established with the assistance of LIEF Grant LE170100200.
+
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+[82] Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In ICML, 2015.   
+[83] Linxi Jiang, Xingjun Ma, Zejia Weng, James Bailey, and Yu-Gang Jiang. Imbalanced gradients: A new cause of overestimated adversarial robustness. arXiv preprint arXiv:2006.13726, 2020.   
+[84] Vikash Sehwag, Shiqi Wang, Prateek Mittal, and Suman Jana. Hydra: Pruning adversarially robust neural networks. NeurIPS, 2020.   
+[85] Pavel Izmailov, Dmitrii Podoprikhin, Timur Garipov, Dmitry Vetrov, and Andrew Gordon Wilson. Averaging weights leads to wider optima and better generalization. In UAI, 2018.
+
+# Checklist
+
+The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example:
+
+• Did you include the license to the code and datasets? [Yes]   
+• Did you include the license to the code and datasets? [No] The code and the data are proprietary.   
+• Did you include the license to the code and datasets? [N/A]
+
+Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below.
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] We briefly discussed the negative impacts of adversarial attacks in the introduction and border impacts.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] See Appendix A.1
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Code available in supplementary material   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix B
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]
+
+(b) Did you mention the license of the assets? [Yes]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/OdklztJBBYH/OdklztJBBYH_content_list.json

@@ -0,0 +1,1252 @@
+[
+    {
+        "type": "text",
+        "text": "Exploring Architectural Ingredients of Adversarially Robust Deep Neural Networks ",
+        "text_level": 1,
+        "bbox": [
+            179,
+            122,
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+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Hanxun Huang1 Yisen Wang2,3 Sarah Erfani1 Quanquan $\\mathbf { G u ^ { 4 } }$ James Bailey1 Xingjun $\\mathbf { M } \\mathbf { a } ^ { \\mathsf { \\pm } }$ ",
+        "bbox": [
+            295,
+            224,
+            738,
+            255
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1School of Computing and Information Systems, The University of Melbourne, Victoria, Australia   \n2Key Lab. of Machine Perception, School of Artificial Intelligence, Peking University, Beijing, China 3Institute for Artificial Intelligence, Peking University, Beijing, China 4University of California, Los Angeles, USA 5School of Computer Science, Fudan University, Shanghai, China ",
+        "bbox": [
+            184,
+            256,
+            848,
+            329
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            364,
+            535,
+            381
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep neural networks (DNNs) are known to be vulnerable to adversarial attacks. A range of defense methods have been proposed to train adversarially robust DNNs, among which adversarial training has demonstrated promising results. However, despite preliminary understandings developed for adversarial training, it is still not clear, from the architectural perspective, what configurations can lead to more robust DNNs. In this paper, we address this gap via a comprehensive investigation on the impact of network width and depth on the robustness of adversarially trained DNNs. Specifically, we make the following key observations: 1) more parameters (higher model capacity) does not necessarily help adversarial robustness; 2) reducing capacity at the last stage (the last group of blocks) of the network can actually improve adversarial robustness; and 3) under the same parameter budget, there exists an optimal architectural configuration for adversarial robustness. We also provide a theoretical analysis explaning why such network configuration can help robustness. These architectural insights can help design adversarially robust DNNs. Code is available at https://github.com/HanxunH/RobustWRN. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            628,
+            310,
+            646
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep neural networks (DNNs) are becoming standard models for many real-world applications such as image classification [1], object detection [2] and natural language processing [3]. However, a line of research has shown that DNNs are vulnerable to adversarial examples (attacks), which can be easily crafted by slightly perturbing the input instance to maximize the model’s prediction error [4–6]. This vulnerability of DNNs has become a major concern for their deployment in security-critical applications such as autonomous driving [7, 8] and medical diagnosis [9, 10]. ",
+        "bbox": [
+            174,
+            661,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A number of defense methods have been proposed to train adversarially robust DNNs [11–14], among which adversarial training has demonstrated the most promising results [15–17]. Adversarial training can be viewed as a type of data augmentation that trains DNNs on adversarial (instead of natural) examples [15–19]. Based on adversarial training, a set of works have been proposed to understand its learning and convergence behaviors, and the key factors for training adversarially robust DNNs. For example, it has been found that adversarial training encourages the model to learn more robust or compact features [20, 21], and it requires more data [22–25] or higher capacity models to gain more robustness [15, 26]. While these understandings have motivated several improved defense methods, it is still not clear, from an architectural perspective, what makes an adversarially robust DNN. ",
+        "bbox": [
+            174,
+            750,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present the first comprehensive investigation on the architectural ingredients of adversarially robust DNNs. Our investigation is based on adversarial training and WideResNet-34-10 (WRN-34-10) [27], one extensively tested architecture in the defense literature. Based on the base architectural configuration of WRN-34-10, we apply a finely-controlled grid search to explore the impact of network width and depth configurations on the robustness of adversarial trained DNNs. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The standard WRN-34-10 consists of 3 stages with each stage being a group of 5 (i.e., depth) residual blocks and each residual block having 2 convolutional layers. We denote the three stages as Stage-1, Stage-2 and Stage-3 following the direction from the input to the output. Each stage is configured by a depth (number of residual blocks) and a width (number of filters) factor. The hyper-parameters for width and depth of each stage control the scale of learnable parameters (capacity). In this paper, we explore different configurations of width and depth for each of the three stages. Based on our explorations, we make the following key observations: ",
+        "bbox": [
+            174,
+            166,
+            826,
+            265
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Simply increasing the number of parameters (model capacity) by upscaling width or depth does not necessarily lead to improved robustness. This contrasts with current beliefs that, under the same type of architecture, more parameters (higher model capacity) can improve adversarial robustness [15, 26, 28]. Adversarial training does require larger capacity models, but there exists a trade-off. We provide both theoretical and empirical evidences that wider/deeper models increase Lipschitzness (larger Lipschitz constant). • For a larger model used in adversarial training, reducing capacity at the last stage (Stage-3) of WRNs can achieve a better trade-off between capacity and Lipschitzness, thus improving adversarial robustness. This can be achieved by reducing either depth or width, with width reduction being slightly more effective. This highlights that smaller DNNs can also have better robustness if the parameter reduction is applied at the right place (i.e., the last stage). • Under the same type of architectures (i.e., WRNs) and parameter budget, there may exist an optimal architectural configuration that can produce the most robust DNN. We show that the same configuration rule can also be applied to improve the robustness of VGGs, DenseNets (DNs), as well as networks found by Differentiable Architecture Search (DARTS). ",
+        "bbox": [
+            217,
+            276,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, we provide a series of understandings for the above findings, which can not only provide useful insights for training more robust models with adversarial training, but also shed new light on the architectural ingredients of adversarially robust DNNs. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            566,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 Adversarial Training ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            598,
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+            613
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Adversarial training has been demonstrated to be the most reliable training method for obtaining adversarially robust DNNs [29, 30]. The standard adversarial training (SAT) can be formulated as a min-max optimization framework as follows: ",
+        "bbox": [
+            174,
+            623,
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+            665
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c268ca84cceff4a5a43b97169a55c12ca36027ae7cad85deb07cc7b2f6b340e7.jpg",
+        "text": "$$\n\\underset { \\pmb { \\theta } } { \\arg \\operatorname* { m i n } } \\mathbb { E } _ { ( \\pmb { x } , \\pmb { y } ) \\sim \\mathbb { D } } \\left[ \\underset { \\pmb { x } ^ { \\prime } } { \\operatorname* { m a x } } \\mathcal { L } ( f _ { \\pmb { \\theta } } , \\pmb { x } ^ { \\prime } , \\pmb { y } ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            671,
+            622,
+            702
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where the inner maximization generates adversarial examples $\\mathbf { x } ^ { \\prime }$ , the outer minimization trains the model on $\\mathbf { x } ^ { \\prime }$ , $f _ { \\theta }$ denote the neural network and $\\mathcal { L } ( \\cdot )$ is the cross entropy (CE) loss. During the inner maximization process, SAT uses PGD to generate adversarial examples [15]: ",
+        "bbox": [
+            174,
+            708,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7242574952d789cec3f58929e406450540823e08153b0ea429e33ae0d0ac3f22.jpg",
+        "text": "$$\n\\pmb { x } _ { k } ^ { \\prime } = \\Pi _ { \\epsilon } ( \\pmb { x } _ { k - 1 } ^ { \\prime } + \\alpha \\cdot \\mathrm { s i g n } ( \\nabla _ { \\pmb { x } } \\mathcal { L } ( f _ { \\pmb { \\theta } } , \\pmb { x } _ { k - 1 } ^ { \\prime } , y ) ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            756,
+            656,
+            775
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathrm { s i g n } ( \\cdot )$ is the sign function, $ { \\boldsymbol { { x } } } _ { k } ^ { \\prime }$ is the adversarial example obtained at the $k$ -th (for overall $K$ steps) perturbation step, $\\alpha$ is the step size, and $\\Pi _ { \\epsilon }$ is a projection (clipping) operation that projects the perturbation back onto the $\\epsilon$ -ball centered around $_ { \\textbf { \\em x } }$ if it goes beyond. ",
+        "bbox": [
+            176,
+            779,
+            825,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Improved variants of SAT have also been proposed, such as the trade-off between adversarial robustness and natural accuracy (TRADES) [16], Dynamic AdveRsarial Training (DART) [17], Friendly Adversarial Training (FAT) [31], Misclassification Aware adveRsarial Training (MART) [18], Robust Self-Training (RST) [23], Unsupervised Adversarial Training (UAT) [32], Guided Adversarial Training (GAT) [33], Max-Margin AT [34], using Max-Mahalanobis Center (MMC) loss [35], accelerated AT [36–38], using pre-training [39], incorporating hypersphere embedding [40], self-progressing robust training [41], Adversarial Weight Perturbation (AWP) [19], Adversarial Distributional Training (ADT) [42], Channel-wise Activation Suppressing (CAS) [21], GeometryAware Instance-Reweighted Adversarial Training (GAIRAT) [43] and robustness distillation [44, 45]. Adversarial Training has also been found to cause robust overfitting [46], but it can be mitigated by smoothing techniques [47]. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            90,
+            826,
+            161
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Understanding Adversarially Trained DNNs ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            176,
+            521,
+            191
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Understanding the working mechanism of adversarial training has been a hot research area. For example, it has been found that adversarial training encourages the model to learn more robust features [20, 48], have good generative ability [49, 50], improve the model’s transferability to downstream tasks [51, 52] and improve performance on clean data [53]. It has also been found that using auxiliary training data with adversarial training can further improve adversarial robustness [22, 23], and that weight decay plays an important role in adversarial training [54]. Another important observation is that using WRNs instead of ResNets (RNs) can bring $\\sim 3 \\% { - } 5 \\%$ more robustness [16–18]. Other works also suggest that adversarial training requires deeper and wider models [15, 26, 55]. Also, the skip-connection operation used in WRN has been found can improve robustness for deeper architectures [56] and there exists a trade-off between depth and width for approximating natural functions [57]. On the other hand, there are also works showing that increasing the number of parameters for the same type of DNN architectures can only lead to limited robustness improvement [28, 58]; and wider networks may cause more perturbation instability [59]. ",
+        "bbox": [
+            174,
+            200,
+            825,
+            381
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Several recent works have applied neural architecture search (NAS) to search for more robust DNN architectures [60]. They found that, 1) densely connected cells result in improved robustness; and 2) under certain computational budget, adding convolution operations to direct connection edge is effective. Other works improve the NAS search strategy by searching on targeted capacity [61], maximizing certified lower bound [62], using the log-normal distribution to approximate the Lipschitz constant [63], using lower and upper confidence bounds in Bandit [64], or using perturbation-based regularization [65]. Another study on hand-crafted versus NAS-based architectures shows that, without adversarial training, NAS-based architectures are more robust for small-scale datasets and simple tasks than hand-crafted architectures, however, hand-crafted architectures are more robust than NAS-based architectures as the dataset size or the task complexity increases [66]. Note that NAS is extremely time-consuming, especially when applied with adversarial training. Previous works using NAS find optimal topological connections within the cell structure [60], but did not investigate depth/width configurations, which arguably has more impact on robustness (e.g., RNs vs. WRNs). In this work, we focus on fine-grained configuration exploration rather than blind search, which can produce more precise understandings of how depth and width affect robustness. ",
+        "bbox": [
+            173,
+            387,
+            825,
+            594
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Wider and Deeper Models Increase Lipschitz Upper Bound ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            613,
+            699,
+            631
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "It has been theoretically shown that high Lipschitzness (larger Lipschitz constant) corresponds to low stability of the model’s output to input perturbations [59]. However, adversarial training does require a larger capacity model (e.g., RN vs. WRN) [15], an empirical finding that goes against the theoretical expectation. In this section, we first theoretically show a trade-off between network capacity (width/depth) and the Lipschitz upper bound. In Section 4, we will empirically examine this trade-off and its relation to the improved adversarial robustness for larger capacity models. ",
+        "bbox": [
+            174,
+            643,
+            825,
+            727
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The Lipschitz constant $L$ of a DNN measures the maximum rate of change in the output with the change in the input, and is closely related to adversarial robustness [4]. Formally, it is $\\| f _ { \\pmb \\theta } ( \\pmb x ) - \\bar { f } _ { \\pmb \\theta } ( \\pmb x ^ { \\prime } ) \\| \\le \\bar { L } \\| \\pmb x - \\pmb x ^ { \\prime } \\|$ . ",
+        "bbox": [
+            176,
+            733,
+            825,
+            776
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 (Lipschitz Constant Upper Bound of a Neural Network with Gaussian Distributed Weights). Consider an $n$ layer DNN $f$ , where the weight parameters $\\pmb \\theta$ are independent Gaussian random variables distributed as $\\mathcal { N } ( 0 , \\sigma _ { \\theta } ^ { 2 } )$ with $\\boldsymbol { \\sigma } _ { \\theta } ^ { 2 }$ denoting the variance of the Gaussian distribution, and where the activation functions are $I$ -Lipschitz. The expected Lipschitz constant of a DNN with hidden layer size $h$ is upper bounded by: ",
+        "bbox": [
+            174,
+            776,
+            825,
+            843
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/91f5aca5b3174b10cae1de0e5a0fd40cd6aca6ed46d2f8b0777986d413f519cd.jpg",
+        "text": "$$\nL ( f _ { \\pmb \\theta } ) \\leq \\prod _ { j = 1 } ^ { n } \\left( \\sqrt { h _ { j - 1 } } + \\sqrt { h _ { j } } \\right) \\cdot \\sigma _ { \\pmb \\theta _ { j } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            844,
+            611,
+            882
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. For a convolutional neural network $f$ , each layer’s convolution operation with feature map size $W \\times m \\times m$ and kernel size $k \\times k$ , where the weight parameters $\\pmb { \\theta }$ are independent Gaussian ",
+        "bbox": [
+            173,
+            883,
+            821,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "random variables distributed as $\\mathcal { N } ( 0 , \\sigma _ { \\theta } ^ { 2 } )$ , the expected Lipschitz constant is upper bounded by: ",
+        "bbox": [
+            171,
+            90,
+            802,
+            107
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9c6f4d0c76fd5a335d1cc1d2e28c982bdc69bc9a6b1d016a7793a37cf76f2809.jpg",
+        "text": "$$\nL ( f _ { \\pmb \\theta } ) \\leq \\prod _ { j = 1 } ^ { n } ( m _ { j } \\sqrt { W _ { j - 1 } } + ( m _ { j } - k _ { j } + 1 ) \\sqrt { W _ { j } } ) \\cdot \\sigma _ { { \\pmb \\theta } _ { j } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            109,
+            668,
+            148
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The proof for Theorem 1 and 2 is inspired by [67–69] and can be found in Appendix A.1. This establishes a connection between the upper bound on the Lipschitz constant of a feed-forward DNN with $n$ layers and width of $h _ { j }$ for each layer. For the $j$ -th layer of convolution operations, the Lipschitz constant upper bound increases with its input dimension $( W _ { j - 1 } \\times m _ { j } \\times m _ { j } )$ and the number of output channels $W _ { j }$ . More simply, it is upper-bounded by the variance of the weight matrix and the input representation’s dimension plus the output representation’s dimension. For the entire network, the Lipschitz constant upper bound grows exponentially with the depth. This suggests that wider and deeper models have a relatively larger change of the output due to the changes in the input, i.e., lower adversarial robustness. ",
+        "bbox": [
+            173,
+            156,
+            825,
+            281
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Several works attempt to regularize the network’s Lipschitz constant by using Parseval tight frames on the weight matrixes [70], enforcing constraints on the singular values of the weight matrixes [71], or via a Lipschitz-margin training [72]. However, a follow-up work points out that there exist both experimental and theoretical limitations for the above approaches [73]. Whilst the Lipschitz constant may not be used as a regularization, it has been widely adopted for analyzing the stability and adversarially robustness of DNNs [4, 59, 74]. ",
+        "bbox": [
+            173,
+            287,
+            825,
+            372
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 Exploring Adversarially Robust Architectures ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            390,
+            591,
+            407
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our exploration of the relationship between DNN architectural configuration, Lipschitzness (size of the Lipschitz constant) and adversarial robustnessare starts with a fine-controlled grid search on the width/depth of the WideResNet (WRN) [27]. In Sections 4.1 and 4.2, we show our exploration results with depth and width, respectively. Based on these results, a pattern of robust depth/width configuration is discovered. In Section 4.4, we examine a linear scaling effect with the discovered robust configuration. In Section 4.5, we provide an analysis on the trade-off between model capacity and Lipschitzness, and the key factors contributing to improved adversarial robustness. ",
+        "bbox": [
+            173,
+            420,
+            825,
+            518
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/2f9baf00f136e5769fb34335b7fe2d63f3c30fcb5a6832f987221d1bd62d1435.jpg",
+        "image_caption": [
+            "Figure 1: (a): Illustration of WRN-34-10 denoted as $^ { \\mathrm { d } } 5  – 5 – 5$ . (b): Grid search results on different depth configurations. The three-digit numbers highlight the depth configurations of only those networks that have either a low $\\mathrm { < = 5 0 . 0 \\% }$ ) or a high $> = 5 2 . 5 \\%$ ) adversarial robustness (against $\\mathrm { P G D ^ { 2 0 } }$ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            222,
+            531,
+            774,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Base architecture. We take the standard WRN-34-10 designed for CIFAR-10 as our base architecture. Figure 1a provides an overview of the architecture and the detailed configurations are summarized in Appendix Table 3. The standard WRN architecture consists of 3 stages (groups) of residual blocks and 4 fixed convolutional layers. Here, we focus on the configuration of the 3 stages, which are the key components of the network. We denote the depth and width configuration for the $i$ -th $( i \\in \\{ 1 , 2 , 3 \\} )$ ) stage as $D _ { i }$ and $W _ { i }$ , respectively. For standard WRN-34-10, $D _ { 1 / 2 / 3 } = 5$ (denoted as $^ { \\mathrm { d } } 5  – 5 – 5 )$ and $W _ { 1 / 2 / 3 } = 1 0$ (denoted as $^ { \\mathrm { w } } 1 0 \\mathrm { - } 1 0 \\mathrm { - } 1 0 $ ). For the rest of this paper, we use $^ { \\mathrm { d } } D _ { 1 } { - } D _ { 2 } { - } D _ { 3 }$ and $^ { \\mathrm { w } } W _ { 1 ^ { - } } W _ { 2 ^ { - } } W _ { 3 }$ to represent the exact width and depth configurations. We explore the stage-wise depth and width configurations while keeping other configurations unchanged. ",
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+        "type": "text",
+        "text": "Experimental settings. We train all explored networks on CIFAR-10 dataset [1] using the standard adversarial training (SAT) with Projected Gradient Descent (PGD) [15] (see definition in equation (2)). Following the typical adversarial training setting, we constrain the $L _ { \\infty }$ -norm of the maximum adversarial perturbation to $\\epsilon = 8 / 2 5 5$ , and use 10-step PGD $( { \\mathrm { P G D } } ^ { 1 0 } )$ with step size $\\alpha = 2 / 2 5 5$ . After training, we test the robustness of the network on PGD adversarial examples crafted on the entire test set of CIFAR-10, under the same perturbation constraint $\\epsilon = 8 / 2 5 5$ . For evaluation, we use the 20-step PGD $( \\mathrm { P G D ^ { 2 0 } } )$ with step size $\\bar { \\alpha } = \\epsilon / 1 0$ . The robustness is measured by the network’s accuracy on the $\\mathrm { P G D ^ { 2 0 } }$ test adversarial examples. More details can be found in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "4.1 Exploring Different Depths ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first explore different depth configurations based on the base WRN-34-10 architecture introduced above. For each of the three stages (e.g., Stage-1, Stage-2, Stage-3), we explore different depth $D _ { i } \\in \\{ 1 , 3 , 5 , 7 , 9 \\}$ . Since each stage has 5 possible depth configurations, the total number of all possible depth configuration for all 3 stages are 125 (5x5x5, permutation with replacement). We first perform a grid search on all the 125 depth configurations, then take a closer look at the impact at each individual stage. The adversarial robustness of the 125 networks (adversarially trained using SAT) against $\\mathrm { P G D ^ { 2 0 } }$ test adversarial examples is plotted in Figure 1b. Note that the depth configuration of standard WRN-34-10 is $^ { \\mathrm { d } } 5  – 5 – 5$ . By investigating the robustness scores along the $\\mathbf { X }$ -axis (number of parameters), we find that more parameters does not necessarily lead to improved robustness. For example, the networks with more than 80M (million) parameters are even less robust than some of those with only 20M parameters. Given the same level of parameters, for example ${ \\sim } 2 0 \\mathbf { M }$ , different depth configurations can lead to $\\sim 6 \\%$ difference in robustness. This implies that, under the same parameter budget, there may exist an optimal depth configuration for adversarial robustness. ",
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+    {
+        "type": "text",
+        "text": "Next, we take a closer look at the above grid search result and investigate the common characteristics of the top-5 most robust networks, the details of which are reported in Figure 2d. Interestingly, we find that the top-5 networks all have a significantly reduced depth of 1 or 3 at the last stage (i.e., Stage-3). This trend indicates that reducing model capacity at the last (deepest) stage can actually improve robustness. The other observation is that, having more residual blocks (higher depth) at the two shallow stages (i.e., Stage-1/2) can also improve robustness. For example, the top-2 networks have 9 residual blocks at Stage-1, and all top-4 networks have 9 or at least 7 residual blocks at Stage-2. This suggests that capacity is more important for the shallow layers. We conjecture this is because the network still needs sufficient capacity to learn the augmented examples by adversarial training. Note that the best performing model $- 9 - 7 - 1$ only uses half of the parameters of the standard WRN-34-10 (d5-5-5), which is only ranked the 45-th out of all 125 models. ",
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+        "img_path": "images/591388548c726a1145488d9a8b95e16b1587118fa8538106e5fb4c9b6f0b9ecc.jpg",
+        "image_caption": [
+            "Figure 2: (a-c) The impact of depth on adversarial robustness at different stages. When studying one stage, the depths of other two stages are fixed to 5. (d) Clean accuracy and adversarial robustness of the top-5 most robust depth configurations discovered in the grid search. All networks are trained using SAT [15] on CIFAR-10. Robustness evaluated using $\\mathrm { P G D ^ { 2 0 } }$ . d5-5-5 is the depth configuration of the baseline WRN-34-10 model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We further explore the distinctive impacts of depth on adversarial robustness at different stages via a control study. Specifically, we add or remove residual blocks from each individual stage of WRN-34-10 (d5-5-5) while keeping the other two stages fixed to depth 5. The robustness results are shown in Figure 2a-2c. As can be observed, reducing depth at the first two stages constantly degrades the robustness, however, it is the other way around at the last stage (i.e., Stage-3). In relation to previous understanding that higher model capacity can lead to more robust models [28, 58], our finding indicates that it is true for the shallow layers but quite the opposite for the deeper layers. In other words, more parameters can improve adversarial robustness only when added to the shallow layers (e.g., layers in Stage-1 and Stage-2). ",
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+    {
+        "type": "text",
+        "text": "4.2 Exploring Different Widths ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We further explore whether width also has a similar effect as depth. The standard WRN-34-10 has a width upscaling factor 10 applied to each stage, that is, $^ { \\mathrm { w } } 1 0 \\mathrm { - } 1 0 \\mathrm { - } 1 0$ . Based on our above findings with the depth in Section 4.1, here we skip the grid search and directly investigate the impact of width at different stages. At each stage, we investigate different width configurations $W _ { i } \\in \\{ 2 , 4 , 6 , 8 , 1 0 \\}$ for $i = { 1 , 2 , 3 }$ . ",
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+        "image_caption": [
+            "Figure 3: (a-c): The impact of width on adversarial robustness at different stages. When studying one stage, the widths of other two stages are fixed to 10. (d): Clean accuracy and adversarial robustness of the networks obtained by reducing width in the last stage (i.e., Stage-3). All networks are trained using SAT [15] on CIFAR-10. Robustness is evaluated using $\\mathrm { P G D ^ { 2 0 } }$ . w10-10-10 is the depth configuration of the baseline WRN-34-10 model. "
+        ],
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+        "type": "text",
+        "text": "The robustness results are illustrated in Figure 3. We find that width reduction generally has a similar effect as depth reduction: reducing width at the first two stages harms robustness until $W _ { \\frac { 1 } { 2 } / 2 } = 4$ , however, the same operation can improve robustness when applied to the last stage. This confirms the importance of high capacity at the shallow layers and low capacity at the deeper layers. Compared to depth reduction, we find that, with the same amount of robustness improvement, width reduction (at the last stage) can lead to smaller models. For example, $^ { \\mathrm { w } } 1 0 ^ { . } 1 0 ^ { . } 4$ (the second row in Figure 3d) achieves a similar robustness $( \\sim 5 4 \\% )$ as $^ { \\mathrm { d } } 9 . 7 . 1$ (the first row in Figure 2d). However, the number of parameters of the $^ { \\mathrm { w } } 1 0 ^ { \\phantom { - } } 1 0 ^ { \\phantom { - } } $ configuration is only 17.05M, which is much less than the 22.19M of the d9-7-1 configuration. ",
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+    {
+        "type": "text",
+        "text": "Another interesting observation is that adversarial robustness does not change much if we reduce the width from $W _ { 1 / 2 } = 4$ to $W _ { 1 / 2 } = 2$ at Stage-1 or Stage-2, whereas the same reduction at Stage-3 hurts robustness. This is somewhat expected since, on one hand, the robustness might not be affected much unless a sufficient number of filters (channels) are removed, which is different to depth that configures the entire residual block. On the other hand, if too many filters are removed at the last stage, the network may lose the capacity required for proper learning, while in our depth exploration, there exists at least one residual block $D _ { 3 } \\geq 1 \\underline { { \\cdot } }$ ) at the last stage. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.3 Exploring Depth-Width Combinations ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Although reducing capacity at the last stage via either depth or width can improve robustness, there exists a limit. For example, if we reduce depth and width at the same time or too much of the width, the network may end up with insufficient capacity for proper learning. We first explore an extreme case that removes the entire Stage-3 as $\\mathrm { d } 5 . { \\dot { 5 } } { - } 0$ . This ends up with $2 \\%$ less robustness than baseline WRN-34-10. This result verifies the necessity of Stage-3. We then reduce depth and width simultaneously by setting the depth to $^ { \\mathrm { d } } 5  – 5 – 1$ and width to $^ { \\mathrm { w } } 1 0 – 1 0 – 2$ . This produces a new network with a similar robustness $( \\sim 5 2 \\% )$ ) to $\\mathrm { P G D ^ { 2 0 } }$ as WRN-34-10. Note that, in this case, comparing to WRN-34-10, the number of parameters has been reduced by $70 \\%$ . We then explore all the 25 possible depth-width combinations between the top-5 depth and width configurations in Figure 2d and 3d, respectively. Surprisingly, we find that none of these models can achieve better robustness than simply reducing the width to $^ { \\mathrm { w } } 1 0 ^ { . } 1 0 ^ { . } 4$ . These models achieved the same level of robustness $( \\sim 5 4 \\% )$ ), but require more computations (FLOPS). For instance, the network with depth $^ { \\mathrm { d } } 7 . 9 . 3$ and width $^ { \\mathrm { w } } 1 0 { - } 1 0 { - } 4$ requires 2 times more FLOPS than WRN-34-10. This does not benefit adversarial training since it is known to be time-consuming. Although a more fine-grained (with decimals) exploration of the width configuration may lead to even more robust WRN models, here we simply take the $^ { \\mathrm { w } } 1 0 { - } 1 0 { - } 4$ configuration as our choice of the optimally-reduced WRN. ",
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+    {
+        "type": "text",
+        "text": "4.4 Scaling with the Discovered Configuration ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Previous works [15, 16] have shown that a wider network like WRN-34-10 can be trained to be more robust than a standard ResNet like RN-34. Here, we investigate if we can obtain more robust models by scaling the discovered width configuration $^ { \\mathrm { w } } 1 0 ^ { \\phantom { - } } 1 0 ^ { \\phantom { - } } 4$ . We test different scaling ratios $\\gamma \\in [ 0 . 2 5 , 2 . 0 ]$ , and show the robustness results in Table 1. Compared to $^ { \\mathrm { w } } 1 0 ^ { \\phantom { - } } 1 0 ^ { \\phantom { - } } $ $( \\gamma = 1 . 0 )$ ), scaling down $\\gamma$ to 0.5 or 0.25 decreases the robustness while scaling up $\\gamma$ can further improve the robustness, although the improvement become less significant when $\\gamma$ goes above 1.5. Note that the network with $\\gamma = 0 . 5$ has 10 times fewer parameters than the baseline WRN-34-10, but can already achieve a better robustness against $\\mathrm { P G D ^ { 2 0 } }$ . The best robustness is achieved at $\\gamma = 2 . 0$ , i.e., $^ { \\mathrm { w } } 2 0 – 2 0 – 8$ . We denote the corresponding WRN-34 network as WRN-34-R, more details can be found in Appendix Table 3. In Section 5, we will apply the $^ { \\mathrm { w } } 1 0 ^ { \\phantom { - } } 1 0 ^ { \\phantom { - } } 4$ configuration rule to more network architectures and evaluate their (along with WRN-34-R) adversarial robustness more systematically. ",
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+    {
+        "type": "table",
+        "img_path": "images/590c848726be0c34f34a0937d0e29df06c5636019ff6290a6ed5f9401c796ce6.jpg",
+        "table_caption": [
+            "Table 1: Clean accuracy and adversarial robustness for scaled $^ { \\mathrm { w } } 1 0 – 1 0 – 4$ configurations. All networks are trained using SAT [15] on CIFAR-10 and evaluated using $\\mathrm { P G D ^ { 2 0 } }$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Scaling Ratio</td><td>Params (M)</td><td>Clean (%)</td><td>PGD20 (%)</td></tr><tr><td>γ= 0.25</td><td>1.07</td><td>80.61</td><td>50.90</td></tr><tr><td>γ= 0.5</td><td>4.27</td><td>84.31</td><td>54.07</td></tr><tr><td>γ=1.0</td><td>17.05</td><td>87.00</td><td>54.99</td></tr><tr><td>γ= 1.5</td><td>38.33</td><td>87.68</td><td>55.33</td></tr><tr><td>Y=2.0</td><td>68.12</td><td>88.12</td><td>55.35</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.5 Empirical Understanding ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We apply two closely related metrics, including Perturbation Stability [59] and Empirical Lipschitz [74] to explore the distinctive impact of the deeper layers to adversarial robustness. They measure the output stability of the neural network. ",
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+    {
+        "type": "text",
+        "text": "Perturbation Stability. Adversarial robustness is typically measured by the percentage of correctly classified adversarial examples, which can be further decomposed into the set of correct clean examples intersect with stable examples [59]. Correct clean examples refer to clean examples that can be correctly classified by the model $\\{ ( \\pmb { x } , y ) \\sim \\mathbb { D } , f _ { \\pmb { \\theta } } ( \\pmb { x } ) \\ \\stackrel { \\ } { = } \\ y \\}$ . Stable examples are defined as, $\\{ \\pmb { x } : \\forall \\pmb { x } ^ { \\prime } \\in \\mathcal { X } , f _ { \\pmb { \\theta } } ( \\pmb { x } ) = f _ { \\pmb { \\theta } } ( \\pmb { x } ^ { \\prime } ) \\}$ , where $\\mathcal { X }$ is the domain of the $\\epsilon$ -ball around $_ { \\textbf { \\em x } }$ . This perturbation stability measures the fraction of examples whose outputs cannot be adversarially perturbed. While many factors can affect the model’s performance on the correct clean examples such as the generalization capability of the model, the perturbation stability only measures if adversarial perturbations can change the prediction of the output. We apply this metric to understand the role of neural network architecture in adversarial robustness. More specifically we are interested to find out whether the improved robustness is a result of improved generalization, stability or both. ",
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+    {
+        "type": "text",
+        "text": "Empirical Lipschitz constant. The empirical Lipschitz constant is defined as [74]: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/85e8eee6745e75f1d447a7cd9f66d36dc87356e2cdf3f06abb5b0b5cb5aece6e.jpg",
+        "text": "$$\n\\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\operatorname* { m a x } _ { \\pmb { x } ^ { \\prime } \\in \\mathcal { X } } \\frac { \\| f _ { \\pmb { \\theta } } ( \\pmb { x } _ { i } ) - f _ { \\pmb { \\theta } } ( \\pmb { x } _ { i } ^ { \\prime } ) \\| _ { 1 } } { \\| \\pmb { x } _ { i } - \\pmb { x } _ { i } ^ { \\prime } \\| _ { \\infty } } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { X }$ is the domain of the $\\epsilon$ -ball around $_ { \\textbf { \\em x } }$ and $\\mathbf { x } ^ { \\prime }$ can be generated by an adversarial attack (i.e., $\\mathrm { P G D ^ { 2 0 } }$ ). A lower value of the empirical Lipschitz constant implies a smoother and more adversarially robust classifier. For our analysis, we measure the empirical Lipschitz constant of the functions represented by the output layers of different residual blocks or the entire network (logits output). e.g., for block-5, we measure the maximum rate of change in its representation output between clean $( { \\pmb x } )$ and adversarial $( { \\pmb x } ^ { \\prime } )$ inputs. ",
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+        "img_path": "images/6ac9f1800b40e21419d866a7203da0c6c55573dc35565fd9e790a026b8ad5f7c.jpg",
+        "image_caption": [
+            "Figure 4: The change of perturbation stability and empirical Lipschitz constant when (a) depth of Stage-3 is reduced, (b) width of Stage-3 is reduced, or (c) linear scaling with $\\gamma$ and $^ { \\mathrm { w } } 1 0 ^ { \\phantom { - } } 1 0 ^ { \\phantom { - } } 4$ . "
+        ],
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+        "type": "text",
+        "text": "The Trade-off Between Capacity and Lipschitzness. We compute the above two metrics on the test set of CIFAR-10 for models with different depth and width configurations explored in Section 4.1 and 4.2. The results are illustrated in Figure 4, where the empirical Lipschitz constant is computed for the entire network. We can observe that, when depth or width for the given network is reduced, the empirical Lipschitz constant is also reduced, and the perturbation stability improves. This is consistent with our theoretical analysis in the Lipschitz constant upper bound. As shown in Figure 4b and 4c, this observation is more obvious for the width reduction. There exists a trade-off between the network capacity and Lipschitzness. For example, with $\\gamma = 0 . 2 5$ scaling, the network achieves a much lower Lipschitzness and better stability, however, it also significantly reduces the clean accuracy. This indicates that adversarial training does require larger capacity models and a better trade-off can be achieved by balancing model capacity and Lipschitzness using proper architectural reduction. ",
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+        "image_caption": [
+            "(a) Reducing width at Stage-1. (b) Reducing width at Stage-2. (c) WRN-34-12 and WRN-34-R Figure 5: Empirical Lipschitz constant of the output layers of different residual blocks (bins 1-15) or the entire network (bin 16). All experiments are run on CIFAR-10. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Figure 5, we plotted the empirical Lipschitz constant of the output layer of each residual block or the entire network. The $f _ { \\pmb \\theta } ( \\pmb x )$ in equation (3) is replaced with the output of each residual block $f _ { \\pmb { \\theta } _ { j } } ( \\pmb { x } )$ (from input to block output), and the last (16-th) bin is the empirical Lipschitz constant of the entire network (from input to logits). From Figure 5, we find that: 1) within each stage, the empirical Lipschitz increases with depth; 2) when transitioning from one stage to the next, the spatial dimension decreases while the empirical Lipschitz decreases; 3) comparing WRN-34-12 with WRN-34-R (Figure 5c), the empirical Lipschitz increase/decrease with the network width. This provides empirical results for our theoretical analysis in Section 3. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Reducing Parameters at Deeper Layers Improves both Perturbation Stability and Lipschitzness. Based on our theoretical analysis, reducing the width at Stage-1 and Stage-2 should improve the Lipschitzness of the corresponding stage as well as the entire network. However, empirically, it is true for the corresponding stage but not necessarily for the entire network. This is because the theoretical analysis in Section 3 only considers the interplay between two adjacent layers (or blocks), not including that of the non-adjacent layers. The empirical results in Figure 5 can fill this gap. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Specifically, Stage-1 width reduction (Figure 5a) lowers the Lipschitzness of Stage-1 blocks but not the overall Lipschitzness. Stage-2 width reduction (Figure 5b) can improve both Stage-2 and the overall Lipschitzness but fails to improve clean accuracy, perturbation stability nor adversarial robustness. WRN-34-R in Figure 5c marks our discovered reduction and scaling rule. Compared with standard WRNs, WRN-34-R not only reduces the width of Stage-3 (decreasing Lipschitzness) but also increases the widths of Stage-1 and Stage-2 (increasing Lipschitzness). Figure 5c shows that the increased Lipschitzness at Stage-1 and Stage-2 of WRN-34-R can be effectively mitigated at Stage-3, leading to decreased overall Lipschitzness and improved robustness (see Table 4.4). This also results in higher clean accuracy and perturbation stability (Figure $_ \\mathrm { 4 c }$ ). We conjecture this is because Stage-3 (the last stage) is closer to the final output, thus has a more direct impact on the overall Lipschitzness. These empirical results provide a more in-depth understanding of the impact of width and depth configurations to the overall Lipschitzness, perturbation stability and adversarial robustness. ",
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+    {
+        "type": "text",
+        "text": "5 Adversarial Robustness Evaluation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we apply the discovered $^ { \\mathrm { w } } 1 0 { - } 1 0 { - } 4$ width configuration rule to VGG, DenseNet (DN), DNNs discovered by NAS, WRN-34-R $^ { \\mathrm { w } } 1 0 – 1 0 – 4$ scaled by $\\gamma = 2 . 0$ ), and evaluate their robustness with various adversarial attacks and defence methods on CIFAR [1] in the white-box setting. Additional results for CIFAR-10 black-box and ImageNet [75] using FastAT [37] can be found in Appendix D. ",
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+        "type": "text",
+        "text": "Experimental Settings. We consider VGG-11 [76], DenseNet-121 (DN-121) [77] and a network found by DARTS [78] with 11 cells. We denote the optimized VGG-11, DN-121 and DARTS networks as VGG-11-R, DN-121-R and DARTS-R (see Appendix B.2 for details), respectively. For a fair comparison between the discovered WRN-34-R (scaled by $\\gamma = 2 . 0$ ) and the standard ",
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+        "type": "text",
+        "text": "WRN, we upscale WRN-34-10 to WRN-34-12 to make sure the two models have a similar amount of parameters. We train all networks using 4 adversarial training methods: Standard Adversarial Training (SAT) [15], TRADES [16], Misclassification Aware adveRsarial Training (MART) [18] and Robust Self-Training (RST) with 500K additional data [23]. More details are in Appendix B.1. ",
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+        "text": "Table 2: White-box robustness results on CIFAR-10 and CIFAR-100. 500K: Additional data used as in [23]. Params: number of parameters. SAT: Adversarial Training [15]; TRADES [16]; MART [18]; GAMA100: 100-step GAMA attack [33]; AA: AutoAttack [30]; $\\mathrm { C W } _ { \\infty }$ : $L _ { \\infty }$ version CW attack [79] optimized by PGD; -R: reconfigured networks following our discovered robust architectural configuration; Last: Results evaluated at the last checkpoint. Best: Results evaluated at the best checkpoint according to $\\mathrm { P G D ^ { 2 0 } }$ . The best results are in bold. ",
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+        "type": "table",
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+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td rowspan=\"2\">Model</td><td rowspan=\"2\">Method</td><td rowspan=\"2\">Params (M)</td><td colspan=\"2\">Clean (%)</td><td colspan=\"2\">FGSM (%）</td><td colspan=\"2\">PGD20 (%)</td><td colspan=\"2\">GAMA100 (%）</td><td colspan=\"2\">CW (%）</td><td colspan=\"2\">AA (%)</td></tr><tr><td>Last</td><td>Best</td><td>Last</td><td>Best</td><td>Last</td><td>Best</td><td>Last</td><td>Best</td><td>Last</td><td>Best</td><td>Last</td><td>Best</td></tr><tr><td rowspan=\"10\">CIFAR-10</td><td>VGG-11</td><td>SAT</td><td>9.23</td><td>79.24</td><td>77.84</td><td>55.98 57.35</td><td>56.68</td><td>42.62</td><td>45.46</td><td>38.59</td><td>40.65</td><td>45.45</td><td>46.29</td><td>37.21 38.44</td><td>39.84</td></tr><tr><td>VGG-11-R</td><td>SAT</td><td>5.83</td><td>79.63</td><td>77.34</td><td></td><td>57.11</td><td>43.93</td><td>45.97</td><td>39.71</td><td>41.31</td><td>46.49</td><td>47.23</td><td></td><td>40.65</td></tr><tr><td>DN-121</td><td>SAT</td><td>6.96</td><td>86.87</td><td>86.07</td><td>65.56</td><td>66.58</td><td>51.67</td><td>54.79</td><td>48.60</td><td>51.00</td><td>52.03</td><td>54.00</td><td>47.16</td><td>50.34</td></tr><tr><td>DN-121-R</td><td>SAT</td><td>6.00</td><td>87.22</td><td>86.01</td><td>67.12</td><td>67.20</td><td>52.52</td><td>55.16</td><td>49.37</td><td>51.44</td><td>53.07</td><td>54.67</td><td>47.75</td><td>50.54</td></tr><tr><td>DARTS</td><td>SAT</td><td>6.58</td><td>86.76</td><td>86.55</td><td>64.48</td><td>67.10</td><td>49.44</td><td>54.23</td><td>46.52</td><td>50.74</td><td>52.03</td><td>54.00</td><td>45.16</td><td>49.98</td></tr><tr><td>DARTS-R</td><td>SAT</td><td>2.53</td><td>87.20</td><td>85.79</td><td>66.74</td><td>66.61</td><td>52.36</td><td>55.01</td><td>48.71</td><td>50.94</td><td>53.07</td><td>54.67</td><td>47.75</td><td>50.54</td></tr><tr><td>WRN-34-12</td><td>SAT</td><td>66.46</td><td>86.71</td><td>87.20</td><td>64.06</td><td>66.26</td><td>49.92</td><td>53.09</td><td>47.45</td><td>50.40</td><td>52.23</td><td>53.58</td><td>46.06</td><td>49.18</td></tr><tr><td>WRN-34-R</td><td>SAT</td><td>68.12</td><td>87.62</td><td>87.85</td><td>66.23</td><td>68.15</td><td>51.08</td><td>55.35</td><td>48.45</td><td>51.36</td><td>52.42</td><td>54.57</td><td>46.75</td><td>50.03</td></tr><tr><td>WRN-34-12</td><td>TRADES</td><td>66.46</td><td>85.84</td><td>84.59</td><td>65.70</td><td>66.85</td><td>53.02</td><td>56.01</td><td>49.60</td><td>52.35</td><td>53.35</td><td>54.72</td><td>48.48</td><td>51.83</td></tr><tr><td>WRN-34-R</td><td>TRADES</td><td>68.12</td><td>86.77</td><td>86.02</td><td>67.99</td><td>68.49</td><td>55.15</td><td>57.66 57.95</td><td>51.92</td><td>53.86</td><td>55.41</td><td>56.30 54.61</td><td>50.90</td><td>53.46</td></tr><tr><td></td><td>WRN-34-12</td><td>MART</td><td>66.46</td><td>85.98</td><td>82.62</td><td>66.85</td><td>67.00</td><td>54.30</td><td></td><td>49.58</td><td>52.20</td><td>52.29</td><td></td><td>47.68</td><td>51.21</td></tr><tr><td>CIFAR-10</td><td>WRN-34-R</td><td>MART</td><td>68.12</td><td>86.09</td><td>83.69</td><td>68.79</td><td>68.18</td><td>56.31</td><td>59.13</td><td>51.40</td><td>53.22</td><td>54.20</td><td>55.44</td><td>49.90</td><td>52.48</td></tr><tr><td>+500K</td><td>WRN-34-12</td><td>RST</td><td>66.46</td><td>90.52</td><td>90.36</td><td>76.01</td><td>76.02</td><td>65.52</td><td>65.56</td><td>61.67</td><td>61.70</td><td>64.30</td><td>64.26</td><td>60.90</td><td>60.96</td></tr><tr><td rowspan=\"2\"></td><td>WRN-34-R</td><td>RST</td><td>68.12</td><td>90.73</td><td>90.56</td><td>76.51</td><td>76.44</td><td>66.46</td><td>66.51</td><td>62.38</td><td>62.49</td><td>65.12</td><td>65.10</td><td>61.49</td><td>61.56</td></tr><tr><td>WRN-34-12</td><td>SAT</td><td>66.53</td><td>59.63</td><td>60.64</td><td>33.67</td><td>37.28</td><td>24.50</td><td>27.61</td><td>23.38</td><td>24.95</td><td>43.78</td><td>42.02</td><td>22.27</td><td>24.42</td></tr><tr><td rowspan=\"6\">CIFAR-100</td><td>WRN-34-R</td><td>SAT</td><td>68.16</td><td>61.17</td><td>61.33</td><td>35.00</td><td>38.72</td><td>25.03</td><td>29.02</td><td>23.38</td><td>25.70</td><td>43.52</td><td>41.46</td><td>22.72</td><td>25.20</td></tr><tr><td>WRN-34-12</td><td>TRADES</td><td>66.53</td><td>55.62</td><td>56.47</td><td>35.35</td><td>36.90</td><td>27.52</td><td>29.48</td><td>24.94</td><td>25.21</td><td>44.75</td><td>46.19</td><td>24.58</td><td>24.85</td></tr><tr><td>WRN-34-R</td><td>TRADES</td><td>68.16</td><td>56.83</td><td>56.75</td><td>36.95</td><td>37.68</td><td>29.17</td><td>29.92</td><td>25.48</td><td>25.48</td><td>45.04</td><td>45.52</td><td>25.13</td><td>25.23</td></tr><tr><td>WRN-34-12</td><td>MART</td><td>66.53</td><td>58.51</td><td>57.29</td><td>36.06</td><td>39.48</td><td>26.50</td><td>32.43</td><td>23.85</td><td>27.64</td><td>41.53</td><td>38.73</td><td>23.33</td><td>26.92</td></tr><tr><td>WRN-34-R</td><td>MART</td><td>68.16</td><td>61.72</td><td>58.27</td><td>39.68</td><td>41.24</td><td>29.94</td><td>34.12</td><td>26.27</td><td>29.33</td><td>39.20</td><td>38.45</td><td>25.60</td><td>28.63</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+        "type": "text",
+        "text": "White-box Robustness. We evaluate the robustness of the networks to 5 adversarial attacks including Fast Gradient Sign Method (FGSM) [5], Projected Gradient Descent (PGD) [15], Carlini and Wagner (CW) [79], Guided Adversarial Margin Attack (GAMA) [33] and AutoAttack (AA) [30]. We apply these attacks on the test sets of CIFAR-10 and CIFAR-100 with the same maximum adversarial perturbation $\\epsilon = 8 / 2 5 5$ as adopted for model training. For PGD, we use the 20-step PGD $( \\mathrm { P G D ^ { 2 0 } } )$ with step size $\\alpha = \\epsilon / 1 0$ . For GAMA attack, we set its perturbation steps to 100 following the original paper. We report the model’s accuracy on the test adversarial examples crafted by these evaluation attacks for models obtained at both the best and the last checkpoints, following [16, 46, 18]. ",
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+    {
+        "type": "text",
+        "text": "The white-box evaluation results including both clean accuracy and adversarial robustness are reported in Table 2. As can be inferred, a robustness gain of $1 \\sim 3 \\%$ can be consistently achieved when the networks are reconfigured following our discovered configuration rule. And the improvements are not restricted to a particular architecture nor adversarial training method, except there is slight decrease for $\\mathrm { C W } _ { \\infty }$ on CIFAR-100 for SAT and MART. This wide range of robustness improvements by a simple architectural reconfiguration confirms that our findings are very general and can be immediately applied to commonly used DNNs to obtain more adversarial robustness. For VGG-11, DN-121 and DARTS, our robust reconfiguration can reduce the parameters by a considerable amount. ",
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+    },
+    {
+        "type": "text",
+        "text": "6 Relation to Existing Understandings ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "One recent work [59] shows that wider networks tend to increase the Lipschitzness, a finding that is consistent with ours. In [59], the theoretical analysis is based on Neural Tangent Kernel (NTK) under the assumption that all layers share the same width. By contrast, our analysis provides a more in-depth understanding related to the width and depth of each individual layer. Whilst in [59], the wider network utilizes a stronger regularization to mitigate the vulnerability (instability) caused by increased Lipschitzness, our work shows that this can be achieved alternatively by a simple reconfiguration of the architecture. ",
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+    {
+        "type": "text",
+        "text": "It has also been found that weight decay plays an important role in adversarial training [54, 80]. This can be explained by our theoretical analysis in Theorem 1 and 2. Considering the weight matrix is normally distributed $\\mathcal { N } ( 0 , \\sigma _ { \\theta } ^ { 2 } )$ , weight decay encourages the model to learn weights of smaller magnitudes, thus reducing the variance $\\sigma _ { \\theta } ^ { 2 }$ of the weight matrix. This will lead to reduced upper bound of the Lipschitz constant and improved robustness. See Appendix E for more discussions. ",
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+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this paper, we explored the architectural ingredients of adversarially robust DNNs via extensive fine-controlled experiments and theoretical analysis. Our findings are: 1) more parameters does not necessarily lead to more adversarially robust models; 2) reducing capacity (up to a limit) via either depth or width at the deeper layers improves adversarial robustness; and 3) under the same type of architectures and parameter budget, there may exist an architectural configuration that can exploit the full robustness potential of the network. We also showed that depth and width offer different levels of flexibility for capacity reduction and robustness improvement. Following a width reduction and scaling rule, we showed that our findings are generic, not restricted to a particular adversarial training method, and can be immediately applied to improve both manually-designed or NAS-discovered DNNs. We also provide a series of empirical understandings on the distinctive impacts of the deeper layers on adversarial robustness. Our work can provide useful insights into the architectural perspective of adversarial robustness, and help design more adversarially robust DNNs. ",
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+    {
+        "type": "text",
+        "text": "Border Impacts ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Adversarial training is currently the most effective defense against adversarial attacks, although its performance is yet to be improved. In this work, we extensively studied the impact of network architecture to adversarial robustness. Our findings suggest that models can be made more robust by even reducing capacity at the deep layers. Such reduction can also help save the training cost of adversarial training which is known to be extremely time-consuming. We will open source the discovered architectural configurations to help future research design more robust architectures. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgment ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Yisen Wang is partially supported by the National Natural Science Foundation of China under Grant 62006153, and Project 2020BD006 supported by PKU-Baidu Fund. This research was undertaken using the LIEF HPC-GPGPU Facility hosted at The University of Melbourne. This Facility was established with the assistance of LIEF Grant LE170100200. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            542,
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+    },
+    {
+        "type": "text",
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Uncovering the limits of adversarial training against norm-bounded adversarial examples. arXiv preprint arXiv:2010.03593, 2020.   \n[81] Ilya Loshchilov and Frank Hutter. SGDR: stochastic gradient descent with warm restarts. In ICLR, 2017.   \n[82] Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In ICML, 2015.   \n[83] Linxi Jiang, Xingjun Ma, Zejia Weng, James Bailey, and Yu-Gang Jiang. Imbalanced gradients: A new cause of overestimated adversarial robustness. arXiv preprint arXiv:2006.13726, 2020.   \n[84] Vikash Sehwag, Shiqi Wang, Prateek Mittal, and Suman Jana. Hydra: Pruning adversarially robust neural networks. NeurIPS, 2020.   \n[85] Pavel Izmailov, Dmitrii Podoprikhin, Timur Garipov, Dmitry Vetrov, and Andrew Gordon Wilson. Averaging weights leads to wider optima and better generalization. In UAI, 2018. ",
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parse/train/Q2iaAc-4I1v/Q2iaAc-4I1v.md

@@ -0,0 +1,316 @@
+# CAUSAL CURIOSITY: RL AGENTS DISCOVERING SELF-SUPERVISED EXPERIMENTS FOR CAUSAL REPRESENTATION LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Humans show an innate ability to learn the regularities of the world through interaction. By performing experiments in our environment, we are able to discern the causal factors of variation and infer how they affect the dynamics of our world. Analogously, here we attempt to equip reinforcement learning agents with the ability to perform experiments that facilitate a categorization of the rolled-out trajectories, and to subsequently infer the causal factors of the environment in a hierarchical manner. We introduce a novel intrinsic reward, called causal curiosity, and show that it allows our agents to learn optimal sequences of actions, and to discover causal factors in the dynamics. The learned behavior allows the agent to infer a binary quantized representation for the ground-truth causal factors in every environment. Additionally, we find that these experimental behaviors are semantically meaningful (e.g., to differentiate between heavy and light blocks, our agents learn to lift them), and are learnt in a self-supervised manner with approximately 2.5 times less data than conventional supervised planners. We show that these behaviors can be re-purposed and fine-tuned (e.g., from lifting to pushing or other downstream tasks). Finally, we show that the knowledge of causal factor representations aids zero-shot learning for more complex tasks.
+
+# 1 INTRODUCTION
+
+Discovering causation in environments an agent might encounter remains an open and challenging problem for causal reinforcement learning (Schölkopf (2015), Bengio et al. (2013), Schölkopf (2019)). Most approaches take the form of BAMDPs (Bayes Adaptive Markov Decision Processes) (Zintgraf et al. (2019)) or Hi-Param MDP (Hidden Parameter MDPs) (Doshi-Velez & Konidaris (2016); Yao et al. (2018); Killian et al. (2017); Perez et al. (2020)) which condition the transition $p ( s _ { t + 1 } | s _ { t } , a _ { t } ; H )$ and/or reward function $R ( r _ { t + 1 } | s _ { t } , a _ { t } , s _ { t + 1 } ; H )$ of each environment on hidden parameters (also referred to as causal factors in some of the above studies). Let $s \in { \mathcal { S } } , a \in { \mathcal { A } }$ $r \in \mathcal { R }$ , $H \in { \mathcal { H } }$ where $s , { \mathcal { A } } , { \mathcal { R } }$ , and $\mathcal { H }$ are the set of states, actions, rewards and feasible hidden parameters. In the physical world and in the case of mechanical systems, examples of the parameter $h _ { j } \in \mathcal { H }$ include gravity, coefficients of friction, masses and sizes of objects. Typically, $H$ is treated as a latent variable for which an embedding is learned during training, using variational methods (Kingma et al. (2014); Ilse et al. (2019)). Let $s _ { 0 : T }$ be the entire state trajectory of length $T$ . Similarly, $a _ { 0 : T }$ is the sequence of actions applied during that trajectory by the agent that results in $s _ { 0 : T }$ . In an environment parameterized by these causal factors, these latent variable approaches define a probability distribution over the entire sequence of (rewards, states, actions) conditioned on a latent $z$ as $p ( r _ { 0 : T } , s _ { 0 : T } , a _ { 0 : T - 1 } ; z )$ that factorizes as
+
+$$
+\prod _ { i = 1 } ^ { T - 1 } p ( r _ { t + 1 } | s _ { t } , a _ { t } , s _ { t + 1 } , z ) p ( s _ { t + 1 } | s _ { t } , a _ { t } , z ) p ( a _ { t } | s _ { t } , z )
+$$
+
+due to the Markov assumption. At test time, the agent infers the causal factor associated with its environment by observing the trajectories produced by its initial actions that can be issued by any policy such as model-based reinforcement learning.
+
+In practice, however, discovering causal factors in a physical environment is prone to various challenges that are caused by the disjointed nature of the influence of these factors on the produced trajectories. More specifically, at each time step, the transition function is affected by a subset of global causal factors. This subset is implicitly defined on the basis of the current state and the action taken. For example, if a body in an environment loses contact with the ground, the coefficient of friction between the body and the ground no longer affects the outcome of any action that is taken. Likewise, the outcome of an upward force applied by the agent to a body on the ground is unaffected by the friction coefficient. We can therefore take advantage of this natural discontinuity to discern causal factors.
+
+Without knowledge of how independent causal mechanisms affect the outcome of a particular action in a given state in an environment, it becomes impossible for the agent to conclude where the variation it encountered came from. Unsurprisingly, Hi-Param and BAMDP approaches fail to learn a disentangled embedding for the causal factors, making their behaviors uninterpretable (Perez et al. (2020)). For example, if, in an environment, a body remains stationary under a particular force, the Hi-Param or BAMDP agent may apply a higher force to achieve its goal of perhaps moving the body, but will be unable to conclude whether the "un-movability" was caused by high friction or high mass of the body. Additionally, these approaches require human-supervised reward engineering, making it difficult to apply them outside of the simulated environments they are tested in.
+
+Our goal is, instead of focusing on maximizing reward for some particular task, to allow agents to discover causal processes through exploratory interaction. During training, our agents discover self-supervised experimental behaviors which they apply to a set of training environments. These behaviors allow them to learn about the various causal mechanisms that govern the transitions in each environment. During inference in a novel environment, they perform these discovered behaviors sequentially and use the outcome of each behavior to infer the embedding for a single causal factor (Figure 1).
+
+The main challenge while learning a disentangled representation for the causal factors of the world is that several causal factors may affect the outcome of behaviors in each environment. For example, when pushing a body on the ground, the outcome, i.e., whether the body moves, or how far the body is pushed, depends on several factors, e.g., mass, shape and size, frictional coefficients, etc. However, if, instead of pushing on the ground, the agent executes a perfect grasp-and-lift behavior, only mass will affect whether the body is lifted off the ground or not.
+
+Thus, it is clear that not all experimental behaviors are created equal and that the outcomes of some behaviors are caused by fewer causal factors than others. Our agents learn these behaviors without supervision using causal curiosity, an intrinsic reward. The outcome of a single such experimental behavior is then used to infer a binary quantized embedding describing the single isolated causal factor. Even though causal factors of variation in a physical world are easily identifiable to humans, a concrete definition is required to back up our proposed method. We conjecture that the causality of a factor of variation depends on the available actions to the agent. If the set of actions that an agent can take is very limited, there is no way for it to discern a diverse set of causal factors in the environment.
+
+Definition 1 (Causal factors). Consider the POMDP $( \mathcal { O } , \mathcal { S } , \mathcal { A } , p , r )$ with observation space $\mathcal { O }$ , state space $s$ , action space $\mathcal { A }$ , the transition function $p$ , and the reward function $r$ . Let $o _ { 0 : T } \in \mathcal { O } ^ { T }$ denotes a trajectory of observations and $T$ be the length of such trajectories. Let $d ( \cdot , \cdot ) : \mathcal { O } ^ { T } \times \mathcal { O } ^ { T } \to \mathbb { R } _ { + }$ be a distance function defined on the space of trajectories of length $T$ . The set $H = \{ h _ { 1 } , h _ { 2 } , \ldots , h _ { k } \}$ is called a set of $\epsilon -$ causal factors if for every $h _ { j } \in H$ , there exists a unique sequence of actions $a _ { 0 : T }$ that clusters the state trajectories into two sets $S$ and $S ^ { \prime }$ such that
+
+$$
+\operatorname* { m i n } \{ d ( o _ { 0 : T } , o _ { 0 : T } ^ { \prime } ) : o _ { 0 : T } \in O , o _ { 0 : T } ^ { \prime } \in O ^ { \prime } \} > \epsilon
+$$
+
+and that $h _ { j }$ is the cause of the trajectory of states obtained i.e.,
+
+$$
+p ( o _ { 0 : T } | d o ( h _ { j } = k ) , a _ { 0 : T } ) \neq p ( o _ { 0 : T } | d o ( h _ { j } = k ^ { \prime } ) , a _ { 0 : T } ) \forall k \neq k ^ { \prime }
+$$
+
+Intuitively, a factor of variation affecting a set of environments is called causal if there exists a sequence of actions available to the agent where the resultant trajectories are clustered into two or more sets (for simplicity here we assume binary clusters). This is analogous to the human ability to conclude whether objects are heavy or light, big or small. For a gentle introduction to the intuition about this definition, we refer the reader to Appendix D.
+
+According to Def. 1, a causal factor is a parameter in the environment whose value, when intervened on (i.e. varied) over a set of values, results in trajectories of states that are divisible into disjoint clusters under a particular sequence of actions. These clusters represent the quantized values of the causal factor. For example, mass, which is a causal factor of a body, under an action sequence of a grasping and lifting motion, results in 2 clusters, liftable (low mass) and not liftable (high mass). However, such an action sequence is not known in advance. Therefore, discovering a causal factor in the environment boils down to finding a sequence of actions that makes the effect of that factor prominent by producing clustered trajectories for different values of that environmental factor.
+
+![](images/bd8a45b5a87224e98efb1cf1c4411698bcdd09b5472eb0da2a01ba7932658e4e.jpg)  
+Figure 1: Overview of Inference. The exploration loop produces a series of $K$ experiments allowing the agent to infer the representations for $K$ causal factors. After exploration, the agent utilizes the acquired knowledge for downstream tasks.
+
+Using the above, we propose an intrinsic reward, which allows our agents to discover experimental behaviors which are semantically meaningful and can be used to re-train for downstream tasks, resulting in high sample efficiency. Our work, therefore, forms an important link between structured representation learning and skill discovery, two largely disjoint fields in RL, which stand to benefit from each other.
+
+The contributions of the work are as follows:
+
+• We equip agents with the ability to perform experiments and behave meaningfully in a set of environments in an unsupervised manner. These behaviors can expose or obfuscate specific independent causal mechanisms that occur in the world of the agent, allowing the agent to learn about each in the absence of the others, an important human behavioral trait.   
+• We introduce an intrinsic reward, causal curiosity, which allows our agents to discover these behaviors without human-engineered rewards. The outcomes of the experiments are used to learn a disentangled quantized binary representation for the causal factors of the environment, analogous to the human ability to conclude whether objects are light/heavy, big/small etc.   
+• Through extensive experiments, we conclude that knowledge of the causal factors aids sample efficiency in two ways - first, that the knowledge of the causal factors aids transfer learning across multiple environments, and, second, that the experimental behaviors acquired can be repurposed for downstream tasks.
+
+# 2 METHOD
+
+Consider a set of $N$ environments $\mathcal { E }$ with $e ^ { ( i ) } \in \mathcal { E }$ where $e ^ { ( i ) }$ denotes the $i ^ { t h }$ environment.
+
+The letter $H$ is overloaded. While $H$ is a set of global causal factors (as defined in Def. 1) such that   
+$h _ { j } \in H$ , each causal factor $h _ { j }$ is itself a random variable which assumes a particular value for every ion of an environment. T. For each environment $e ^ { ( i ) }$ is represented by a set of causal factorsepresents the disentangled embedding   
+$\{ h _ { j } ^ { ( i ) } \forall j \}$ $e ^ { ( i ) }$ $( z _ { ( 0 ) } ^ { ( i ) } , z _ { ( 1 ) } ^ { ( i ) } . . . z _ { ( K - 1 ) } ^ { ( i ) } )$   
+vector, such that z(i)(j) $h _ { j } ^ { ( i ) }$ .
+
+# Algorithm 1 Training Scheme
+
+1: Initialize $j = 0$ 2: Initialize training environment set Envs 3: for iteration m to M do . Experiment Planner Training Loop 4: Sample experimental behavior $a _ { 0 : T } \sim \mathbf { C E M } ( \cdot )$ 5: for $i ^ { \star h }$ env in Envs do 6: Apply $a _ { 0 : T }$ to env 7: $\mathbf { C o l l e c t } \ S ^ { ( } i ) = O _ { 0 : T } ^ { ( i ) }$ 8: Reset env 9: Calculate $- L ( S | M )$ given that $M$ is bimodal clustering model $\triangleright$ Calculate Curiosity   
+10: Update CEM(·) distribution with highest reward trajectories   
+11: Use learnt $q _ { M } ( z | S )$ for cluster assignment of each env in Envs i.e. $z _ { j } ^ { ( i ) } = q _ { M } ( z | S ^ { ( i ) } )$   
+12: Update $j = j + 1$   
+13: Repeat from step 2, first setting $E n v s = \{ e ^ { ( i ) } : z _ { j - 1 } ^ { ( i ) } = 0 \}$ and then, setting $E n v s = \{ e ^ { ( i ) } : z _ { j - 1 } ^ { ( i ) } = 1 \}$
+
+# 2.1 TRAINING THE EXPERIMENT PLANNER
+
+To learn about causal processes through interaction, the agent must produce a sequence of actions $a _ { 0 : T - 1 }$ that we call experimental behavior, which, when applied to environment $e ^ { ( i ) } \in \mathcal { E }$ , produces a sequence of observations (state) $s ^ { ( i ) } = [ o _ { 0 } ^ { ( i ) } , o _ { 1 } ^ { ( i ) } . . o _ { T } ^ { ( i ) } ]$ , which is then used to infer the value of the embedding for a single causal factor $z _ { ( j ) } ^ { ( i ) }$ .
+
+We motivate this using model selection criterion. Normally in model selection applications, the observations are fixed and the goal is to find a model $M ^ { * }$ that is closest to reality, as represented by:
+
+$$
+M ^ { * } = \arg \operatorname* { m i n } _ { M } ( L ( M ) + L ( S | M ) )
+$$
+
+where $L ( \cdot )$ is the description length. However, here, the situation is reversed. A simple bi-modal clustering model is fixed, motivated by Definition 1. Then, the agent is motivated to produce actions that result in observations that are best explained by this model. These discovered action sequences are the experimental behaviors we desire.
+
+$$
+\begin{array} { r } { \boldsymbol { a } _ { 0 : T } ^ { * } = \underset { \boldsymbol { a } _ { 0 : T } } { \arg \operatorname* { m i n } } ( L ( \boldsymbol { M } ) + L ( \boldsymbol { S } | \boldsymbol { M } ) ) } \end{array}
+$$
+
+where each observed trajectory $S = S ( a _ { 0 : T } )$ is a function of the action sequence. As mentioned earlier, the model is fixed in this formulation; hence, the first term $L ( M )$ is constant and not a function of the actions. $- L ( S | M )$ that is fed back to the RL agent as a reward function to maximize. We regard this reward function as causal curiosity.
+
+Note that since each causal factor has its own independent causal mechanism that causes $S$ , the MDL of $S$ will be higher if multiple causal factors cause $S$ . On the contrary, if the agent produces actions which result in an $S$ that is easily explained by a low-capacity bi-modal model $M$ , then it will imply that $S$ is caused by fewer causal factors. Consequently, the causal curiosity reward for such an action sequence, $- L ( S | M )$ , will be high. Therefore, causal curiosity favors experimental behaviors that result in observations caused by few causal factors - thereby allowing us to use $S$ to infer a representation for a single causal factor. For details, please refer Appendix A.
+
+# 2.2 CAUSAL INFERENCE MODULE
+
+By maximizing the causal curiosity reward it is possible to achieve behaviors which result in trajectories of states only caused by a single hidden parameter. However, we wish to use the outcome of performing these experimental behaviors in each environment to infer a representation for the causal factor isolated by the experiment in question.
+
+We achieve this through cluster membership. After training the Model Predictive Control Planner (Camacho & Alba (2013)), we sample from an action sequence $a _ { 0 : T }$ and apply it to each of the training environments. The learnt clustering model $M$ is then used to infer a representation for each environment using the collected outcome $\bar { S } ^ { ( i ) }$ obtained by applying $a _ { 0 : T }$ to each environment.
+
+$$
+z _ { j } ^ { ( i ) } = q _ { M } ( z | S ^ { ( i ) } )
+$$
+
+This corresponds to Step 11 of Algorithm (1). The representation learnt is binary in nature corresponding to the quantization of the continuous spectrum of values a causal factor takes in the training set into high and low values. Note however that a binary quantized embedding is not a necessary part of our method. A dense embedding may alternatively be learnt here similar to (Perez et al. (2020); Zintgraf et al. (2019)) using approximate variational inference. However, performing interventions on a dense embedding (Section 2.3) increases the computational complexity exponentially. Balancing space and time complexity, we report results using the quantized binary form of Equation (6). We discuss the implications of increasing the complexity of $z _ { j } ^ { ( i ) }$ in the discussion.
+
+# 2.3 INTERVENTIONS ON BELIEFS
+
+Having learnt about the effects of a single causal factor of the environment we wish to learn such experimental behaviors for each of the remaining hidden parameters that may vary in an environment. To achieve this, in an ideal setting, the agent would require access to the generative mechanism of the environments it encounters. Ideally, it would hold the values of the causal factor already learnt about constant i.e. $d o ( h _ { j } = c o n s t a n t )$ , and intervene over (vary the value of) another causal factor over a set of values $K$ i.e. $d o ( h _ { j } = k )$ such that $k \in K$ . For example, if a human scientist were to study the effects of a causal factor, say mass of a body, she would hold the values of all causal factors constant, (interact with cubes of the same size and external texture) and vary only mass to see how it affects the outcome of specific behaviors she applies to each body.
+
+However, in the real-world the agent does not have access to the generative mechanism of the environments it encounters, but merely has the ability to act in them. Thus, it can intervene on the representations of a causal factor of the environment i.e. $d o ( z _ { i } = c o n s t a n t )$ . For, example having learnt about gravity, the agent picks all environments it believes have low gravity, and uses them to learn about a separate causal factor say, friction.
+
+This corresponds to Step 13 of Algorithm (1). Thus, to learn about the $j ^ { t h }$ causal factor, we repeat steps 3 onwards on each of the clusters obtained for the $j - 1 ^ { t h }$ .
+
+$$
+E n v s = \{ e ^ { ( i ) } : z _ { j - 1 } ^ { ( i ) } = k \} , k \in \{ 0 , 1 \}
+$$
+
+This process continues in the form of a tree (Figure 4), where for each cluster of environments, a new experiment learns to split the cluster into 2 sub-clusters depending on the value of another hidden parameter. At level $n$ , the agent produces $2 ^ { n }$ experiments and inference models, having already intervened on the binary quantized representations of $n$ causal factors.
+
+# 3 RELATED WORK
+
+Doshi-Velez & Konidaris (2016) define a class Markov Decision Processes where transition probabilities $p ( s _ { t + 1 } | s _ { t } , a _ { t } ; \theta )$ depend on a hidden parameter $\theta$ , whose value is not observed, but its effects are felt. Killian et al. (2017) and Yao et al. (2018) utilize these Hidden Parameter MDPs (Markov Decision Processes) to enable efficient policy transfer, assuming that transition probabilities across states are a function of hidden parameters. Perez et al. (2020) relax this assumption, allowing both transition probabilities and reward functions to be functions of hidden parameters. Zintgraf et al. (2019) approach the problem from a Bayes-optimal policy standpoint, defining transition probabilities and reward functions to be dependent on a hidden parameter characteristic of the MDP in consideration. We utilize this setup to define causal factors.
+
+Substantial attempts have been made at unsupervised disentanglement, most notably, the $\beta$ -VAE Higgins et al. Burgess et al. (2018), where a combination of factored priors and the information bottleneck force disentangled representations. Kim & Mnih (2018) enforce explicit factorization of the prior without compromising on the mutual information between the data and latent variables, a shortcoming of the $\beta$ -VAE. Chen et al. (2018) factor the KL divergence into a more explicit form, highlighting an improved objective function and a classifier-agnostic disentanglement metric. Locatello et al. (2018) show theoretically that unsupervised disentanglement (in the absence of inductive biases) is impossible and highly unstable, susceptible to random seed values. They follow this up with Locatello et al. (2020) where they show, both theoretically and experimentally, that pair-wise images provide sufficient inductive bias to disentangle causal factors of variation. However, these works have been applied to supervised learning problems whereas we attempt to disentangle the effects of hidden variables in dynamical environments, a relatively untouched question.
+
+Curiosity for robotics is not a new area of research. Schmidhuber (2006), Ngo et al. (2012), Pathak et al. (2017) describe curiosity as the motivation behind the behavior of an agent in an environment for which the outcome is unpredictable, i.e., an intrinsic reward that motivates the agent to explore the unseen portions of the state space (and subsequent transitions). While causal curiosity is an intrinsic reward, it differs from these traditional definitions of curiosity in that it motivates the agent to produce structure in the outcome of its behavior.
+
+# 4 EXPERIMENTS
+
+Our work has 2 main thrusts - the discovered experimental behaviors and the representations obtained from the outcome of the behaviors in environments. The experimental behaviors are tied to contributions 1 and 2 in the Introduction. The causal factors allow us to achieve contribution 3 in the Introduction. We visualize these learnt behaviors and verify that they are indeed semantically meaningful and interpretable. We quantify the utility of the learned behaviors by using the behaviors as pre-training for a downstream task. In our experimental setup, we verify that these behaviors are indeed invariant to all other causal factors except one.
+
+We visualize the representations obtained using these behaviors and verify that they are indeed the binary quantized representations for each of the ground truth causal factors that we manipulated in our experiments. Finally, we verify that the knowledge of the representation does indeed aid transfer learning and zero-shot generalizability in downstream tasks.
+
+Causal World We use the Causal World Simulation (Ahmed et al. (Under submission 2020)) based on the Pybullet Physics engine to test our approach. The simulator consists of a 3-fingered robot, with 3 joints on each finger. We constrain each environment to consist of a single object that the agent can interact with. The causal factors that we manipulate for each of the objects are size, shape and mass of the blocks. The simulator allows us to capture and track the positions and velocities of each of the movable objects in an environment. While, for most experiments, the 3D position and 3D pose of the blocks is used as the state at each time step, we perform ablation studies where less information is provided to the agent.
+
+# 4.1 VISUALIZING DISCOVERED BEHAVIORS
+
+We would like to analyze whether the discovered experimental behaviors are human interpretable, i.e., are the experimental behaviors discovered in each of the setups semantically meaningful? We find that our agents learn to perform several useful behaviors without any supervision. For instance, to differentiate between objects with varying mass, we find that they acquire a perfect grasp-and-lift behavior with an upward force. In other random seed experiments, the agents learn to lift the blocks by using the wall of the environment for support. To differentiate between cubes and spheres, the agent discovers a pushing behavior which gently rolls the spheres along a horizontal direction. Qualitatively, we find that these behaviors are stable and predictable. See videos of discovered behaviors here (website under construction).
+
+Concurrent with the objective they are trained on, we find that the acquired behaviors impose structure on the outcome when applied to each of the training environments. The outcome of each experimental behavior on the set of training environments results in dividing it into 2 subsets. These subsets correspond to the binary quantized values of a single factor, e.g., large or small, while being invariant to the values of other causal factors of the environments. We also perform ablation studies where instead of providing the full state vector, we provide only one coordinate (e.g., only x, y or z coordinate of the block). We find that causal curiosity results in behaviors that differentiate the environments based on outcomes along the direction provided. For example, when only the $\mathbf { X }$ coordinate was provided, the agent learned to evaluate mass by applying a pushing behavior along the x direction. Similarly, a lifting behavior was obtained when only the z coordinate was supplied to the curiosity module (Figure 2).
+
+![](images/88d18b1b70f86f723c5c9a304545f39a38019220f7b369ab3dd9c9c67412ae78.jpg)  
+Figure 2: Examples of discovered behaviors. The agent discovers experimental behaviors that allow it to characterize each environmental object in a binary manner, e.g., heavy/light, big small, rollable/not rollable, etc. These behaviors are acquired without any external supervision by maximizing the causal curiosity reward. A, B, C correspond to self-discovered toss, lift-and-spin and roll behaviors respectively. D shows an ablation study, where the agent is only provided the z coordinate of the block in every environment. Each line corresponds to one environment and the z coordinate of the block is plotted with time when the discovered behavior is applied. It learns a lifting behavior, where cluster 1 represents the heavy blocks (z coordinate does not change much) and cluster 2 represents the light blocks $\mathbf { z }$ increases as block is lifted and then falls when dropped and subsequently increases again when it bounces).
+
+# 4.2 UTILITY OF LEARNED BEHAVIORS FOR DOWNSTREAM TASKS
+
+While the behaviors acquired are semantically meaningful, we would like to quantify their utility as pre-training for downstream tasks. We analyze the performance on Lifting where the agent must grasp and lift a block to a predetermined height and Travel, where the agent must impart a velocity to the block along a predetermined direction. We re-train the learnt planner using an external reward for these tasks (Curious). We implement a baseline vanilla Cross Entropy Method optimized Model Predictive Control Planner (De Boer et al. (2005)) trained using the identical reward function and compare the rewards per trajectory during training. We also run a baseline (Additive reward) which explores whether the agent recieves both the causal curiosity reward and the external reward. We find high zero-shot generalizability and quicker convergence as compared to the vanilla CEM planner (Figure ??). We find that maximizing the curiosity reward in addition to simultaneously maximizing external rewards results in suboptimal performance due to our formulation of the curiosity reward. To maximize curiosity, the agent must discover behaviors that divide environments into 2 clusters. Thus in the context of the experimental setups, this corresponds to acquiring a lifting/pushing behavior that allows the agent to lift/impart horizontal velocity to blocks in half of the environments, while not being able to do so in the remaining environments. However, the explicit external reward incentivizes the agent to lift/impart horizontal velocity blocks in all environments. Thus these competing objectives result in sub-par performance.
+
+# 4.3 VISUALIZATION OF HIERARCHICAL BINARY LATENT SPACE
+
+Our agents discover a disentangled latent space such that they are able to isolate the sources of causation of the variability they encounters in their environments. For every environment, they learn a disentangled embedding vector which describes each of the causal factors.
+
+To show this, we use 3 separate experimental setups - Mass, SizeMass and ShapeSizeMass where each of the causal factors are allowed to vary over a range of discrete values. During Mass, the agent is allowed access to 5 environments with objects having the same shape (cuboids) and size but differing only in mass. During SizeMass, the agent has access to 30 environments with cuboids having sizes and masses ranging over 6 and 5 values respectively. Finally, during ShapeSizeMass, the agent has access to 60 environments with objects having shapes, sizes and masses ranging over 2, 6, 5, and values respectively.
+
+![](images/8d0cb9d67f34c3e142de473044f931902b4ebb86b24b7f09c85f4164f62cf705.jpg)
+
+![](images/26a347cf44dd8d6fb015c4ab944719c1ed82d32789723592ad4cc3d3c364699d.jpg)  
+Figure 3: Utility of discovered behaviors. We find that the behaviors discovered by the agents while optimizing causal curiosity show high zero-shot generalizability and converge to the same performance as conventional planners for downstream tasks. We also analyze the worst case performance and find that the pre-training ensures better performance than random initialization. The table compares the time-steps of training required on an average to acquire a skill with the time steps required to learn a similar behavior using external reward. We find that the unsupervised experimental behaviors are approximately 2.5 times more sample efficient. We also find that maxizing both curiosity and external reward in our experimental setups results in sub-optimal results.   
+Figure 4: Discovered hierarchical latent space. The agent learns experiments that differentiate the full set of blocks in ShapeSizeMass into hierarchical binary clusters. At each level, the environments are divided into 2 clusters on the basis of the value of a single causal factor. We also show the principal components of the trajectories in the top left. For brevity, the full of extent of the tree is not depicted here. For each level of hierarchy $k$ , there are $2 ^ { k }$ number of clusters.
+
+During training, the agent discovers a hierarchical binary latent space (Figure 4), where each level of hierarchy corresponds to a single causal factor. The binary values at each level of hierarchy correspond to the high/low values of the causal factor in question. To our knowledge, we obtain the first interpretable latent space describing the various causal processes in the environment of an agent. This implies that it learns to quantify each physical attribute of the blocks it encounters in a completely unsupervised manner.
+
+# 4.4 KNOWLEDGE OF CAUSAL FACTORS AIDS TRANSFER
+
+Next, we test whether knowledge of the causal factors does indeed aid transfer and zero-shot generalizability. To this end, we supply the representations obtained by the agent during the experimental behavior phase as input to a policy network in addition to the state of the simulator, and train it for a place-and-orient downstream task (Figure 1). We define 2 experimental setups - TransferMass and TransferSizeMass. In Mass, the agent is given access to 10 environments, with 10 varying values of mass. In TransferSizeMass, the agent is allowed access to 10 environments, with 2 and 5 values of size and mass respectively. In both setups, the agent learns about the varying causal mechanisms by optimizing causal curiosity. Subsequently, using the causal representation along with the state for each environment, it is trained to maximize external reward. For details of the setup, please see Appendix B.
+
+After training, the agents are exposed to a set of unseen test environments, where we analyze their zero-shot generalizability. These test environments consist of unseen masses and sizes and their unseen combinations. This corresponds to "Strong Generalization" as defined by Perez et al. (2020). We report results averaged over 10 random seeds.
+
+For each setup, we train a PPO-optimized Actor-Critic Policy (referred to as Causally-curious agent) with access to the causal representations and a 56 dimensional state vector from the environment i.e., $a _ { t } \sim \pi ( \cdot | s _ { t } , z _ { 0 : K } )$ (thus, a total of 57 dimensional input for TransferMass, and a 58 dimensional for TransferSizeMass). Similar to Perez et al. (2020), we implement 2 baselines - the Generalist and the Specialist. The Specialist consists of an agent with identical architecture as Causally-curious agent, but without access to causal representations (i.e., receives a 56 dimensional state vector). It is initialized randomly and is trained only on the test environments, serving as a benchmark for complexity of the test tasks. It performs poorly, indicating that the test tasks are complex. The architecture of the Generalist is identical to the Specialist. Like the Specialist, the Generalist also does not have access to the causal representations, but is trained on the same set of training environments that the Causally-curious agent is trained on. The poor performance of the generalist indicates that the tasks distribution of training and test tasks differs significantly and that memorization of behaviors does not yield good transfer. We find that causally-curious agents significantly outperform the both baselines indicating that indeed, knowledge of the causal representation does aid zero-shot generalizability.
+
+![](images/daa580bc3bf3fbefcb268128d5ed8334c86cc7c830011eff056ed3ca882801b0.jpg)  
+Figure 5: Knowledge of causal factors aids transfer. We find that knowledge of the causal representation allows agents to generalize to unseen environments with high zero-shot performance. The table depicts the extra timesteps required by the Generalist in each experimental setup to match the zero-shot performance of causally-curious agent. We find that as the number of varying causal factors increase, the difference in zero-shot performance of the Causally-curious agent and the Generalist increases, showing that the CC agents are indeed robust to multiple varying causal factors.
+
+# 5 CONCLUSION
+
+We introduce causal curiosity, an intrinsic reward that allows agents to discover binary quantized representations for the causal factors that affect environments an RL agent may encounter. We show that optimizing causal curiosity rewards results in the agent performing self-supervised experiments. We find that these experiments happen to be semantically meaningful and can be used as pre-training for downstream tasks. While our work learns binary quantized causal representations, a dense encoding may improve the amount of encoded information about the causal mechanisms of the environments. We leave this to future work.
+
+# REFERENCES
+
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+
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+
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+
+# A IMPLEMENTATION DETAILS FOR EXPERIMENT DISCOVERY
+
+# A.1 PLANNER
+
+The Experiment Planner consisted of a uniform distribution planner for a horizon of 6 control signals. The planner was trained using the Cross Entropy Method Model Predictive Control (Camacho & Alba (2013); De Boer et al. (2005)) on the true environment. We sampled 40 plans per iteration from the distribution initialized to uniform U(controlLow, controlHigh). Each of the sampled plans are applied to each of the training environments and the top $10 \%$ of the plans are used to update the distribution.
+
+![](images/aed96a8d6c6d91e88dd7adbfebccc0327a701f19d62f6844b005e1919b787e3b.jpg)  
+Figure 6: Overview of training. The experiment planner generates a trajectory of actions which is applied to each of the environments with varying causal factors namely mass, shape and size of blocks. For each environment, an observation trajectory or state $S ^ { ( i ) } \in \mathbb { S }$ is obtained. A simple model with fixed low expressive power is used to approximate the generative model for $S$ . The "information overflow" $L ( S | M )$ is returned as negative reward forcing $\mathbb { S }$ to be caused by few causal factors.
+
+# A.2 TRAINING ENVIRONMENTS
+
+The training environments vary in each experiment. In Section 4.3, we utilize 3 setups, Mass, SizeMass and ShapeSizeMass. For Mass, we allow the agent to access 5 environments with masses varying from $0 . 1 ~ \mathrm { k g }$ to $0 . 5 ~ \mathrm { k g }$ . In SizeMass, the agent has access to 30 environments with masses varying uniformly from 0.1 to $0 . 5 \mathrm { k g }$ and sizes from 0.05 to 0.1 meters. Finally, in ShapeSizeMass, the agent has access to 60 environments, with masses varying uniformly from 0.1 to $0 . 5 \mathrm { k g }$ , sizes from 0.05 to 0.1 meters and shapes either being cubes or spheres. During experiment discovery, in each environment, the agent has access to the position of the block in the environment along with its quaternion orientation.
+
+The total number of causal causal factors of each environment are rather large in number due to the fact that the simulator is a complex realistic physics engine. Examples of the causal factors in the environment include gravity, friction coefficients between all on interacting surfaces, shapes, sizes and masses of blocks, control signal frequencies of the environment. However, we only vary 1 during Mass, 2 during SizeMass and 3 during ShapeSizeMass.
+
+# A.3 CURIOSITY REWARD CALCULATION
+
+We predetermine the minimum description length of the clustering model $L ( M )$ by assuming that the observations $O _ { 0 : T }$ , obtained by applying experimental behavior $a _ { 0 : T }$ are produced by a bi-modal generator distribution, where each mode corresponds to either a low or high (quantized) value of a causal factor. This also ensures that $L ( M )$ is as small as possible. The planner, eq. (5) solves the following optimization problem:
+
+$$
+\begin{array} { r l } { \underset { a _ { 0 : T } \in A ^ { T } } { \mathrm { a r g } \mathrm { m a x } } [ \mathrm { m i n } \{ d ( o _ { 0 : T } , o _ { 0 : T } ^ { \prime } ) : o _ { 0 : T } \in O , o _ { 0 : T } ^ { \prime } \in O ^ { \prime } \} - } & { \mathrm { } \mathrm { m a x } \{ d ( o _ { 0 : T } , o _ { 0 : T } ^ { \prime \prime } ) : o _ { 0 : T } ^ { \prime \prime } , o _ { 0 : T } \in O \} - } \\ { \mathrm { m a x } \{ d ( o _ { 0 : T } ^ { \prime } , o _ { 0 : T } ^ { \prime \prime } ) : o _ { 0 : T } ^ { \prime } , o _ { 0 : T } ^ { \prime \prime \prime } \in O ^ { \prime } \} ] } \end{array}
+$$
+
+the distance function $d ( \cdot , \cdot )$ in the space of trajectories is set to be Soft Dynamic Time Warping (Cuturi $\&$ Blondel (2017)). The trajectory length $T$ is 6 control steps long. The objective is a modified version of the Silhouette Score (Rousseeuw (1987)).
+
+Intuitively, Objective (8) expresses the ability of a low complexity model, assumed to be bi-modal, to encode the state $S = o _ { 0 : T }$ . If multiple causal factors control $S$ , then the Minimum Description Length of $L ( S )$ will be high. Subsequently, since $M$ is a simple model, the deviation of $S$ from $M$ will be high i.e. $L ( S | M )$ will be high resulting in a low value of the optimization objective. $O$ and $O ^ { \prime }$ correspond to clusters of outcomes which quantize the values of a causal factor isolated by $a _ { 0 : T }$ $O _ { 0 : T } , o _ { 0 : T } ^ { \prime \prime } \in S$ correspond to trajectories of states i.e. observations obtained by applying $a _ { 0 : T }$ to environments with say, low values of a causal factor while $o _ { 0 : T } ^ { \prime } , o _ { 0 : T } ^ { \prime \prime \prime } \in O ^ { \prime }$ correspond to trajectories of observations i.e. state obtained by applying $a _ { 0 : T }$ to environments with say, high values of the same causal factor. Objective (8) attempts to ensure that these clusters are far apart from each other and are tight i.e. a simple model $M$ encodes $S$ well.
+
+We further motivate how this formulation allows disentanglement of causal factors. A central assumption is that causal factors are independent, by definition, i.e. Independent Mechanisms Assumption Peters et al. (2017). Consider the outcome $S$ obtained by applying an action sequence $a _ { 0 : T }$ to a set of environments. If the action sequence $a _ { 0 : T }$ results in multiple causal factors affecting the outcome $S$ , the Kolmogorov complexity of $S$ will be high. The reason for this is that each causal factor has its own independent causal mechanism (Peters et al. (2017); Parascandolo et al. (2018)) that affects $S$ . Thus, given this independence, the information in $S$ will be a sum of the information “injected” into it from the multiple causes. Conversely, if the outcome $S$ obtained by applying an action sequence $a _ { 0 : T }$ has a lower Kolmogorov Complexity, then $S$ is caused by fewer causal factors. Causal Curiosity attempts to reduce this complexity of $S$ , by assuming a simple generative model $M$ is sufficient to encode $S$ . Thus for experimental behaviors which allow several causes to affect $S$ , the “overflow” of $S$ from $M$ will be high and subsequently the causal curiosity reward will be low. Thus, post-optimization of the objective, we arrive at an action sequence that allows for disentanglement of the causal factors.
+
+# B IMPLEMENTATION DETAILS FOR TRANSFER
+
+In Section 4.4, we show the utility of learning causal representations in 2 separate experimental setups. During TransferMass, the agent has access to 10 environments during training, with masses ranging from 0.1 to $0 . 5 \mathrm { k g }$ . At test time, the agent is required to perform the place-and-orient task masses 2 masses - $0 . 7 \mathrm { k g }$ and $0 . 7 5 \mathrm { k g }$ . During TransferSizeMass, the agent has access to 10 environments during training, with sizes from either 0.01 or $0 . 0 5 \mathrm { m }$ and masses ranging from 0.1 to $0 . 5 \mathrm { k g }$ . At test time the agent is asked to perform the task on 2 environments with masses $0 . 7 \mathrm { k g }$ and $0 . 7 5 \mathrm { k g }$ with sizes $= 0 . 0 5 \mathrm { m }$ .
+
+We find that testing with large and light blocks increase the chances of accidental goal completions. Thus, during test-time, we use environments with high masses for out-of-distribution testing. The causal representation is concatenated to the state of the environment as a contextual input and supplied to a PPO-Optimized Actor-Critic Policy. The policy network consists of 2 hidden layers with 256 and 128 units respectively. The experiments are parallelized on 10 CPUs and implemented using stable baselines (Hill et al. (2018)).
+
+The agent receives a dense reward at each time step during the maximizing external reward phase (Figure 1), the negative of the distance of the block from the goal position scaled by factor of 1000. The control signal was repeated 10 times to the actuators of the motors on each finger.
+
+# C IMPLEMENTATION DETAILS FOR SECTION 4.2
+
+In section 4.2, we study how the acquired experimental behaviors obtained through Causal Curiosity can be used as pre-training for a variety of downstream tasks. The Vanilla CEM depicts the cost of training an experiment planner from scratch to maximize an external dense reward where the agent minimizes the distance between the position of a block in an environment from the goal in the Lifting setup and imparts a velocity to the block along a particular direction in the Travel setup.
+
+$$
+R ( a _ { 0 : T } ) = - \sum _ { t } d i s t ( g o a l _ { t } - b l o c k _ { t } )
+$$
+
+The second baseline (Additive Reward) studies the setup when the agent receives both the curiosity signal and the external reward and attempts to maximize both. The agent receives access all the
+
+training environments with varying causal factors and must simultaneously maximize both curiosity and the task reward. The equation below shows the reward maximized for the Lifting task.
+
+$$
+\begin{array} { r l r } { \displaystyle R ( a _ { 0 : T } ) = \sum _ { e n v s } \sum _ { t } ^ { T } - d i s t ( g o a l _ { t } - b l o c k _ { t } ) + } & \\ { \displaystyle [ \operatorname* { m i n } \{ d ( o _ { 0 : T } , o _ { 0 : T } ^ { \prime } ) : o _ { 0 : T } \in O , o _ { 0 : T } ^ { \prime } \in O ^ { \prime } \} - } & { \operatorname* { m a x } \{ d ( o _ { 0 : T } , o _ { 0 : T } ^ { \prime \prime } ) : o _ { 0 : T } ^ { \prime \prime } , o _ { 0 : T } \in O \} - } & \\ { \operatorname* { m a x } \{ d ( o _ { 0 : T } ^ { \prime } , o _ { 0 : T } ^ { \prime \prime \prime } ) : o _ { 0 : T } ^ { \prime } , o _ { 0 : T } ^ { \prime \prime \prime } \in O ^ { \prime } \} ] } & \end{array}
+$$
+
+The curious agent first acquired the experimental behavior by interacting with multiple environments with varying causal factors. The lifting skill was obtained during Mass, when the agent attempted to differentiate between multiple blocks of varying mass. The curious agent trained for 600,000 time steps on the curiosity reward. The acquired behavior was then applied to the downstream lifting task and fine tuned to external rewards. The Vanilla CEM baseline had an identical structure to that of the Curious agent, and received only external reward as in Equation (9). The additive agent simultaneously optimized both external reward and the curiosity reward as in Equation (10).
+
+# D INTUITION FOR DEFINITION OF CAUSAL FACTORS
+
+We begin with a simple example of a person walking on earth. This person experiences various physical processes while interacting in her world, for example gravity, friction, wind etc. These physical processes affect the outcome of interactions of the person with her environment. For example, while jumping on earth, the human experiences gravity which affects the outcome of her jump, the fact that she falls back to the ground. Additionally, these physical processes (or causal mechanisms) are parameterized by causal factors, for example, acceleration constant due to gravity $g = 9 . 8 m / s ^ { 2 }$ on earth, or coefficients of friction between her feet and the ground which assume particular numerical values.
+
+These causal factors may vary across multiple environments. For example, the person may walk on sand or on ice, surfaces with varying frictional values. Thus the outcome of running on such surfaces will vary, running on sand will require significant effort, while running on ice may result in the person slipping. Thus the coefficient of friction between the person’s feet and the surface she walks on affects the outcome of a particular behavior in said environment. In our definition, $h _ { j }$ are causal factors such friction with some particular coefficient of friction, or gravity with acceleration constant $g$ or other. $H$ is the global set containing all such causal factors.
+
+Now we ask the question (which we subsequently answer), given multiple environments, how would a human characterize each of them depending on the value of a causal factor? Through experimental behaviors. The human in the above example would attempt to run in each of the environments she encountered, be it on sand, on ice, in mud etc. If she slipped in an environment, she would characterize it as slippery. If she didn’t, she would characterize it as non-slippery. We attempt to equip our agent with similar logic. The “sequence of actions” $\left( { { a _ { 0 : T } } } \right)$ described in our paper corresponds to the human running. The state $S ^ { ( i ) }$ in the environment $e ^ { ( i ) }$ consisting of the sequence of observations $\left( o _ { 0 : T } \right)$ corresponds to the outcome of running. $S$ might belong to either of the clusters of outcomes $S$ or $S ^ { \prime }$ corresponding to slipping or not slipping.
+
+# E SCALABILITY LIMITATION
+
+We utilize the extremely popular One-Factor-at-a-time (OFAT) general paradigm of scientific investigation, as an inspiration for our method. In the case of many hundreds of causal factors, the complexity of this method will scale exponentially. However, we believe that this would indeed be the case given a human experimenter attempting to discover the causation in any system she is studying. Learning about causation is a computationally expensive affair. We point the reader towards a wealth of material on the design of scientific experiments and more specifically the lack of scalability of OFAT (Fisher (1936); Hicks (1964); Czitrom (1999)). Nevertheless, OFAT remains the de facto standard for scientific investigation.
+
+1: Input: Unseen Test Environment env, trained Planner and Causal Inference Module   
+2: Initialize causal $R e p = [ ]$   
+3: Initialize training environment set Envs   
+4: for $\mathbf { k }$ in range(K) do   
+5: Reset env   
+6: Sample experimental behavior $a _ { 0 : T } \sim \mathrm { C E M } ( \cdot | \ c a u s a l R e p )$   
+7: Apply $a _ { 0 : T }$ to env . Exploration Phase   
+8: Collect $S = o _ { 0 : T }$   
+9: Use learnt $q _ { M } ( z | S )$ for cluster assignment i.e. $z _ { k } = q _ { M } ( z | s , c a u s a l R e p )$   
+10: Append $z _ { k }$ to causalRep . Causal Inference Module   
+11: Learn a policy conditioned on causal factors $a _ { t } \sim \pi ( \cdot | o _ { t } , z _ { 0 : K } )$ to maximize external reward.

parse/train/Q2iaAc-4I1v/Q2iaAc-4I1v_content_list.json

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+    {
+        "type": "text",
+        "text": "CAUSAL CURIOSITY: RL AGENTS DISCOVERING SELF-SUPERVISED EXPERIMENTS FOR CAUSAL REPRESENTATION LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            260,
+            544,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Humans show an innate ability to learn the regularities of the world through interaction. By performing experiments in our environment, we are able to discern the causal factors of variation and infer how they affect the dynamics of our world. Analogously, here we attempt to equip reinforcement learning agents with the ability to perform experiments that facilitate a categorization of the rolled-out trajectories, and to subsequently infer the causal factors of the environment in a hierarchical manner. We introduce a novel intrinsic reward, called causal curiosity, and show that it allows our agents to learn optimal sequences of actions, and to discover causal factors in the dynamics. The learned behavior allows the agent to infer a binary quantized representation for the ground-truth causal factors in every environment. Additionally, we find that these experimental behaviors are semantically meaningful (e.g., to differentiate between heavy and light blocks, our agents learn to lift them), and are learnt in a self-supervised manner with approximately 2.5 times less data than conventional supervised planners. We show that these behaviors can be re-purposed and fine-tuned (e.g., from lifting to pushing or other downstream tasks). Finally, we show that the knowledge of causal factor representations aids zero-shot learning for more complex tasks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+            564
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Discovering causation in environments an agent might encounter remains an open and challenging problem for causal reinforcement learning (Schölkopf (2015), Bengio et al. (2013), Schölkopf (2019)). Most approaches take the form of BAMDPs (Bayes Adaptive Markov Decision Processes) (Zintgraf et al. (2019)) or Hi-Param MDP (Hidden Parameter MDPs) (Doshi-Velez & Konidaris (2016); Yao et al. (2018); Killian et al. (2017); Perez et al. (2020)) which condition the transition $p ( s _ { t + 1 } | s _ { t } , a _ { t } ; H )$ and/or reward function $R ( r _ { t + 1 } | s _ { t } , a _ { t } , s _ { t + 1 } ; H )$ of each environment on hidden parameters (also referred to as causal factors in some of the above studies). Let $s \\in { \\mathcal { S } } , a \\in { \\mathcal { A } }$ $r \\in \\mathcal { R }$ , $H \\in { \\mathcal { H } }$ where $s , { \\mathcal { A } } , { \\mathcal { R } }$ , and $\\mathcal { H }$ are the set of states, actions, rewards and feasible hidden parameters. In the physical world and in the case of mechanical systems, examples of the parameter $h _ { j } \\in \\mathcal { H }$ include gravity, coefficients of friction, masses and sizes of objects. Typically, $H$ is treated as a latent variable for which an embedding is learned during training, using variational methods (Kingma et al. (2014); Ilse et al. (2019)). Let $s _ { 0 : T }$ be the entire state trajectory of length $T$ . Similarly, $a _ { 0 : T }$ is the sequence of actions applied during that trajectory by the agent that results in $s _ { 0 : T }$ . In an environment parameterized by these causal factors, these latent variable approaches define a probability distribution over the entire sequence of (rewards, states, actions) conditioned on a latent $z$ as $p ( r _ { 0 : T } , s _ { 0 : T } , a _ { 0 : T - 1 } ; z )$ that factorizes as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/5f07918a1ce73cbad639a0da140097422f02d51aeeb8e0638df921e361568b4c.jpg",
+        "text": "$$\n\\prod _ { i = 1 } ^ { T - 1 } p ( r _ { t + 1 } | s _ { t } , a _ { t } , s _ { t + 1 } , z ) p ( s _ { t + 1 } | s _ { t } , a _ { t } , z ) p ( a _ { t } | s _ { t } , z )\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "due to the Markov assumption. At test time, the agent infers the causal factor associated with its environment by observing the trajectories produced by its initial actions that can be issued by any policy such as model-based reinforcement learning. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In practice, however, discovering causal factors in a physical environment is prone to various challenges that are caused by the disjointed nature of the influence of these factors on the produced trajectories. More specifically, at each time step, the transition function is affected by a subset of global causal factors. This subset is implicitly defined on the basis of the current state and the action taken. For example, if a body in an environment loses contact with the ground, the coefficient of friction between the body and the ground no longer affects the outcome of any action that is taken. Likewise, the outcome of an upward force applied by the agent to a body on the ground is unaffected by the friction coefficient. We can therefore take advantage of this natural discontinuity to discern causal factors. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Without knowledge of how independent causal mechanisms affect the outcome of a particular action in a given state in an environment, it becomes impossible for the agent to conclude where the variation it encountered came from. Unsurprisingly, Hi-Param and BAMDP approaches fail to learn a disentangled embedding for the causal factors, making their behaviors uninterpretable (Perez et al. (2020)). For example, if, in an environment, a body remains stationary under a particular force, the Hi-Param or BAMDP agent may apply a higher force to achieve its goal of perhaps moving the body, but will be unable to conclude whether the \"un-movability\" was caused by high friction or high mass of the body. Additionally, these approaches require human-supervised reward engineering, making it difficult to apply them outside of the simulated environments they are tested in. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our goal is, instead of focusing on maximizing reward for some particular task, to allow agents to discover causal processes through exploratory interaction. During training, our agents discover self-supervised experimental behaviors which they apply to a set of training environments. These behaviors allow them to learn about the various causal mechanisms that govern the transitions in each environment. During inference in a novel environment, they perform these discovered behaviors sequentially and use the outcome of each behavior to infer the embedding for a single causal factor (Figure 1). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The main challenge while learning a disentangled representation for the causal factors of the world is that several causal factors may affect the outcome of behaviors in each environment. For example, when pushing a body on the ground, the outcome, i.e., whether the body moves, or how far the body is pushed, depends on several factors, e.g., mass, shape and size, frictional coefficients, etc. However, if, instead of pushing on the ground, the agent executes a perfect grasp-and-lift behavior, only mass will affect whether the body is lifted off the ground or not. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Thus, it is clear that not all experimental behaviors are created equal and that the outcomes of some behaviors are caused by fewer causal factors than others. Our agents learn these behaviors without supervision using causal curiosity, an intrinsic reward. The outcome of a single such experimental behavior is then used to infer a binary quantized embedding describing the single isolated causal factor. Even though causal factors of variation in a physical world are easily identifiable to humans, a concrete definition is required to back up our proposed method. We conjecture that the causality of a factor of variation depends on the available actions to the agent. If the set of actions that an agent can take is very limited, there is no way for it to discern a diverse set of causal factors in the environment. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Definition 1 (Causal factors). Consider the POMDP $( \\mathcal { O } , \\mathcal { S } , \\mathcal { A } , p , r )$ with observation space $\\mathcal { O }$ , state space $s$ , action space $\\mathcal { A }$ , the transition function $p$ , and the reward function $r$ . Let $o _ { 0 : T } \\in \\mathcal { O } ^ { T }$ denotes a trajectory of observations and $T$ be the length of such trajectories. Let $d ( \\cdot , \\cdot ) : \\mathcal { O } ^ { T } \\times \\mathcal { O } ^ { T } \\to \\mathbb { R } _ { + }$ be a distance function defined on the space of trajectories of length $T$ . The set $H = \\{ h _ { 1 } , h _ { 2 } , \\ldots , h _ { k } \\}$ is called a set of $\\epsilon -$ causal factors if for every $h _ { j } \\in H$ , there exists a unique sequence of actions $a _ { 0 : T }$ that clusters the state trajectories into two sets $S$ and $S ^ { \\prime }$ such that ",
+        "bbox": [
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+        "img_path": "images/6dbdf27818f1a75374388ab52359782af6a86e2e974d8a56f8b9c45ab6ad7fe5.jpg",
+        "text": "$$\n\\operatorname* { m i n } \\{ d ( o _ { 0 : T } , o _ { 0 : T } ^ { \\prime } ) : o _ { 0 : T } \\in O , o _ { 0 : T } ^ { \\prime } \\in O ^ { \\prime } \\} > \\epsilon\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "and that $h _ { j }$ is the cause of the trajectory of states obtained i.e., ",
+        "bbox": [
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+    {
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+        "img_path": "images/fb96b4ed1f6b5b8d9e540a07c1941c8af72956bc332b79b894efb3ac455b9721.jpg",
+        "text": "$$\np ( o _ { 0 : T } | d o ( h _ { j } = k ) , a _ { 0 : T } ) \\neq p ( o _ { 0 : T } | d o ( h _ { j } = k ^ { \\prime } ) , a _ { 0 : T } ) \\forall k \\neq k ^ { \\prime }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Intuitively, a factor of variation affecting a set of environments is called causal if there exists a sequence of actions available to the agent where the resultant trajectories are clustered into two or more sets (for simplicity here we assume binary clusters). This is analogous to the human ability to conclude whether objects are heavy or light, big or small. For a gentle introduction to the intuition about this definition, we refer the reader to Appendix D. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "According to Def. 1, a causal factor is a parameter in the environment whose value, when intervened on (i.e. varied) over a set of values, results in trajectories of states that are divisible into disjoint clusters under a particular sequence of actions. These clusters represent the quantized values of the causal factor. For example, mass, which is a causal factor of a body, under an action sequence of a grasping and lifting motion, results in 2 clusters, liftable (low mass) and not liftable (high mass). However, such an action sequence is not known in advance. Therefore, discovering a causal factor in the environment boils down to finding a sequence of actions that makes the effect of that factor prominent by producing clustered trajectories for different values of that environmental factor. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/bd8a45b5a87224e98efb1cf1c4411698bcdd09b5472eb0da2a01ba7932658e4e.jpg",
+        "image_caption": [
+            "Figure 1: Overview of Inference. The exploration loop produces a series of $K$ experiments allowing the agent to infer the representations for $K$ causal factors. After exploration, the agent utilizes the acquired knowledge for downstream tasks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Using the above, we propose an intrinsic reward, which allows our agents to discover experimental behaviors which are semantically meaningful and can be used to re-train for downstream tasks, resulting in high sample efficiency. Our work, therefore, forms an important link between structured representation learning and skill discovery, two largely disjoint fields in RL, which stand to benefit from each other. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The contributions of the work are as follows: ",
+        "bbox": [
+            176,
+            522,
+            468,
+            537
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• We equip agents with the ability to perform experiments and behave meaningfully in a set of environments in an unsupervised manner. These behaviors can expose or obfuscate specific independent causal mechanisms that occur in the world of the agent, allowing the agent to learn about each in the absence of the others, an important human behavioral trait.   \n• We introduce an intrinsic reward, causal curiosity, which allows our agents to discover these behaviors without human-engineered rewards. The outcomes of the experiments are used to learn a disentangled quantized binary representation for the causal factors of the environment, analogous to the human ability to conclude whether objects are light/heavy, big/small etc.   \n• Through extensive experiments, we conclude that knowledge of the causal factors aids sample efficiency in two ways - first, that the knowledge of the causal factors aids transfer learning across multiple environments, and, second, that the experimental behaviors acquired can be repurposed for downstream tasks. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 METHOD ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            785,
+            282,
+            801
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Consider a set of $N$ environments $\\mathcal { E }$ with $e ^ { ( i ) } \\in \\mathcal { E }$ where $e ^ { ( i ) }$ denotes the $i ^ { t h }$ environment. ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The letter $H$ is overloaded. While $H$ is a set of global causal factors (as defined in Def. 1) such that   \n$h _ { j } \\in H$ , each causal factor $h _ { j }$ is itself a random variable which assumes a particular value for every ion of an environment. T. For each environment $e ^ { ( i ) }$ is represented by a set of causal factorsepresents the disentangled embedding   \n$\\{ h _ { j } ^ { ( i ) } \\forall j \\}$ $e ^ { ( i ) }$ $( z _ { ( 0 ) } ^ { ( i ) } , z _ { ( 1 ) } ^ { ( i ) } . . . z _ { ( K - 1 ) } ^ { ( i ) } )$   \nvector, such that z(i)(j) $h _ { j } ^ { ( i ) }$ . ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Training Scheme ",
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+        "text": "1: Initialize $j = 0$ 2: Initialize training environment set Envs 3: for iteration m to M do . Experiment Planner Training Loop 4: Sample experimental behavior $a _ { 0 : T } \\sim \\mathbf { C E M } ( \\cdot )$ 5: for $i ^ { \\star h }$ env in Envs do 6: Apply $a _ { 0 : T }$ to env 7: $\\mathbf { C o l l e c t } \\ S ^ { ( } i ) = O _ { 0 : T } ^ { ( i ) }$ 8: Reset env 9: Calculate $- L ( S | M )$ given that $M$ is bimodal clustering model $\\triangleright$ Calculate Curiosity   \n10: Update CEM(·) distribution with highest reward trajectories   \n11: Use learnt $q _ { M } ( z | S )$ for cluster assignment of each env in Envs i.e. $z _ { j } ^ { ( i ) } = q _ { M } ( z | S ^ { ( i ) } )$   \n12: Update $j = j + 1$   \n13: Repeat from step 2, first setting $E n v s = \\{ e ^ { ( i ) } : z _ { j - 1 } ^ { ( i ) } = 0 \\}$ and then, setting $E n v s = \\{ e ^ { ( i ) } : z _ { j - 1 } ^ { ( i ) } = 1 \\}$ ",
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+    {
+        "type": "text",
+        "text": "2.1 TRAINING THE EXPERIMENT PLANNER ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To learn about causal processes through interaction, the agent must produce a sequence of actions $a _ { 0 : T - 1 }$ that we call experimental behavior, which, when applied to environment $e ^ { ( i ) } \\in \\mathcal { E }$ , produces a sequence of observations (state) $s ^ { ( i ) } = [ o _ { 0 } ^ { ( i ) } , o _ { 1 } ^ { ( i ) } . . o _ { T } ^ { ( i ) } ]$ , which is then used to infer the value of the embedding for a single causal factor $z _ { ( j ) } ^ { ( i ) }$ . ",
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+    {
+        "type": "text",
+        "text": "We motivate this using model selection criterion. Normally in model selection applications, the observations are fixed and the goal is to find a model $M ^ { * }$ that is closest to reality, as represented by: ",
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+        "type": "equation",
+        "img_path": "images/674320465fbf664b9f9d6fc63df220b40acb91a1a1b9f51d56d1755078c571b4.jpg",
+        "text": "$$\nM ^ { * } = \\arg \\operatorname* { m i n } _ { M } ( L ( M ) + L ( S | M ) )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $L ( \\cdot )$ is the description length. However, here, the situation is reversed. A simple bi-modal clustering model is fixed, motivated by Definition 1. Then, the agent is motivated to produce actions that result in observations that are best explained by this model. These discovered action sequences are the experimental behaviors we desire. ",
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+    {
+        "type": "equation",
+        "img_path": "images/66d979c69227e9deb138fe1487097ef339ba92be5022f866fa33b2a2abbc751e.jpg",
+        "text": "$$\n\\begin{array} { r } { \\boldsymbol { a } _ { 0 : T } ^ { * } = \\underset { \\boldsymbol { a } _ { 0 : T } } { \\arg \\operatorname* { m i n } } ( L ( \\boldsymbol { M } ) + L ( \\boldsymbol { S } | \\boldsymbol { M } ) ) } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where each observed trajectory $S = S ( a _ { 0 : T } )$ is a function of the action sequence. As mentioned earlier, the model is fixed in this formulation; hence, the first term $L ( M )$ is constant and not a function of the actions. $- L ( S | M )$ that is fed back to the RL agent as a reward function to maximize. We regard this reward function as causal curiosity. ",
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+        "type": "text",
+        "text": "Note that since each causal factor has its own independent causal mechanism that causes $S$ , the MDL of $S$ will be higher if multiple causal factors cause $S$ . On the contrary, if the agent produces actions which result in an $S$ that is easily explained by a low-capacity bi-modal model $M$ , then it will imply that $S$ is caused by fewer causal factors. Consequently, the causal curiosity reward for such an action sequence, $- L ( S | M )$ , will be high. Therefore, causal curiosity favors experimental behaviors that result in observations caused by few causal factors - thereby allowing us to use $S$ to infer a representation for a single causal factor. For details, please refer Appendix A. ",
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+    {
+        "type": "text",
+        "text": "2.2 CAUSAL INFERENCE MODULE ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "By maximizing the causal curiosity reward it is possible to achieve behaviors which result in trajectories of states only caused by a single hidden parameter. However, we wish to use the outcome of performing these experimental behaviors in each environment to infer a representation for the causal factor isolated by the experiment in question. ",
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+        "type": "text",
+        "text": "We achieve this through cluster membership. After training the Model Predictive Control Planner (Camacho & Alba (2013)), we sample from an action sequence $a _ { 0 : T }$ and apply it to each of the training environments. The learnt clustering model $M$ is then used to infer a representation for each environment using the collected outcome $\\bar { S } ^ { ( i ) }$ obtained by applying $a _ { 0 : T }$ to each environment. ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/bf8c27cdf0e531a4b0528bac7ee0b50ba01df0e0a0015f5ac8d5cbc7c69b4a76.jpg",
+        "text": "$$\nz _ { j } ^ { ( i ) } = q _ { M } ( z | S ^ { ( i ) } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This corresponds to Step 11 of Algorithm (1). The representation learnt is binary in nature corresponding to the quantization of the continuous spectrum of values a causal factor takes in the training set into high and low values. Note however that a binary quantized embedding is not a necessary part of our method. A dense embedding may alternatively be learnt here similar to (Perez et al. (2020); Zintgraf et al. (2019)) using approximate variational inference. However, performing interventions on a dense embedding (Section 2.3) increases the computational complexity exponentially. Balancing space and time complexity, we report results using the quantized binary form of Equation (6). We discuss the implications of increasing the complexity of $z _ { j } ^ { ( i ) }$ in the discussion. ",
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+    {
+        "type": "text",
+        "text": "2.3 INTERVENTIONS ON BELIEFS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Having learnt about the effects of a single causal factor of the environment we wish to learn such experimental behaviors for each of the remaining hidden parameters that may vary in an environment. To achieve this, in an ideal setting, the agent would require access to the generative mechanism of the environments it encounters. Ideally, it would hold the values of the causal factor already learnt about constant i.e. $d o ( h _ { j } = c o n s t a n t )$ , and intervene over (vary the value of) another causal factor over a set of values $K$ i.e. $d o ( h _ { j } = k )$ such that $k \\in K$ . For example, if a human scientist were to study the effects of a causal factor, say mass of a body, she would hold the values of all causal factors constant, (interact with cubes of the same size and external texture) and vary only mass to see how it affects the outcome of specific behaviors she applies to each body. ",
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+    {
+        "type": "text",
+        "text": "However, in the real-world the agent does not have access to the generative mechanism of the environments it encounters, but merely has the ability to act in them. Thus, it can intervene on the representations of a causal factor of the environment i.e. $d o ( z _ { i } = c o n s t a n t )$ . For, example having learnt about gravity, the agent picks all environments it believes have low gravity, and uses them to learn about a separate causal factor say, friction. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This corresponds to Step 13 of Algorithm (1). Thus, to learn about the $j ^ { t h }$ causal factor, we repeat steps 3 onwards on each of the clusters obtained for the $j - 1 ^ { t h }$ . ",
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+    {
+        "type": "equation",
+        "img_path": "images/ab456d01140a053c903457f38c5c5b312456e2a5d8114c0d19ff26d781e35c6d.jpg",
+        "text": "$$\nE n v s = \\{ e ^ { ( i ) } : z _ { j - 1 } ^ { ( i ) } = k \\} , k \\in \\{ 0 , 1 \\}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This process continues in the form of a tree (Figure 4), where for each cluster of environments, a new experiment learns to split the cluster into 2 sub-clusters depending on the value of another hidden parameter. At level $n$ , the agent produces $2 ^ { n }$ experiments and inference models, having already intervened on the binary quantized representations of $n$ causal factors. ",
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+    {
+        "type": "text",
+        "text": "3 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Doshi-Velez & Konidaris (2016) define a class Markov Decision Processes where transition probabilities $p ( s _ { t + 1 } | s _ { t } , a _ { t } ; \\theta )$ depend on a hidden parameter $\\theta$ , whose value is not observed, but its effects are felt. Killian et al. (2017) and Yao et al. (2018) utilize these Hidden Parameter MDPs (Markov Decision Processes) to enable efficient policy transfer, assuming that transition probabilities across states are a function of hidden parameters. Perez et al. (2020) relax this assumption, allowing both transition probabilities and reward functions to be functions of hidden parameters. Zintgraf et al. (2019) approach the problem from a Bayes-optimal policy standpoint, defining transition probabilities and reward functions to be dependent on a hidden parameter characteristic of the MDP in consideration. We utilize this setup to define causal factors. ",
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+    {
+        "type": "text",
+        "text": "Substantial attempts have been made at unsupervised disentanglement, most notably, the $\\beta$ -VAE Higgins et al. Burgess et al. (2018), where a combination of factored priors and the information bottleneck force disentangled representations. Kim & Mnih (2018) enforce explicit factorization of the prior without compromising on the mutual information between the data and latent variables, a shortcoming of the $\\beta$ -VAE. Chen et al. (2018) factor the KL divergence into a more explicit form, highlighting an improved objective function and a classifier-agnostic disentanglement metric. Locatello et al. (2018) show theoretically that unsupervised disentanglement (in the absence of inductive biases) is impossible and highly unstable, susceptible to random seed values. They follow this up with Locatello et al. (2020) where they show, both theoretically and experimentally, that pair-wise images provide sufficient inductive bias to disentangle causal factors of variation. However, these works have been applied to supervised learning problems whereas we attempt to disentangle the effects of hidden variables in dynamical environments, a relatively untouched question. ",
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+        "type": "text",
+        "text": "Curiosity for robotics is not a new area of research. Schmidhuber (2006), Ngo et al. (2012), Pathak et al. (2017) describe curiosity as the motivation behind the behavior of an agent in an environment for which the outcome is unpredictable, i.e., an intrinsic reward that motivates the agent to explore the unseen portions of the state space (and subsequent transitions). While causal curiosity is an intrinsic reward, it differs from these traditional definitions of curiosity in that it motivates the agent to produce structure in the outcome of its behavior. ",
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+        "type": "text",
+        "text": "4 EXPERIMENTS ",
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+        "type": "text",
+        "text": "Our work has 2 main thrusts - the discovered experimental behaviors and the representations obtained from the outcome of the behaviors in environments. The experimental behaviors are tied to contributions 1 and 2 in the Introduction. The causal factors allow us to achieve contribution 3 in the Introduction. We visualize these learnt behaviors and verify that they are indeed semantically meaningful and interpretable. We quantify the utility of the learned behaviors by using the behaviors as pre-training for a downstream task. In our experimental setup, we verify that these behaviors are indeed invariant to all other causal factors except one. ",
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+        "type": "text",
+        "text": "We visualize the representations obtained using these behaviors and verify that they are indeed the binary quantized representations for each of the ground truth causal factors that we manipulated in our experiments. Finally, we verify that the knowledge of the representation does indeed aid transfer learning and zero-shot generalizability in downstream tasks. ",
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+        "type": "text",
+        "text": "Causal World We use the Causal World Simulation (Ahmed et al. (Under submission 2020)) based on the Pybullet Physics engine to test our approach. The simulator consists of a 3-fingered robot, with 3 joints on each finger. We constrain each environment to consist of a single object that the agent can interact with. The causal factors that we manipulate for each of the objects are size, shape and mass of the blocks. The simulator allows us to capture and track the positions and velocities of each of the movable objects in an environment. While, for most experiments, the 3D position and 3D pose of the blocks is used as the state at each time step, we perform ablation studies where less information is provided to the agent. ",
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+    {
+        "type": "text",
+        "text": "4.1 VISUALIZING DISCOVERED BEHAVIORS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We would like to analyze whether the discovered experimental behaviors are human interpretable, i.e., are the experimental behaviors discovered in each of the setups semantically meaningful? We find that our agents learn to perform several useful behaviors without any supervision. For instance, to differentiate between objects with varying mass, we find that they acquire a perfect grasp-and-lift behavior with an upward force. In other random seed experiments, the agents learn to lift the blocks by using the wall of the environment for support. To differentiate between cubes and spheres, the agent discovers a pushing behavior which gently rolls the spheres along a horizontal direction. Qualitatively, we find that these behaviors are stable and predictable. See videos of discovered behaviors here (website under construction). ",
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+        "type": "text",
+        "text": "Concurrent with the objective they are trained on, we find that the acquired behaviors impose structure on the outcome when applied to each of the training environments. The outcome of each experimental behavior on the set of training environments results in dividing it into 2 subsets. These subsets correspond to the binary quantized values of a single factor, e.g., large or small, while being invariant to the values of other causal factors of the environments. We also perform ablation studies where instead of providing the full state vector, we provide only one coordinate (e.g., only x, y or z coordinate of the block). We find that causal curiosity results in behaviors that differentiate the environments based on outcomes along the direction provided. For example, when only the $\\mathbf { X }$ coordinate was provided, the agent learned to evaluate mass by applying a pushing behavior along the x direction. Similarly, a lifting behavior was obtained when only the z coordinate was supplied to the curiosity module (Figure 2). ",
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+    {
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+        "img_path": "images/88d18b1b70f86f723c5c9a304545f39a38019220f7b369ab3dd9c9c67412ae78.jpg",
+        "image_caption": [
+            "Figure 2: Examples of discovered behaviors. The agent discovers experimental behaviors that allow it to characterize each environmental object in a binary manner, e.g., heavy/light, big small, rollable/not rollable, etc. These behaviors are acquired without any external supervision by maximizing the causal curiosity reward. A, B, C correspond to self-discovered toss, lift-and-spin and roll behaviors respectively. D shows an ablation study, where the agent is only provided the z coordinate of the block in every environment. Each line corresponds to one environment and the z coordinate of the block is plotted with time when the discovered behavior is applied. It learns a lifting behavior, where cluster 1 represents the heavy blocks (z coordinate does not change much) and cluster 2 represents the light blocks $\\mathbf { z }$ increases as block is lifted and then falls when dropped and subsequently increases again when it bounces). "
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+    {
+        "type": "text",
+        "text": "4.2 UTILITY OF LEARNED BEHAVIORS FOR DOWNSTREAM TASKS ",
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+        "type": "text",
+        "text": "While the behaviors acquired are semantically meaningful, we would like to quantify their utility as pre-training for downstream tasks. We analyze the performance on Lifting where the agent must grasp and lift a block to a predetermined height and Travel, where the agent must impart a velocity to the block along a predetermined direction. We re-train the learnt planner using an external reward for these tasks (Curious). We implement a baseline vanilla Cross Entropy Method optimized Model Predictive Control Planner (De Boer et al. (2005)) trained using the identical reward function and compare the rewards per trajectory during training. We also run a baseline (Additive reward) which explores whether the agent recieves both the causal curiosity reward and the external reward. We find high zero-shot generalizability and quicker convergence as compared to the vanilla CEM planner (Figure ??). We find that maximizing the curiosity reward in addition to simultaneously maximizing external rewards results in suboptimal performance due to our formulation of the curiosity reward. To maximize curiosity, the agent must discover behaviors that divide environments into 2 clusters. Thus in the context of the experimental setups, this corresponds to acquiring a lifting/pushing behavior that allows the agent to lift/impart horizontal velocity to blocks in half of the environments, while not being able to do so in the remaining environments. However, the explicit external reward incentivizes the agent to lift/impart horizontal velocity blocks in all environments. Thus these competing objectives result in sub-par performance. ",
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+    {
+        "type": "text",
+        "text": "4.3 VISUALIZATION OF HIERARCHICAL BINARY LATENT SPACE ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our agents discover a disentangled latent space such that they are able to isolate the sources of causation of the variability they encounters in their environments. For every environment, they learn a disentangled embedding vector which describes each of the causal factors. ",
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+    {
+        "type": "text",
+        "text": "To show this, we use 3 separate experimental setups - Mass, SizeMass and ShapeSizeMass where each of the causal factors are allowed to vary over a range of discrete values. During Mass, the agent is allowed access to 5 environments with objects having the same shape (cuboids) and size but differing only in mass. During SizeMass, the agent has access to 30 environments with cuboids having sizes and masses ranging over 6 and 5 values respectively. Finally, during ShapeSizeMass, the agent has access to 60 environments with objects having shapes, sizes and masses ranging over 2, 6, 5, and values respectively. ",
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+        "image_caption": [
+            "Figure 3: Utility of discovered behaviors. We find that the behaviors discovered by the agents while optimizing causal curiosity show high zero-shot generalizability and converge to the same performance as conventional planners for downstream tasks. We also analyze the worst case performance and find that the pre-training ensures better performance than random initialization. The table compares the time-steps of training required on an average to acquire a skill with the time steps required to learn a similar behavior using external reward. We find that the unsupervised experimental behaviors are approximately 2.5 times more sample efficient. We also find that maxizing both curiosity and external reward in our experimental setups results in sub-optimal results. ",
+            "Figure 4: Discovered hierarchical latent space. The agent learns experiments that differentiate the full set of blocks in ShapeSizeMass into hierarchical binary clusters. At each level, the environments are divided into 2 clusters on the basis of the value of a single causal factor. We also show the principal components of the trajectories in the top left. For brevity, the full of extent of the tree is not depicted here. For each level of hierarchy $k$ , there are $2 ^ { k }$ number of clusters. "
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+    {
+        "type": "text",
+        "text": "During training, the agent discovers a hierarchical binary latent space (Figure 4), where each level of hierarchy corresponds to a single causal factor. The binary values at each level of hierarchy correspond to the high/low values of the causal factor in question. To our knowledge, we obtain the first interpretable latent space describing the various causal processes in the environment of an agent. This implies that it learns to quantify each physical attribute of the blocks it encounters in a completely unsupervised manner. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.4 KNOWLEDGE OF CAUSAL FACTORS AIDS TRANSFER ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Next, we test whether knowledge of the causal factors does indeed aid transfer and zero-shot generalizability. To this end, we supply the representations obtained by the agent during the experimental behavior phase as input to a policy network in addition to the state of the simulator, and train it for a place-and-orient downstream task (Figure 1). We define 2 experimental setups - TransferMass and TransferSizeMass. In Mass, the agent is given access to 10 environments, with 10 varying values of mass. In TransferSizeMass, the agent is allowed access to 10 environments, with 2 and 5 values of size and mass respectively. In both setups, the agent learns about the varying causal mechanisms by optimizing causal curiosity. Subsequently, using the causal representation along with the state for each environment, it is trained to maximize external reward. For details of the setup, please see Appendix B. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "After training, the agents are exposed to a set of unseen test environments, where we analyze their zero-shot generalizability. These test environments consist of unseen masses and sizes and their unseen combinations. This corresponds to \"Strong Generalization\" as defined by Perez et al. (2020). We report results averaged over 10 random seeds. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "For each setup, we train a PPO-optimized Actor-Critic Policy (referred to as Causally-curious agent) with access to the causal representations and a 56 dimensional state vector from the environment i.e., $a _ { t } \\sim \\pi ( \\cdot | s _ { t } , z _ { 0 : K } )$ (thus, a total of 57 dimensional input for TransferMass, and a 58 dimensional for TransferSizeMass). Similar to Perez et al. (2020), we implement 2 baselines - the Generalist and the Specialist. The Specialist consists of an agent with identical architecture as Causally-curious agent, but without access to causal representations (i.e., receives a 56 dimensional state vector). It is initialized randomly and is trained only on the test environments, serving as a benchmark for complexity of the test tasks. It performs poorly, indicating that the test tasks are complex. The architecture of the Generalist is identical to the Specialist. Like the Specialist, the Generalist also does not have access to the causal representations, but is trained on the same set of training environments that the Causally-curious agent is trained on. The poor performance of the generalist indicates that the tasks distribution of training and test tasks differs significantly and that memorization of behaviors does not yield good transfer. We find that causally-curious agents significantly outperform the both baselines indicating that indeed, knowledge of the causal representation does aid zero-shot generalizability. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/daa580bc3bf3fbefcb268128d5ed8334c86cc7c830011eff056ed3ca882801b0.jpg",
+        "image_caption": [
+            "Figure 5: Knowledge of causal factors aids transfer. We find that knowledge of the causal representation allows agents to generalize to unseen environments with high zero-shot performance. The table depicts the extra timesteps required by the Generalist in each experimental setup to match the zero-shot performance of causally-curious agent. We find that as the number of varying causal factors increase, the difference in zero-shot performance of the Causally-curious agent and the Generalist increases, showing that the CC agents are indeed robust to multiple varying causal factors. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We introduce causal curiosity, an intrinsic reward that allows agents to discover binary quantized representations for the causal factors that affect environments an RL agent may encounter. We show that optimizing causal curiosity rewards results in the agent performing self-supervised experiments. We find that these experiments happen to be semantically meaningful and can be used as pre-training for downstream tasks. While our work learns binary quantized causal representations, a dense encoding may improve the amount of encoded information about the causal mechanisms of the environments. We leave this to future work. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A IMPLEMENTATION DETAILS FOR EXPERIMENT DISCOVERY ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "A.1 PLANNER ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            132,
+            287,
+            147
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The Experiment Planner consisted of a uniform distribution planner for a horizon of 6 control signals. The planner was trained using the Cross Entropy Method Model Predictive Control (Camacho & Alba (2013); De Boer et al. (2005)) on the true environment. We sampled 40 plans per iteration from the distribution initialized to uniform U(controlLow, controlHigh). Each of the sampled plans are applied to each of the training environments and the top $10 \\%$ of the plans are used to update the distribution. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/aed96a8d6c6d91e88dd7adbfebccc0327a701f19d62f6844b005e1919b787e3b.jpg",
+        "image_caption": [
+            "Figure 6: Overview of training. The experiment planner generates a trajectory of actions which is applied to each of the environments with varying causal factors namely mass, shape and size of blocks. For each environment, an observation trajectory or state $S ^ { ( i ) } \\in \\mathbb { S }$ is obtained. A simple model with fixed low expressive power is used to approximate the generative model for $S$ . The \"information overflow\" $L ( S | M )$ is returned as negative reward forcing $\\mathbb { S }$ to be caused by few causal factors. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2 TRAINING ENVIRONMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The training environments vary in each experiment. In Section 4.3, we utilize 3 setups, Mass, SizeMass and ShapeSizeMass. For Mass, we allow the agent to access 5 environments with masses varying from $0 . 1 ~ \\mathrm { k g }$ to $0 . 5 ~ \\mathrm { k g }$ . In SizeMass, the agent has access to 30 environments with masses varying uniformly from 0.1 to $0 . 5 \\mathrm { k g }$ and sizes from 0.05 to 0.1 meters. Finally, in ShapeSizeMass, the agent has access to 60 environments, with masses varying uniformly from 0.1 to $0 . 5 \\mathrm { k g }$ , sizes from 0.05 to 0.1 meters and shapes either being cubes or spheres. During experiment discovery, in each environment, the agent has access to the position of the block in the environment along with its quaternion orientation. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The total number of causal causal factors of each environment are rather large in number due to the fact that the simulator is a complex realistic physics engine. Examples of the causal factors in the environment include gravity, friction coefficients between all on interacting surfaces, shapes, sizes and masses of blocks, control signal frequencies of the environment. However, we only vary 1 during Mass, 2 during SizeMass and 3 during ShapeSizeMass. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.3 CURIOSITY REWARD CALCULATION ",
+        "text_level": 1,
+        "bbox": [
+            178,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We predetermine the minimum description length of the clustering model $L ( M )$ by assuming that the observations $O _ { 0 : T }$ , obtained by applying experimental behavior $a _ { 0 : T }$ are produced by a bi-modal generator distribution, where each mode corresponds to either a low or high (quantized) value of a causal factor. This also ensures that $L ( M )$ is as small as possible. The planner, eq. (5) solves the following optimization problem: ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5049f33b313a712831f085b55f235e8088258c15bb8a24536c68962cf023e710.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\underset { a _ { 0 : T } \\in A ^ { T } } { \\mathrm { a r g } \\mathrm { m a x } } [ \\mathrm { m i n } \\{ d ( o _ { 0 : T } , o _ { 0 : T } ^ { \\prime } ) : o _ { 0 : T } \\in O , o _ { 0 : T } ^ { \\prime } \\in O ^ { \\prime } \\} - } & { \\mathrm { } \\mathrm { m a x } \\{ d ( o _ { 0 : T } , o _ { 0 : T } ^ { \\prime \\prime } ) : o _ { 0 : T } ^ { \\prime \\prime } , o _ { 0 : T } \\in O \\} - } \\\\ { \\mathrm { m a x } \\{ d ( o _ { 0 : T } ^ { \\prime } , o _ { 0 : T } ^ { \\prime \\prime } ) : o _ { 0 : T } ^ { \\prime } , o _ { 0 : T } ^ { \\prime \\prime \\prime } \\in O ^ { \\prime } \\} ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "the distance function $d ( \\cdot , \\cdot )$ in the space of trajectories is set to be Soft Dynamic Time Warping (Cuturi $\\&$ Blondel (2017)). The trajectory length $T$ is 6 control steps long. The objective is a modified version of the Silhouette Score (Rousseeuw (1987)). ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, Objective (8) expresses the ability of a low complexity model, assumed to be bi-modal, to encode the state $S = o _ { 0 : T }$ . If multiple causal factors control $S$ , then the Minimum Description Length of $L ( S )$ will be high. Subsequently, since $M$ is a simple model, the deviation of $S$ from $M$ will be high i.e. $L ( S | M )$ will be high resulting in a low value of the optimization objective. $O$ and $O ^ { \\prime }$ correspond to clusters of outcomes which quantize the values of a causal factor isolated by $a _ { 0 : T }$ $O _ { 0 : T } , o _ { 0 : T } ^ { \\prime \\prime } \\in S$ correspond to trajectories of states i.e. observations obtained by applying $a _ { 0 : T }$ to environments with say, low values of a causal factor while $o _ { 0 : T } ^ { \\prime } , o _ { 0 : T } ^ { \\prime \\prime \\prime } \\in O ^ { \\prime }$ correspond to trajectories of observations i.e. state obtained by applying $a _ { 0 : T }$ to environments with say, high values of the same causal factor. Objective (8) attempts to ensure that these clusters are far apart from each other and are tight i.e. a simple model $M$ encodes $S$ well. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We further motivate how this formulation allows disentanglement of causal factors. A central assumption is that causal factors are independent, by definition, i.e. Independent Mechanisms Assumption Peters et al. (2017). Consider the outcome $S$ obtained by applying an action sequence $a _ { 0 : T }$ to a set of environments. If the action sequence $a _ { 0 : T }$ results in multiple causal factors affecting the outcome $S$ , the Kolmogorov complexity of $S$ will be high. The reason for this is that each causal factor has its own independent causal mechanism (Peters et al. (2017); Parascandolo et al. (2018)) that affects $S$ . Thus, given this independence, the information in $S$ will be a sum of the information “injected” into it from the multiple causes. Conversely, if the outcome $S$ obtained by applying an action sequence $a _ { 0 : T }$ has a lower Kolmogorov Complexity, then $S$ is caused by fewer causal factors. Causal Curiosity attempts to reduce this complexity of $S$ , by assuming a simple generative model $M$ is sufficient to encode $S$ . Thus for experimental behaviors which allow several causes to affect $S$ , the “overflow” of $S$ from $M$ will be high and subsequently the causal curiosity reward will be low. Thus, post-optimization of the objective, we arrive at an action sequence that allows for disentanglement of the causal factors. ",
+        "bbox": [
+            173,
+            250,
+            825,
+            444
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B IMPLEMENTATION DETAILS FOR TRANSFER ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            463,
+            571,
+            479
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In Section 4.4, we show the utility of learning causal representations in 2 separate experimental setups. During TransferMass, the agent has access to 10 environments during training, with masses ranging from 0.1 to $0 . 5 \\mathrm { k g }$ . At test time, the agent is required to perform the place-and-orient task masses 2 masses - $0 . 7 \\mathrm { k g }$ and $0 . 7 5 \\mathrm { k g }$ . During TransferSizeMass, the agent has access to 10 environments during training, with sizes from either 0.01 or $0 . 0 5 \\mathrm { m }$ and masses ranging from 0.1 to $0 . 5 \\mathrm { k g }$ . At test time the agent is asked to perform the task on 2 environments with masses $0 . 7 \\mathrm { k g }$ and $0 . 7 5 \\mathrm { k g }$ with sizes $= 0 . 0 5 \\mathrm { m }$ . ",
+        "bbox": [
+            174,
+            494,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We find that testing with large and light blocks increase the chances of accidental goal completions. Thus, during test-time, we use environments with high masses for out-of-distribution testing. The causal representation is concatenated to the state of the environment as a contextual input and supplied to a PPO-Optimized Actor-Critic Policy. The policy network consists of 2 hidden layers with 256 and 128 units respectively. The experiments are parallelized on 10 CPUs and implemented using stable baselines (Hill et al. (2018)). ",
+        "bbox": [
+            173,
+            599,
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The agent receives a dense reward at each time step during the maximizing external reward phase (Figure 1), the negative of the distance of the block from the goal position scaled by factor of 1000. The control signal was repeated 10 times to the actuators of the motors on each finger. ",
+        "bbox": [
+            176,
+            690,
+            823,
+            732
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C IMPLEMENTATION DETAILS FOR SECTION 4.2 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            752,
+            588,
+            768
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In section 4.2, we study how the acquired experimental behaviors obtained through Causal Curiosity can be used as pre-training for a variety of downstream tasks. The Vanilla CEM depicts the cost of training an experiment planner from scratch to maximize an external dense reward where the agent minimizes the distance between the position of a block in an environment from the goal in the Lifting setup and imparts a velocity to the block along a particular direction in the Travel setup. ",
+        "bbox": [
+            174,
+            782,
+            825,
+            853
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5b7e0efbe4c05bb2dcf3162407d6c3e80216034225d0dc2e5317b011991601d3.jpg",
+        "text": "$$\nR ( a _ { 0 : T } ) = - \\sum _ { t } d i s t ( g o a l _ { t } - b l o c k _ { t } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            858,
+            629,
+            891
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The second baseline (Additive Reward) studies the setup when the agent receives both the curiosity signal and the external reward and attempts to maximize both. The agent receives access all the ",
+        "bbox": [
+            174,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "training environments with varying causal factors and must simultaneously maximize both curiosity and the task reward. The equation below shows the reward maximized for the Lifting task. ",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e341f661b04b7699d8a4b76c0bbaf8dff370fd02442a4be9cf8f945eef6687bf.jpg",
+        "text": "$$\n\\begin{array} { r l r } { \\displaystyle R ( a _ { 0 : T } ) = \\sum _ { e n v s } \\sum _ { t } ^ { T } - d i s t ( g o a l _ { t } - b l o c k _ { t } ) + } & \\\\ { \\displaystyle [ \\operatorname* { m i n } \\{ d ( o _ { 0 : T } , o _ { 0 : T } ^ { \\prime } ) : o _ { 0 : T } \\in O , o _ { 0 : T } ^ { \\prime } \\in O ^ { \\prime } \\} - } & { \\operatorname* { m a x } \\{ d ( o _ { 0 : T } , o _ { 0 : T } ^ { \\prime \\prime } ) : o _ { 0 : T } ^ { \\prime \\prime } , o _ { 0 : T } \\in O \\} - } & \\\\ { \\operatorname* { m a x } \\{ d ( o _ { 0 : T } ^ { \\prime } , o _ { 0 : T } ^ { \\prime \\prime \\prime } ) : o _ { 0 : T } ^ { \\prime } , o _ { 0 : T } ^ { \\prime \\prime \\prime } \\in O ^ { \\prime } \\} ] } & \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            196,
+            137,
+            777,
+            222
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The curious agent first acquired the experimental behavior by interacting with multiple environments with varying causal factors. The lifting skill was obtained during Mass, when the agent attempted to differentiate between multiple blocks of varying mass. The curious agent trained for 600,000 time steps on the curiosity reward. The acquired behavior was then applied to the downstream lifting task and fine tuned to external rewards. The Vanilla CEM baseline had an identical structure to that of the Curious agent, and received only external reward as in Equation (9). The additive agent simultaneously optimized both external reward and the curiosity reward as in Equation (10). ",
+        "bbox": [
+            173,
+            226,
+            825,
+            324
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D INTUITION FOR DEFINITION OF CAUSAL FACTORS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            343,
+            627,
+            359
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We begin with a simple example of a person walking on earth. This person experiences various physical processes while interacting in her world, for example gravity, friction, wind etc. These physical processes affect the outcome of interactions of the person with her environment. For example, while jumping on earth, the human experiences gravity which affects the outcome of her jump, the fact that she falls back to the ground. Additionally, these physical processes (or causal mechanisms) are parameterized by causal factors, for example, acceleration constant due to gravity $g = 9 . 8 m / s ^ { 2 }$ on earth, or coefficients of friction between her feet and the ground which assume particular numerical values. ",
+        "bbox": [
+            173,
+            375,
+            825,
+            486
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "These causal factors may vary across multiple environments. For example, the person may walk on sand or on ice, surfaces with varying frictional values. Thus the outcome of running on such surfaces will vary, running on sand will require significant effort, while running on ice may result in the person slipping. Thus the coefficient of friction between the person’s feet and the surface she walks on affects the outcome of a particular behavior in said environment. In our definition, $h _ { j }$ are causal factors such friction with some particular coefficient of friction, or gravity with acceleration constant $g$ or other. $H$ is the global set containing all such causal factors. ",
+        "bbox": [
+            174,
+            493,
+            825,
+            592
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Now we ask the question (which we subsequently answer), given multiple environments, how would a human characterize each of them depending on the value of a causal factor? Through experimental behaviors. The human in the above example would attempt to run in each of the environments she encountered, be it on sand, on ice, in mud etc. If she slipped in an environment, she would characterize it as slippery. If she didn’t, she would characterize it as non-slippery. We attempt to equip our agent with similar logic. The “sequence of actions” $\\left( { { a _ { 0 : T } } } \\right)$ described in our paper corresponds to the human running. The state $S ^ { ( i ) }$ in the environment $e ^ { ( i ) }$ consisting of the sequence of observations $\\left( o _ { 0 : T } \\right)$ corresponds to the outcome of running. $S$ might belong to either of the clusters of outcomes $S$ or $S ^ { \\prime }$ corresponding to slipping or not slipping. ",
+        "bbox": [
+            173,
+            598,
+            825,
+            726
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "E SCALABILITY LIMITATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            746,
+            429,
+            761
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We utilize the extremely popular One-Factor-at-a-time (OFAT) general paradigm of scientific investigation, as an inspiration for our method. In the case of many hundreds of causal factors, the complexity of this method will scale exponentially. However, we believe that this would indeed be the case given a human experimenter attempting to discover the causation in any system she is studying. Learning about causation is a computationally expensive affair. We point the reader towards a wealth of material on the design of scientific experiments and more specifically the lack of scalability of OFAT (Fisher (1936); Hicks (1964); Czitrom (1999)). Nevertheless, OFAT remains the de facto standard for scientific investigation. ",
+        "bbox": [
+            173,
+            776,
+            826,
+            888
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "1: Input: Unseen Test Environment env, trained Planner and Causal Inference Module   \n2: Initialize causal $R e p = [ ]$   \n3: Initialize training environment set Envs   \n4: for $\\mathbf { k }$ in range(K) do   \n5: Reset env   \n6: Sample experimental behavior $a _ { 0 : T } \\sim \\mathrm { C E M } ( \\cdot | \\ c a u s a l R e p )$   \n7: Apply $a _ { 0 : T }$ to env . Exploration Phase   \n8: Collect $S = o _ { 0 : T }$   \n9: Use learnt $q _ { M } ( z | S )$ for cluster assignment i.e. $z _ { k } = q _ { M } ( z | s , c a u s a l R e p )$   \n10: Append $z _ { k }$ to causalRep . Causal Inference Module   \n11: Learn a policy conditioned on causal factors $a _ { t } \\sim \\pi ( \\cdot | o _ { t } , z _ { 0 : K } )$ to maximize external reward. ",
+        "bbox": [
+            179,
+            440,
+            826,
+            599
+        ],
+        "page_idx": 14
+    }
+]

parse/train/Skvd-myR-/Skvd-myR-.md

@@ -0,0 +1,327 @@
+# LEARNING NON-METRIC VISUAL SIMILARITY FOR IMAGE RETRIEVAL
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Measuring visual (dis)similarity between two or more instances within a data distribution is a fundamental task in many applications, especially in image retrieval. Theoretically, non-metric distances are able to generate a more complex and accurate similarity model than metric distances, provided that the non-linear data distribution is precisely captured by the similarity model. In this work, we analyze a simple approach for deep learning networks to be used as an approximation of non-metric similarity functions and we study how these models generalize across different image retrieval datasets.
+
+# 1 INTRODUCTION
+
+For humans, deciding whether two images are visually similar or not is, to some extent, a natural task. However, in computer vision, this is a challenging problem and algorithms do not always succeed in matching pictures that contain similar-looking elements. This is mainly because of the well-known semantic gap problem, which refers to the difference or gap between low-level image pixels and high-level semantic concepts. Estimating visual similarity is a fundamental task that seeks to break this semantic gap by accurately evaluating how alike two or more pictures are. Visual similarity is crucial for many computer vision areas including image retrieval, image classification and object recognition, among others.
+
+Given a query image, content-based image retrieval systems rank pictures in a dataset according to how similar they are with respect to the input. This can be broken into two fundamental tasks: 1) computing meaningful image representations that capture the most salient visual information from pixels and 2) measuring accurate visual similarity between these image representations to rank images according to a similarity score.
+
+In the last years, several methods to represent visual information from raw pixels in images have been proposed, first by designing handcrafted features such as SIFT Lowe (2004), then by compacting these local features into a single global image descriptor using different techniques such as Fisher Vectors Perronnin et al. (2010) and more recently by extracting deep image representations from neural networks (Babenko et al. (2014)). However, once two images are described by feature vectors, visual similarity is commonly measured by computing a standard metric between them. Although regular distance metrics, such as Euclidean distance or cosine similarity, are fast and easy to implement, they do not take into account the possible interdependency within the dataset, which means that even if a strong nonlinear data dependency is occurring in the visual collection, they might not be able to capture it. This suggests that learning a similarity estimation directly from visual data can improve the performance on image retrieval tasks, provided that the likely nonlinearity dependencies within the dataset are precisely learned by the similarity function.
+
+Visual similarity learning is closely related to distance metric learning. Traditionally, distance metric learning algorithms were based on linear metrics such as the Mahalanobis distance. However, if the visual data presents any nonlinear interdependency, better results are expected when using nonlinear approaches. According to some studies Tan et al. (2006), standard metric axioms are not valid for human perception of visual similarity and hence, visual similarity functions should not necessarily satisfy distance metric conditions. Deep learning-based similarity learning methods are mostly focused on learning an optimal mapping from pixels to a linear space in which Euclidean distance can be applied. Instead, we propose a simple approach based on neural networks to learn a non-metric similarity score in the feature space.
+
+![](images/56365ceee1d5c2b6bb70d64f61a9f02ba28a75df5d0963025682de396cd8ad10.jpg)  
+Figure 1: System overview. The feature extraction block computes visual representations of images whereas the visual similarity block estimates a similarity score using a neural network.
+
+![](images/43d42d966b7dc716b383d60d8c886ae8ebf5004f97f3b1cd7ac190fa449f1dcf.jpg)  
+Figure 2: Siamese architectures (left) map pixels into high-quality vector representations. Our similarity network (right) learns a similarity function on top of the vector representations.
+
+Figure 1 shows an overview of the proposed approach. By training a deep learning model, we can estimate a visual similarity function that outperforms methods based on standard metric computations. One convolutional neural network extracts image representations from input images, while a second neural network computes the visual similarity score. The visual similarity neural network is trained using both pairs of similar and dissimilar images in three stages. The output score of the similarity network can be directly applied as a similarity estimation to rank images in an image retrieval task. Experimental results on standard datasets show that our network is able to discriminate when a pair of images is similar or dissimilar and improve standard metrics score on top of that.
+
+# 2 RELATED WORK
+
+Content-Based Image Retrieval. Content-based image retrieval searches for images by considering their visual content. Given a query image, pictures in a collection are ranked according to their visual similarity with respect to the query. Early methods represent the visual content of images by a set of hand-crafted features, such as SIFT Lowe (2004). As a single image may contain hundreds of these features, aggregation techniques like bag-of-words (BOW) Sivic et al. (2003), Fisher Vectors Perronnin et al. (2010) or VLAD Jegou et al. (2010) encode local descriptors into a compact ´ vector, thereby improving computational efficiency and scalability. Recently, because of the latest advancements on deep learning, features obtained from convolutional neural networks (CNN) have rapidly become the new state-of-the-art in image retrieval.
+
+Deep Learning for Image Retrieval. Deep image retrieval extracts activations from CNNs as image representations. At first, some methods Babenko et al. (2014); Sharif Razavian et al. (2014); Wan et al. (2014); Liu et al. (2015) proposed to use representations from one of the last fully connected layers of networks pre-trained on the classification ImageNet dataset Russakovsky et al. (2015). When deeper networks such as GoogLeNet Szegedy et al. (2015) and VGG Simonyan & Zisserman (2014) appeared, some authors Babenko & Lempitsky (2015); Yue-Hei $\mathrm { N g }$ et al. (2015); Sharif Razavian et al. (2014); Xie et al. (2015) showed that mid-layer representations obtained from the convolutional layers performed better in the retrieval task. Since then, there have been several attempts to aggregate these high-dimensional convolutional representations into a compact vector. For example, Gong et al. (2014); Yue-Hei Ng et al. (2015) compacted deep features by using VLAD, Mohedano et al. (2016) encoded the neural codes into an histogram of words, Babenko & Lempitsky (2015); Kalantidis et al. (2016) applied sum-pooling to obtain a compact representation and Razavian et al. (2016); Tolias et al. (2016) aggregated deep features by max-pooling them into a new vector. A different approach is to train the network to directly learn compact binary codes end-to-end (Erin Liong et al., 2015; Lin et al., 2015). Some authors have shown that fine-tunning the networks with similar data to the target task increases the performance significantly (Babenko et al., 2014; Gordo et al., 2016; Radenovic et al., 2016; Salvador et al., 2016; Gordo et al., 2017). ´ Finally, recent work has shown that adding attention models to select meaningful features can be also beneficial for image retrieval (Jimenez et al., 2017; Noh et al., 2017). ´
+
+All of these methods are focused on finding high quality features to represent visual content efficiently and visual similarity is computed by simply applying a standard metric distance. General metrics, such as Euclidean distance or cosine similarity, however, might be failing to consider the inner data structure of these visual representations. Learning a similarity function directly from data may help to capture the human perception of visual similarity in a better way.
+
+Similarity Learning. Some of the most popular similarity learning work, such as OASIS Chechik et al. (2010) and MLR McFee & Lanckriet (2010), are based on linear metric learning by optimizing the weights of a linear transformation matrix. Although linear methods are easier to optimize and less prone to overfitting, nonlinear algorithms are expected to achieve higher accuracy modeling the possible nonlinearities of data. Nonlinear similarity learning based on deep learning has been recently applied to many different visual contexts. In low-level image matching, CNNs have been trained to match pairs of patches for stereo matching Zagoruyko & Komodakis (2015); Luo et al. (2016) and optical flow Fischer et al. (2015); Thewlis et al. (2016). In high-level image matching, deep learning techniques have been proposed to learn low-dimensional embedding spaces in face verification Chopra et al. (2005), retrieval Wu et al. (2013); Wang et al. (2014), classification Hoffer & Ailon (2015); Qian et al. (2015); Oh Song et al. (2016) and product search Bell & Bala (2015), either by using siamese Chopra et al. (2005) or triplet Wang et al. (2014) architectures.
+
+In general, these methods rely on learning a mapping from image pixels to a low dimensional target space to compute the final similarity decision by using a standard metric. They are designed to find the best projection in which a linear distance can be successfully applied. Instead of projecting the visual data into some linear space, that may or may not exist, our approach seeks to learn the nonmetric visual similarity score itself. Similarly, Li et al. (2014) and Han et al. (2015) used a CNN to decide whether or not two input images are a match, applied to pedestrian reindentification and patch matching, respectively. In these methods, the networks are trained as a binary classification problem (i.e. same or different pedestrian/patch), whereas in an image retrieval ranking problem, a regression score is required. Inspired by the results of Wan et al. (2014), which showed that combining deep features with similarity learning techniques can be very beneficial for the performance of image retrieval systems, we propose to train a deep learning algorithm to learn non-metric similarities for image retrieval and improve results in top of high quality image representation methods.
+
+# 3 LEARNING VISUAL SIMILARITY
+
+# 3.1 DEFINITION
+
+Visual similarity is the task that measures how related two images are by using their visual content. Given $n$ samples in the training image collection $I$ , for each image $I _ { i } \in I$ with $i \in [ 1 , n ]$ , a global $d$ -dimensional representation $\bar { x } _ { i } \in \bar { \mathbb { R } } ^ { d }$ is obtained as $x _ { i } = f ( I _ { i } , w _ { f } )$ , where $f$ is the function that maps images into global features and $w _ { f }$ is the set of parameters of $f$ . We define $s _ { i , j }$ as the similarity score which measures how alike two images $I _ { i }$ and $I _ { j }$ are. The higher $s _ { i , j }$ is, the more similar $I _ { i }$ and $I _ { j }$ are. The aim is to learn a visual similarity function $S$ that computes the similarity score from global image representations as:
+
+$$
+\begin{array} { r } { s _ { i , j } = S ( x _ { i } , x _ { j } ) = g ( f ( I _ { i } , w _ { f } ) , f ( I _ { j } , w _ { f } ) , w _ { g } ) } \\ { s . t . \quad s _ { i , j } > s _ { i , k } \to I _ { i } , I _ { j } \mathrm { a r e ~ m o r e ~ s i m i l a r ~ t h a n ~ } I _ { i } , I _ { k } } \end{array}
+$$
+
+where $g$ is a nonlinear function and $w _ { g }$ is the set of parameters to optimize.
+
+Note that $g$ does not have to be a metric in order to be a similarity function and thus, it is not required to satisfy the rigid constraints of metric axioms, i.e. non-negativity, identity of indiscernibles, symmetry and triangle inequality. Some non-metric similarity works such as Tan et al. (2006) suggest that these restrictions are not compatible with human perception. As an example, they showed that although a centaur might be visually similar to both a person and a horse, the person and the horse are not similar to each other. A possible explanation for this phenomenon is that when comparing two images, human beings may pay more attention to similarities and thus, similar portions of the images may be more discriminative than dissimilar parts. To overcome the issues associated with applying strong rigid constraints to visual similarity, we propose to learn the non-metric similarity function $g$ using a neural network approach.
+
+# 3.2 IMAGE REPRESENTATION
+
+Here we describe the image representation method, $f$ , we use. As this work aims to learn a nonmetric similarity estimation from visual data, our efforts are not focused on improving existing image representation methods, but to learn how to compare them. Without loss of generality, we use the RMAC descriptor proposed in Tolias et al. (2016) as image representation, although any other image representation method can be considered as well.
+
+Table 1: Network architectures. Fully connected layers (FC-{filters}) are always followed by a ReLU layer except for the last one. Training: 22.5 million pairs. Validation: 7.5 million pairs.   
+
+<table><tr><td colspan="2"></td><td rowspan="2"></td><td colspan="2">Training Data</td><td colspan="2">Validation Data</td></tr><tr><td>Config</td><td>Params</td><td>MSE</td><td>p</td><td>MSE</td><td>p</td></tr><tr><td>A</td><td>FC-1024,FC-1024,FC-1</td><td>2.1M</td><td>0.00021</td><td>0.946</td><td>0.00035</td><td>0.909</td></tr><tr><td>B</td><td>FC-4096,FC-4096,FC-1</td><td>21M</td><td>0.00008</td><td>0.978</td><td>0.00019</td><td>0.965</td></tr><tr><td>C</td><td>FC-8192,FC-8192,FC-1</td><td>76M</td><td>0.00007</td><td>0.982</td><td>0.00012</td><td>0.974</td></tr><tr><td>D</td><td>FC-4096,FC-4096,FC-4096,FC-1</td><td>38M</td><td>0.00009</td><td>0.978</td><td>0.00019</td><td>0.964</td></tr></table>
+
+RMAC is a deep global image representation obtained from the last convolutional layer of a pretrained CNN on ImageNet classification task Russakovsky et al. (2015). When an image is fed into the network, the last convolutional layer outputs a $W \times H \times K$ response, where $K$ is the number of filters and $W$ and $H$ are the spatial width and height of the output, respectively, that depend on the network architecture as well as on the size of the input image. The response of the $k$ -th filter of the last convolutional layer can be represented by $\Omega _ { k }$ , a 2D tensor of size $W \times H$ . If $\Omega _ { k } ( \boldsymbol { p } )$ is the response at a particular position $p$ , and $R$ is a spatial region within the feature map, the regional feature vector $f _ { R }$ is defined as:
+
+$$
+f _ { R } = [ f _ { R , 1 } \ldots f _ { R , k } \ldots f _ { R , K } ] ^ { \top }
+$$
+
+where $f _ { R , k } = \operatorname* { m a x } _ { p \in R } \Omega _ { k } ( p )$ . Thus, $f _ { R }$ consists of the maximum activation of each filter inside the region $R$ . Several regional features are extracted at different multi-scale overlapping regions. Each of these regional vectors is independently post-processed with $\ell 2$ -normalization, PCA-whitening and $\ell 2$ -normalization, as suggested in Jegou & Chum (2012). Finally, regional vectors are summed ´ and $\ell { 2 }$ -normalized once again to obtain the final compact vector. The size of the final vector is $K$ , which is independent of the size of the input image, its aspect ratio or the number of regions used.
+
+# 3.3 SIMILARITY NETWORK
+
+To compare two images and obtain a visual similarity score we learn the similarity function $g$ by training a deep learning architecture. Given two input images $I _ { i }$ and $I _ { j }$ , we first extract their representations $x _ { i }$ and $x _ { j }$ , respectively, as explained in Section 3.2. The two $K$ -dimensional global vectors are concatenated and fed into the similarity network, as shown in Figure 1. This process is different to the standard siamese architecture Chopra et al. (2005) because the latter maps images into vector representations and updates the shared weights according to the learning protocol and our approach trains and updates the similarity network on top of high-quality vector representations. Moreover, in the similarity network architecture, weights in the image representation block are not necessarily shared. Figure 2 shows the difference between both approaches.
+
+The similarity network is composed by a set of fully connected layers, each one of them followed by a non-linear function, such as ReLU Krizhevsky et al. (2012). The input of the network is fixed to be of $1 \times K \times 2$ size, so the size of the first layer is $1 \times K \times 2 \times C h$ , where $C h$ is the number of channels. We consider hidden layers of size $1 \times C h \times 2 \times C h$ . Finally, the output layer is of size $1 \times C h \times 2 \times 1$ and it is not followed by a ReLU layer, as the output similarity score is expected to cover a full range of values, both positive and negative. The regression loss function, $L$ , penalizes when the predicted score of the network is far away from an annotated similarity score, such as:
+
+$$
+L ( I _ { i } , I _ { j } ) = | s _ { i , j } - y _ { i , j } | = | g ( x _ { i } , x _ { j } , w _ { g } ) - y _ { i , j } |
+$$
+
+where $s _ { i , j }$ is the network output and $y _ { i , j }$ is the annotated score. Four configurations A-D with different number of filters $C h$ and number of hidden layers are proposed and tested during our experiments, as shown in Table 1.
+
+# 3.4 TRAINING SIMILARITY
+
+The visual similarity network is trained in three stages. In each stage the weights are initialized by the trained weights of the previous stage while the learned task gets progressively more difficult.
+
+![](images/f14d25dacad17f5f47fe3d267ac011ec55bc343bee77586d81f7d1f71f5a521d.jpg)  
+Figure 3: Misclassified pairs. (Upper) Lower row: (dissimilar) similar images in which the network score is (lower) higher than the cosine similarity.
+
+![](images/38e7e36e2a240dc779a7a5dd10ddba811943cb7fd5980e4bba2bbff4f467c6a1.jpg)  
+Figure 4: mAP versus $\Delta$ . Rigid lines are DeepSimH scores, dashed lines are cosine similarity scores.
+
+# STAGE 1: STANDARD METRIC
+
+In Stage 1, the network learns a standard similarity function based on the cosine similarity. We generate random pairs of vectors, $x _ { i }$ and $x _ { j }$ , and we assign the cosine similarity between them as the score label yi,j :
+
+$$
+y _ { i , j } = { \frac { x _ { i } \cdot x _ { j } } { \| x _ { i } \| \| x _ { j } \| } }
+$$
+
+In order to train the model in the full range of possible values, pairs are produced so that the cosine similarity is uniformly distributed within the training set.
+
+STAGE 2: VISUAL SIMILARITY
+
+In Stage 2, the basic similarity network learns to increase the similarity score when given two matching images and to decrease it when a pair of images is not a match. The weights in this training stage are initialized by the weights obtained during Stage 1. We now use pairs of image representation vectors $x _ { i }$ and $x _ { j }$ , randomly chosen from our training image dataset. The score label is set to:
+
+$$
+y _ { i , j } = \left\{ \begin{array} { l l } { \frac { x _ { i } \cdot x _ { j } } { \| x _ { i } \| \| x _ { j } \| } + \Delta , } & { \mathrm { i f } x _ { i } \mathrm { a n d } x _ { j } \mathrm { a r e } \mathrm { s i m i l a r } } \\ { \frac { x _ { i } \cdot x _ { j } } { \| x _ { i } \| \| x _ { j } \| } - \Delta , } & { \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+
+where $\Delta$ is the margin parameter. Thus, the model learns to discriminate when a pair of images are similar (dissimilar) and assigns it a higher (lower) value than the standard score.
+
+In this stage, the model learns how to compute a similarity score from examples of images that are known to be matching or non-matching. Therefore a relevant dataset to the final retrieval task should be used. Similarity between pairs might be decided using different techniques, such as image classes, score based on local features or manual labeling, among others. Without loss of generality, we consider two images as similar when they belong to the same class and as dissimilar when they belong to different classes.
+
+# STAGE 3: HARD EXAMPLES
+
+In the Stage 3, the similarity network is refined by training it specifically by using difficult pairs of images. Previous works Gordo et al. (2016); Radenovic et al. (2016) have shown that fine-tunning ´ neural networks using difficult samples is very helpful in terms of performance. This is easy to understand: if the network is only trained by using easy pairs (e.g. a car and a dog), it will not be able to discriminate between difficult pairs (e.g. a car and a van). To choose the set of hard pairs we compute the scores of a random set of image pairs by using the network trained in Stage 2. Those pairs in which the network output is worse than the cosine similarity measure are selected as difficult pairs for retraining1. Examples of difficult image pairs can be seen in Figure 3.
+
+# 4 EXPERIMENTS
+
+# 4.1 TESTING DATASETS
+
+Our approach is evaluated on the standard image retrieval datasets described below.
+
+Oxford5k Philbin et al. (2007): a dataset that consists of 5,062 images of 11 different Oxford landmarks. The query set contains 55 annotated images, 5 per landmark.
+
+Paris6k Philbin et al. (2008): a datasets that consists of 6,412 images of 11 different Paris landmarks. The query set contains 55 annotated images, 5 per landmark.
+
+Land5k: a validation subset of the Landmarks database Babenko et al. (2014). It consists of the 4,915 validation images from 529 classes. A random selection of 45 images is used as queries.
+
+Oxford105k, Paris106k: the large-scale versions of Oxford5k and Paris6k, respectively. They include 100,000 distractor images from Flickr Philbin et al. (2007).
+
+In both the Oxford5k and the Paris6k collections query images are cropped according to the region of interest provided by the authors of the datasets. Evaluation is performed by computing the mean Average Precision (mAP), using the provided ground truth and algorithms. For Land5k we consider an image to be relevant to the query when it belongs to the same class.
+
+# 4.2 TRAINING DATASETS
+
+For the purposes of this work, having a training dataset as similar as possible to the final similarity task is essential. We create several versions of the training dataset to evaluate the effect of using different samples in the training process.
+
+Landmarks Gordo et al. (2016): an automatically cleaned subset of the full Landmarks Babenko et al. (2014) dataset which officially contains about 49,000 images from 586 landmarks. However, due to broken URLs, we could only download 33,119 training images and 4,915 validation images. This dataset does not contain images from classes that overlap with Oxford5k and Paris6k datasets as they were manually removed.
+
+Landmarks-extra500: the Landmarks collection plus 250 random images from each of the Oxford5k and Paris6k datasets. In total, it contains 33,619 training images.
+
+Landmarks-extra: the Landmarks collection in addition to about 500 images from Oxford5k and 1,700 images from Paris6k classes. In total, it contains 35,342 training images belonging to 605 different landmarks. Note that query images are not added in any case and they remain unseen by the system.
+
+# 4.3 EXPERIMENTAL DETAILS
+
+Image Representation. To compute RMAC representations we use the VGG16 network Simonyan & Zisserman (2014), which has been previously pre-trained on the ImageNet dataset Russakovsky et al. (2015). Unless otherwise stated, we use the default values proposed in Tolias et al. (2016) to obtain 512-dimensional RMAC vectors. VGG16 network is used off-the-shelf without any retraining or fine-tunning performed on top of it. Experimental results have shown that RMAC representations are very sensitive to the PCA matrices used in the post-processing step. As we are keeping query images unseen by the system and not using them in the PCA matrices computation as in Tolias et al. (2016), our results are slightly different to theirs.
+
+Visual Similarity Learning. Similarity learning is trained using almost a million of random pairs, of which half of the pairs are visual matches and the other half are non-matches. PCA whitening is done using Paris5k images. As RMAC representation performs better in high resolution images, we re-scale all the images up to 1024 pixels, keeping the original aspect ratio of the pictures. For the similarity network, four different configurations A-D (Table 1) are explored during our experiments. The network is optimized using backpropagation and stochastic gradient descent. We use a learning rate of 0.001, a batch size of 100, a weight decay of 0.0005 and momentum of 0.9.
+
+Table 2: mAP when using different training configurations and $\Delta$ (in brackets) values.   
+
+<table><tr><td rowspan="2"></td><td colspan="3">Landmarks</td><td colspan="3">Landmarks-extra500</td><td colspan="3">Landmarks-extra</td></tr><tr><td>Ox5k</td><td>Pa6k</td><td>La5k</td><td>Ox5k</td><td>Pa6k</td><td>La5k</td><td>Ox5k</td><td>Pa6k</td><td>La5k</td></tr><tr><td>Cosine</td><td>0.665</td><td>0.638</td><td>0.564</td><td>0.665</td><td>0.638</td><td>0.564</td><td>0.665</td><td>0.638</td><td>0.564</td></tr><tr><td>DeepCosine</td><td>0.638</td><td>0.596</td><td>0.549</td><td>0.638</td><td>0.596</td><td>0.549</td><td>0.638</td><td>0.596</td><td>0.549</td></tr><tr><td>OASIS</td><td>0.514</td><td>0.385</td><td>0.578</td><td>0.570</td><td>0.651</td><td>0.589</td><td>0.619</td><td>0.853</td><td>0.579</td></tr><tr><td>Linear (0.2)</td><td>0.598</td><td>0.660</td><td>0.508</td><td>0.611</td><td>0.632</td><td>0.514</td><td>0.602</td><td>0.581</td><td>0.502</td></tr><tr><td>DeepSim (0.2)</td><td>0.658</td><td>0.460</td><td>0.669</td><td>0.717</td><td>0.654</td><td>0.671</td><td>0.718</td><td>0.757</td><td>0.668</td></tr><tr><td>DeepSimH(0.2)</td><td>0.655</td><td>0.503</td><td>0.697</td><td>0.719</td><td>0.677</td><td>0.693</td><td>0.786</td><td>0.860</td><td>0.662</td></tr><tr><td>DeepSimH (0.4)</td><td>0.637</td><td>0.504</td><td>0.737</td><td>0.703</td><td>0.701</td><td>0.745</td><td>0.794</td><td>0.878</td><td>0.706</td></tr><tr><td>DeepSimH(0.6)</td><td>0.613</td><td>0.514</td><td>0.776</td><td>0.703</td><td>0.716</td><td>0.776</td><td>0.789</td><td>0.885</td><td>0.735</td></tr><tr><td>DeepSimH (0.8)</td><td>0.600</td><td>0.511</td><td>0.783</td><td>0.685</td><td>0.710</td><td>0.803</td><td>0.808</td><td>0.891</td><td>0.758</td></tr></table>
+
+Computational cost. Standard metrics are relatively fast and computationally cheap. Our visual similarity network involves the use of millions of parameters that inevitable increase the computational cost. However, it is still feasible to compute in a reasonable amount of time. In our experiments, training time is about 5 hours in a GeForce GTX 1080 GPU and testing time for a pair of images is $1 . 2 5 ~ \mathrm { m s }$ on average (0.35 ms when using cosine similarity).
+
+# 5 RESULTS
+
+# 5.1 ARCHITECTURE DISCUSSION
+
+Four different configurations A-D for the similarity neural network are proposed. We compare the performance of each one during Stage 1, when the network is trained with the standard cosine similarity measurement. If $s _ { l }$ is the network score and $y _ { l }$ is the cosine similarity of the $l$ -th pair with $l = 1 . . L$ , we evaluate each network by computing the mean squared error, MSE, and the correlation coefficient, $\rho$ , as:
+
+$$
+M S E = \frac { 1 } { L } \sum _ { l = 1 } ^ { L } ( s _ { l } - y _ { l } ) ^ { 2 } \rho = \frac { 1 } { L - 1 } \sum _ { l = 1 } ^ { L } \frac { s _ { l } - \mu _ { s } } { \sigma _ { s } } \frac { y _ { l } - \mu _ { y } } { \sigma _ { y } }
+$$
+
+where $\mu _ { s }$ and $\sigma _ { s }$ are the mean and standard deviation of the vector of network scores $s$ , and $\mu _ { y }$ and $\sigma _ { y }$ are the mean and standard deviation of the vectors of cosine similarities $y$ .
+
+Results are shown in Table 1. Unsurprisingly, the configuration with bigger number of parameters, C, achieves the best MSE and $\rho$ results, both in training and validation sets. However, the performance of networks B and $\mathrm { D }$ is very close to the performance of network C. As network B requires only 21 million parameters and network C requires 76 million parameters, we keep configuration B as our default architecture for the rest of the experiments.
+
+# 5.2 EVALUATION OF THE SIMILARITY NETWORK
+
+In this section, we study the benefits of using a non-metric distance function trained with neural networks. In order to isolate the contribution of the visual similarity computation and perform a fair comparison between different distance functions, we only train the similarity network part. However, an end-to-end training of the whole image retrieval pipeline is explored in Appendix B.
+
+To evaluate our similarity network, we compute the mAP at each stage of the training process (Section 4.2). Results when using different training datasets can be found in Table 2. Cosine similarity is computed as a baseline. We denote as DeepCosine the results obtained after the first stage, when the network is trained to mimic cosine similarity. Naturally, DeepCosine performs worse than the cosine similarity, as it is an estimation of the cosine metric. DeepSim refers to the results obtained after the second stage, when the network is fine-tunned to learn visual similarity with random pairs of images. DeepSimH are the results after the last stage, when the network is trained by using both random and hard pairs of images. We compare our approach against the standard similarity learning algorithm OASIS Chechik et al. (2010). Finally, we also conduct experiments on linear metric learning, which are denoted as Linear in Table 2, by training an affine transformation of the feature vectors using the same training protocol as described in Equation 5.
+
+Table 3: mAP results for different state-of-the-art methods. Dim corresponds to the dimensionality of the feature representation. Similarity is the similarity function.   
+
+<table><tr><td></td><td>Method</td><td>Dim</td><td>Similarity</td><td>Ox5k</td><td>Ox105k</td><td>Pa6k</td><td>Pa106k</td></tr><tr><td rowspan="11">Jltr-ertiti</td><td>Babenko et al. (2014)</td><td>512</td><td>L2</td><td>0.435</td><td>0.392</td><td>=</td><td>1</td></tr><tr><td>Sharif Razavian et al. (2014)</td><td>4096</td><td>Averaged L2</td><td>0.322</td><td>1</td><td>0.495</td><td>1</td></tr><tr><td>Wan et al. (2014)</td><td>4096</td><td>OASIS</td><td>0.466</td><td>1</td><td>0.867</td><td>=</td></tr><tr><td>Babenko &amp; Lempitsky (2015)</td><td>256</td><td>Cosine</td><td>0.657</td><td>0.642</td><td></td><td>=</td></tr><tr><td>Yue-Hei Ng et al. (2015)</td><td>128</td><td>L2</td><td>0.593</td><td>1</td><td>0.59</td><td>=</td></tr><tr><td>Kalantidis et al. (2016)</td><td>512</td><td>L2</td><td>0.708</td><td>0.653</td><td>0.797</td><td>0.722</td></tr><tr><td>Mohedano et al. (2016)</td><td>25k</td><td>Cosine</td><td>0.739</td><td>0.593</td><td>0.82</td><td>0.648</td></tr><tr><td>Salvador et al. (2016)</td><td>512</td><td>Cosine</td><td>0.588</td><td>1</td><td>0.656</td><td>1</td></tr><tr><td>Tolias et al. (2016)</td><td>512</td><td>Cosine</td><td>0.669</td><td>0.616</td><td>0.83</td><td>0.757</td></tr><tr><td>Jiménez et al. (2017)</td><td>512</td><td>Cosine</td><td>0.712</td><td>0.672</td><td>0.805</td><td>0.733</td></tr><tr><td>Ours (△ = 0.8)</td><td>512</td><td>DeepSimH</td><td>0.808</td><td>0.772</td><td>0.891</td><td>0.818</td></tr><tr><td rowspan="7">Binunnau</td><td>Babenko et al. (2014)</td><td>512</td><td>L2</td><td>0.557</td><td>0.522</td><td>1</td><td>1</td></tr><tr><td>Gordo et al. (2016)</td><td>512</td><td>Cosine</td><td>0.831</td><td>0.786</td><td>0.871</td><td>0.797</td></tr><tr><td>Wan et al. (2014)</td><td>4096</td><td>OASIS</td><td>0.783</td><td>1</td><td>0.947</td><td>1</td></tr><tr><td>Radenovic et al. (2016)</td><td>512</td><td>Cosine</td><td>0.77</td><td>0.692</td><td>0.838</td><td>0.764</td></tr><tr><td>Salvador et al. (2016)</td><td>512</td><td>Cosine</td><td>0.71</td><td></td><td>0.798</td><td></td></tr><tr><td>Gordo et al. (2017)</td><td>2048</td><td>Cosine</td><td>0.861</td><td>0.828</td><td>0.945</td><td>0.906</td></tr><tr><td>Ours (△ = 0.8)</td><td>512</td><td>DeepSimH</td><td>0.882</td><td>0.821</td><td>0.882</td><td>0.829</td></tr></table>
+
+Our similarity networks outperform OASIS in all the testing datasets. Moreover when using Landmarks-clean-extra as training dataset, results are boosted with respect to the standard metric, achieving improvements ranging from $20 \%$ (Oxford5k) to $40 \%$ (Pairs6k). When using a small subset of images from Oxford5k and Paris6k classes, i.e. Landmarks-clean-extra-500 dataset, our similarity networks also improve mAP with respect to the cosine similarity in the three testing datasets. This indicates that visual similarity can be learnt even when using a reduced subset of the target image domain. Experiments on affine transformations show that, unlike our proposed methods, simple linear metrics are not able to properly fit Equation 5. However, visual similarity does not transfer well across domains when no images of the target domain are used during training. An extended discussion about the effects of the training dataset can be found in Appendix A.
+
+Overall, these results suggests that our network is able to learn whether two images are similar or not and provide a similarity score accordingly. Figure 4 shows how the mAP is affected when using different values of $\Delta$ . Except when $\Delta = 0$ (i.e. visual similarity is not learned), DeepSimH always improves mAP with respect to the standard cosine similarity.
+
+# 5.3 COMPARISON WITH THE STATE OF THE ART.
+
+Finally, we compare our method against several state-of-the-art techniques (Table 3). As standard practice, works are split into two main groups: off-the-shelf and fine-tunning approaches. Off-theshelf are techniques that extract visual representations by using CNNs trained on ImageNet dataset Russakovsky et al. (2015) without modifying the network. On the other hand, fine-tunning methods retrain the network to compute more accurate visual representation. For a fair comparison, we only consider methods that represent each image with a single compact vector and do not apply query expansion or image re-ranking. When using off-the-shelf RMAC features, our DeepSimH approach outperforms previous methods in every dataset. To compare against fine-tunned methods, we compute RMAC vectors using the fine-tunned version of VGG16 proposed in Radenovic et al. (2016) ´ and training our DeepSimH exactly in the same way as in the off-the-shelf version. Accuracy is significantly improved when using our similarity network instead of the analogous cosine similarity method Radenovic et al. (2016). DeepSimH achieves the best mAP precision in´ $_ { \mathrm { O X } 5 \mathrm { k } }$ dataset and comes second in $_ { \mathrm { O X 1 0 5 k } }$ and $\mathrm { P a l 0 6 k }$ after Gordo et al. (2017), which uses the more complex and higher-dimensional ResNet He et al. (2016) instead of a VGG16 network for image representation.
+
+# 6 CONCLUSIONS
+
+We have presented a method for learning visual similarity directly from visual data. Instead of using a rigid metric distance, such as the standard cosine similarity, we propose to train a neural network model to learn a similarity estimation between a pair of visual representations previously extracted from input images. Our method outperforms state-of-the-art approaches based on rigid distances in standard image retrieval collection of images and experimental results showed that learning a nonmetric visual similarity function is beneficial in image retrieval tasks provided that a small subset of images of the same domain are available during training. Standard image retrieval techniques that are commonly applied after cosine similarity computation, such as query expansion or image re-ranking, might also be applied on top of the similarity network. Finally, we end with an open question, which is the subject of planned future work, concerning efficient computation of exact or approximate K-nearest neighbours based on the learned network similarity function.
+
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+Ali Sharif Razavian, Josephine Sullivan, Stefan Carlsson, and Atsuto Maki. Visual instance retrieval with deep convolutional networks. ITE Transactions on Media Technology and Applications, 4 (3), 2016.
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+Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li Fei-Fei. ImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision (IJCV), 115(3), 2015.
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+Amaia Salvador, Xavier Giro-i Nieto, Ferran Marqu ´ es, and Shin’ichi Satoh. Faster r-cnn features ´ for instance search. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, 2016.
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+Ali Sharif Razavian, Hossein Azizpour, Josephine Sullivan, and Stefan Carlsson. Cnn features offthe-shelf: an astounding baseline for recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, 2014.
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+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. CoRR, 2014.
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+Josef Sivic, Andrew Zisserman, et al. Video google: A text retrieval approach to object matching in videos. In Proceedings of the IEEE International Conference on Computer Vision, 2003.
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+Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2015.
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+Xiaoyang Tan, Songcan Chen, Jun Li, and Zhi-Hua Zhou. Learning non-metric partial similarity based on maximal margin criterion. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, volume 1, 2006.
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+J. Thewlis, S. Zheng, P. Torr, and A. Vedaldi. Fully-trainable deep matching. In Proceedings of the British Machine Vision Conference, 2016.
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+Giorgos Tolias, Ronan Sicre, and Herve J ´ egou. Particular object retrieval with integral max-pooling ´ of cnn activations. In Proceedings of the International Conference on Learning Representations, 2016.
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+Laurens Van Der Maaten. Accelerating t-sne using tree-based algorithms. Journal of machine learning research, 15(1), 2014.
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+Ji Wan, Dayong Wang, Steven Chu Hong Hoi, Pengcheng Wu, Jianke Zhu, Yongdong Zhang, and Jintao Li. Deep learning for content-based image retrieval: A comprehensive study. In Proceedings of the 22nd ACM international conference on Multimedia. ACM, 2014.
+
+Jiang Wang, Yang Song, Thomas Leung, Chuck Rosenberg, Jingbin Wang, James Philbin, Bo Chen, and Ying Wu. Learning fine-grained image similarity with deep ranking. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2014.
+
+Pengcheng Wu, Steven CH Hoi, Hao Xia, Peilin Zhao, Dayong Wang, and Chunyan Miao. Online multimodal deep similarity learning with application to image retrieval. In Proceedings of the 21st ACM international conference on Multimedia. ACM, 2013.
+
+Lingxi Xie, Richang Hong, Bo Zhang, and Qi Tian. Image classification and retrieval are one. In Proceedings of the 5th ACM on International Conference on Multimedia Retrieval. ACM, 2015.
+
+Joe Yue-Hei Ng, Fan Yang, and Larry S Davis. Exploiting local features from deep networks for image retrieval. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, 2015.
+
+Sergey Zagoruyko and Nikos Komodakis. Learning to compare image patches via convolutional neural networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2015.
+
+![](images/4319978d225706a0e00ce9bc4ebe0fc5c78e6a37ce74916c2747867da9e0e095.jpg)  
+Figure 5: mAP when using different number of target samples in the training set.
+
+# APPENDIX A TRAINING ON TARGET DATASET
+
+In this appendix, a further discussion about the influence of the dataset used to train the similarity network and estimate the visual similarity between a pair of images is carried out.
+
+As we already noted in Section 5.2, visual similarity does not transfer well across domains. A subset of samples from the target dataset is required during training to learn a meaningful similarity function. This is a well-known problem in the field of metric learning (Kulis et al. (2013)). In Figure 5, we explore the effect on performance when we use different subsets of samples from the target collection in addition to the Landmarks dataset (Gordo et al. (2016)) during the second stage of our training (Section 3.4).
+
+Figure 5 shows that there is a clear correlation between the similarity network performance and the number of samples from the target dataset used during training. Indeed, in agreement with previous work in metric learning (Kulis et al. (2013)), we observe that not considering samples from the target dataset to train a similarity function might be harmful. The similarity network, however, outperforms standard metric results even when a small number of samples from the target collection is used during training: only 100 images from $_ { \mathrm { O X } 5 \mathrm { k } }$ and 250 images from Pa6k are required to outperform cosine similarity in $_ { \mathrm { O X } 5 \mathrm { k } }$ and Pa6k datasets, respectively. This fact suggests that the similarity network is able to generalize from a small subset of target samples and is not memorizing the distances in the training collection.
+
+Finally, we present some visual results of our findings. Figure 6 and Figure 7 show the t-Distributed Stochastic Neighbor Embedding (t-SNE) (Van Der Maaten, 2014) representation of $_ { \mathrm { O X } 5 \mathrm { k } }$ images when using RMAC as image representation, and cosine similarity or our similarity network as similarity function, respectively. Although RMAC descriptor with a standard metric is already performing well in terms of visual similarity (e.g., in Figure 6 images from Radcliffe camera are grouped together in the right bottom corner), performance can be pushed even more when our similarity network is used instead (Figure 7. In summary, these results indicate the benefit of training a similarity network over a standard metric function such as cosine similarity for the image retrieval task.
+
+![](images/3b75d7f71a158f658f55774c1226cc9c9123f97cee7be8c809a112f09444badc.jpg)  
+Figure 6: t-SNE plot for a subset of $5 0 0 \mathrm { O x } 5 \mathrm { k }$ images when using RMAC and cosine similarity.
+
+![](images/7e8787416ced5649f289dc8099d01f497114584fbe5615b98704a60c25ec1ac3.jpg)  
+Figure 7: t-SNE plot for a subset of $5 0 0 \mathrm { O x } 5 \mathrm { k }$ images when using RMAC and DeepSim.
+
+# APPENDIX B END-TO-END TRAINING
+
+So far, we have isolated the similarity computation part in the image retrieval pipeline by only training the similarity network. In this way, it is easy to see that the improvement in the testing datasets compare to when using other similarity methods (Section 5.2) is, in fact, due to the visual similarity network function. In this appendix, however, we explore a real end-to-end approach for image retrieval. The end-to-end approach consists on feeding the system with pixels to obtain a visual similarity score between a pair of images. The whole pipeline is presented in Figure 8. For the feature extraction part, we adopt the MAC compact image representation, following Radenovic´ et al. (2016) work. For the visual similarity part, we use our visual similarity network DeepSim. The whole approach is end-to-end differentiable so backpropagation can be applied during training.
+
+![](images/1a82b4116c061c1069df5e09b436350cd8a0fa681c672ea9e6ac9a5ddd6e8703.jpg)  
+Figure 8: End-to-End architecture. The feature extraction part consists on a VGG16 network followed by a max-pooling and a l2-normalization layers. In the visual similarity part, two compact vectors are concatenated and forwarded to the DeepSim network to obtain a similarity score.
+
+In this case, we use MAC Tolias et al. (2016) as compact image representation. After feeding a VGG16 network Simonyan & Zisserman (2014) with a pre-processed image, the feature maps from the last convolutional layer are obtained. These feature maps are then max-pooled over the whole region to obtain a compact vector, which is l2-normalized. The final dimensionality of the MAC vector does not depend on the input image size, but in the number of filters in the last convolutional layer. Image pre-processing includes resizing the image to 720 pixels on its largest side (maintaining aspect ratio) and mean subtraction.
+
+We initialize the VGG16 network with the weights trained on ImageNet dataset. We then learn the weights of the similarity network by freezing VGG16 weights and applying Stage 1 and Stage 2, as described in Section 3.4. Finally, for the end-to-end training, we unfreeze all the weights of the architecture and fine-tune all the layers one last time. As all the layers have been already pre-trained, the final end-to-end fine-tunning is performed in about 200,000 pairs of images from Landarmarksextra dataset (Section 4.2) for just 5,000 iterations. Note that we adopt MAC Tolias et al. (2016) instead of RMAC as it is easier to train and thus, the results are slightly worst. From Table 4 we note, firstly, a boost in performance when using DeepSim instead of the cosine similarity and finally, a significant improvement when the architecture is trained end-to-end with respect to both the baseline and when only training the visual similarity part.
+
+The results are unsurprising as fine-tuning the entire architecture allows us to fit better to a particular dataset. However the key message of the paper is that fine-tuning the final similarity computation, instead on relying on cosines as researchers have been doing so far, may be a worthwhile step that can push accuracy results higher irrespective of the feature vector computation.
+
+Table 4: mAP when training different parts of the image retrieval pipeline. In blue, the modules that are fine-tunned in every experiment.   
+
+<table><tr><td>Features</td><td>Similarity</td><td>Oxford5k</td><td>Paris6k</td><td>Landmarks5k</td></tr><tr><td>MAC</td><td>Cosine</td><td>0.481</td><td>0.539</td><td>0.494</td></tr><tr><td>MAC</td><td>DeepSim</td><td>0.509</td><td>0.683</td><td>0.589</td></tr><tr><td>MAC</td><td>DeepSim</td><td>0.555</td><td>0.710</td><td>0.685</td></tr></table>

parse/train/Skvd-myR-/Skvd-myR-_content_list.json

@@ -0,0 +1,1761 @@
+[
+    {
+        "type": "text",
+        "text": "LEARNING NON-METRIC VISUAL SIMILARITY FOR IMAGE RETRIEVAL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
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+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            236,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Measuring visual (dis)similarity between two or more instances within a data distribution is a fundamental task in many applications, especially in image retrieval. Theoretically, non-metric distances are able to generate a more complex and accurate similarity model than metric distances, provided that the non-linear data distribution is precisely captured by the similarity model. In this work, we analyze a simple approach for deep learning networks to be used as an approximation of non-metric similarity functions and we study how these models generalize across different image retrieval datasets. ",
+        "bbox": [
+            232,
+            263,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            398,
+            336,
+            415
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "For humans, deciding whether two images are visually similar or not is, to some extent, a natural task. However, in computer vision, this is a challenging problem and algorithms do not always succeed in matching pictures that contain similar-looking elements. This is mainly because of the well-known semantic gap problem, which refers to the difference or gap between low-level image pixels and high-level semantic concepts. Estimating visual similarity is a fundamental task that seeks to break this semantic gap by accurately evaluating how alike two or more pictures are. Visual similarity is crucial for many computer vision areas including image retrieval, image classification and object recognition, among others. ",
+        "bbox": [
+            174,
+            429,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Given a query image, content-based image retrieval systems rank pictures in a dataset according to how similar they are with respect to the input. This can be broken into two fundamental tasks: 1) computing meaningful image representations that capture the most salient visual information from pixels and 2) measuring accurate visual similarity between these image representations to rank images according to a similarity score. ",
+        "bbox": [
+            174,
+            547,
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+            617
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In the last years, several methods to represent visual information from raw pixels in images have been proposed, first by designing handcrafted features such as SIFT Lowe (2004), then by compacting these local features into a single global image descriptor using different techniques such as Fisher Vectors Perronnin et al. (2010) and more recently by extracting deep image representations from neural networks (Babenko et al. (2014)). However, once two images are described by feature vectors, visual similarity is commonly measured by computing a standard metric between them. Although regular distance metrics, such as Euclidean distance or cosine similarity, are fast and easy to implement, they do not take into account the possible interdependency within the dataset, which means that even if a strong nonlinear data dependency is occurring in the visual collection, they might not be able to capture it. This suggests that learning a similarity estimation directly from visual data can improve the performance on image retrieval tasks, provided that the likely nonlinearity dependencies within the dataset are precisely learned by the similarity function. ",
+        "bbox": [
+            173,
+            625,
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+            791
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Visual similarity learning is closely related to distance metric learning. Traditionally, distance metric learning algorithms were based on linear metrics such as the Mahalanobis distance. However, if the visual data presents any nonlinear interdependency, better results are expected when using nonlinear approaches. According to some studies Tan et al. (2006), standard metric axioms are not valid for human perception of visual similarity and hence, visual similarity functions should not necessarily satisfy distance metric conditions. Deep learning-based similarity learning methods are mostly focused on learning an optimal mapping from pixels to a linear space in which Euclidean distance can be applied. Instead, we propose a simple approach based on neural networks to learn a non-metric similarity score in the feature space. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/56365ceee1d5c2b6bb70d64f61a9f02ba28a75df5d0963025682de396cd8ad10.jpg",
+        "image_caption": [
+            "Figure 1: System overview. The feature extraction block computes visual representations of images whereas the visual similarity block estimates a similarity score using a neural network. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            93,
+            486,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/43d42d966b7dc716b383d60d8c886ae8ebf5004f97f3b1cd7ac190fa449f1dcf.jpg",
+        "image_caption": [
+            "Figure 2: Siamese architectures (left) map pixels into high-quality vector representations. Our similarity network (right) learns a similarity function on top of the vector representations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            513,
+            90,
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+            193
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Figure 1 shows an overview of the proposed approach. By training a deep learning model, we can estimate a visual similarity function that outperforms methods based on standard metric computations. One convolutional neural network extracts image representations from input images, while a second neural network computes the visual similarity score. The visual similarity neural network is trained using both pairs of similar and dissimilar images in three stages. The output score of the similarity network can be directly applied as a similarity estimation to rank images in an image retrieval task. Experimental results on standard datasets show that our network is able to discriminate when a pair of images is similar or dissimilar and improve standard metrics score on top of that. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            417,
+            344,
+            434
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Content-Based Image Retrieval. Content-based image retrieval searches for images by considering their visual content. Given a query image, pictures in a collection are ranked according to their visual similarity with respect to the query. Early methods represent the visual content of images by a set of hand-crafted features, such as SIFT Lowe (2004). As a single image may contain hundreds of these features, aggregation techniques like bag-of-words (BOW) Sivic et al. (2003), Fisher Vectors Perronnin et al. (2010) or VLAD Jegou et al. (2010) encode local descriptors into a compact ´ vector, thereby improving computational efficiency and scalability. Recently, because of the latest advancements on deep learning, features obtained from convolutional neural networks (CNN) have rapidly become the new state-of-the-art in image retrieval. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Deep Learning for Image Retrieval. Deep image retrieval extracts activations from CNNs as image representations. At first, some methods Babenko et al. (2014); Sharif Razavian et al. (2014); Wan et al. (2014); Liu et al. (2015) proposed to use representations from one of the last fully connected layers of networks pre-trained on the classification ImageNet dataset Russakovsky et al. (2015). When deeper networks such as GoogLeNet Szegedy et al. (2015) and VGG Simonyan & Zisserman (2014) appeared, some authors Babenko & Lempitsky (2015); Yue-Hei $\\mathrm { N g }$ et al. (2015); Sharif Razavian et al. (2014); Xie et al. (2015) showed that mid-layer representations obtained from the convolutional layers performed better in the retrieval task. Since then, there have been several attempts to aggregate these high-dimensional convolutional representations into a compact vector. For example, Gong et al. (2014); Yue-Hei Ng et al. (2015) compacted deep features by using VLAD, Mohedano et al. (2016) encoded the neural codes into an histogram of words, Babenko & Lempitsky (2015); Kalantidis et al. (2016) applied sum-pooling to obtain a compact representation and Razavian et al. (2016); Tolias et al. (2016) aggregated deep features by max-pooling them into a new vector. A different approach is to train the network to directly learn compact binary codes end-to-end (Erin Liong et al., 2015; Lin et al., 2015). Some authors have shown that fine-tunning the networks with similar data to the target task increases the performance significantly (Babenko et al., 2014; Gordo et al., 2016; Radenovic et al., 2016; Salvador et al., 2016; Gordo et al., 2017). ´ Finally, recent work has shown that adding attention models to select meaningful features can be also beneficial for image retrieval (Jimenez et al., 2017; Noh et al., 2017). ´ ",
+        "bbox": [
+            173,
+            583,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "All of these methods are focused on finding high quality features to represent visual content efficiently and visual similarity is computed by simply applying a standard metric distance. General metrics, such as Euclidean distance or cosine similarity, however, might be failing to consider the inner data structure of these visual representations. Learning a similarity function directly from data may help to capture the human perception of visual similarity in a better way. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Similarity Learning. Some of the most popular similarity learning work, such as OASIS Chechik et al. (2010) and MLR McFee & Lanckriet (2010), are based on linear metric learning by optimizing the weights of a linear transformation matrix. Although linear methods are easier to optimize and less prone to overfitting, nonlinear algorithms are expected to achieve higher accuracy modeling the possible nonlinearities of data. Nonlinear similarity learning based on deep learning has been recently applied to many different visual contexts. In low-level image matching, CNNs have been trained to match pairs of patches for stereo matching Zagoruyko & Komodakis (2015); Luo et al. (2016) and optical flow Fischer et al. (2015); Thewlis et al. (2016). In high-level image matching, deep learning techniques have been proposed to learn low-dimensional embedding spaces in face verification Chopra et al. (2005), retrieval Wu et al. (2013); Wang et al. (2014), classification Hoffer & Ailon (2015); Qian et al. (2015); Oh Song et al. (2016) and product search Bell & Bala (2015), either by using siamese Chopra et al. (2005) or triplet Wang et al. (2014) architectures. ",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In general, these methods rely on learning a mapping from image pixels to a low dimensional target space to compute the final similarity decision by using a standard metric. They are designed to find the best projection in which a linear distance can be successfully applied. Instead of projecting the visual data into some linear space, that may or may not exist, our approach seeks to learn the nonmetric visual similarity score itself. Similarly, Li et al. (2014) and Han et al. (2015) used a CNN to decide whether or not two input images are a match, applied to pedestrian reindentification and patch matching, respectively. In these methods, the networks are trained as a binary classification problem (i.e. same or different pedestrian/patch), whereas in an image retrieval ranking problem, a regression score is required. Inspired by the results of Wan et al. (2014), which showed that combining deep features with similarity learning techniques can be very beneficial for the performance of image retrieval systems, we propose to train a deep learning algorithm to learn non-metric similarities for image retrieval and improve results in top of high quality image representation methods. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 LEARNING VISUAL SIMILARITY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "3.1 DEFINITION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Visual similarity is the task that measures how related two images are by using their visual content. Given $n$ samples in the training image collection $I$ , for each image $I _ { i } \\in I$ with $i \\in [ 1 , n ]$ , a global $d$ -dimensional representation $\\bar { x } _ { i } \\in \\bar { \\mathbb { R } } ^ { d }$ is obtained as $x _ { i } = f ( I _ { i } , w _ { f } )$ , where $f$ is the function that maps images into global features and $w _ { f }$ is the set of parameters of $f$ . We define $s _ { i , j }$ as the similarity score which measures how alike two images $I _ { i }$ and $I _ { j }$ are. The higher $s _ { i , j }$ is, the more similar $I _ { i }$ and $I _ { j }$ are. The aim is to learn a visual similarity function $S$ that computes the similarity score from global image representations as: ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/eca8c5aa628846781e6fca514d4cc4ff9965135714c82fc6ee3ed7d268ab46fa.jpg",
+        "text": "$$\n\\begin{array} { r } { s _ { i , j } = S ( x _ { i } , x _ { j } ) = g ( f ( I _ { i } , w _ { f } ) , f ( I _ { j } , w _ { f } ) , w _ { g } ) } \\\\ { s . t . \\quad s _ { i , j } > s _ { i , k } \\to I _ { i } , I _ { j } \\mathrm { a r e ~ m o r e ~ s i m i l a r ~ t h a n ~ } I _ { i } , I _ { k } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            631,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $g$ is a nonlinear function and $w _ { g }$ is the set of parameters to optimize. ",
+        "bbox": [
+            176,
+            675,
+            665,
+            690
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Note that $g$ does not have to be a metric in order to be a similarity function and thus, it is not required to satisfy the rigid constraints of metric axioms, i.e. non-negativity, identity of indiscernibles, symmetry and triangle inequality. Some non-metric similarity works such as Tan et al. (2006) suggest that these restrictions are not compatible with human perception. As an example, they showed that although a centaur might be visually similar to both a person and a horse, the person and the horse are not similar to each other. A possible explanation for this phenomenon is that when comparing two images, human beings may pay more attention to similarities and thus, similar portions of the images may be more discriminative than dissimilar parts. To overcome the issues associated with applying strong rigid constraints to visual similarity, we propose to learn the non-metric similarity function $g$ using a neural network approach. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 IMAGE REPRESENTATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            854,
+            390,
+            869
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here we describe the image representation method, $f$ , we use. As this work aims to learn a nonmetric similarity estimation from visual data, our efforts are not focused on improving existing image representation methods, but to learn how to compare them. Without loss of generality, we use the RMAC descriptor proposed in Tolias et al. (2016) as image representation, although any other image representation method can be considered as well. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/ac1e82bf198a844741af08578b8c39f38c184c0cd99d544bafe3817063b73e3c.jpg",
+        "table_caption": [
+            "Table 1: Network architectures. Fully connected layers (FC-{filters}) are always followed by a ReLU layer except for the last one. Training: 22.5 million pairs. Validation: 7.5 million pairs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\"></td><td rowspan=\"2\"></td><td colspan=\"2\">Training Data</td><td colspan=\"2\">Validation Data</td></tr><tr><td>Config</td><td>Params</td><td>MSE</td><td>p</td><td>MSE</td><td>p</td></tr><tr><td>A</td><td>FC-1024,FC-1024,FC-1</td><td>2.1M</td><td>0.00021</td><td>0.946</td><td>0.00035</td><td>0.909</td></tr><tr><td>B</td><td>FC-4096,FC-4096,FC-1</td><td>21M</td><td>0.00008</td><td>0.978</td><td>0.00019</td><td>0.965</td></tr><tr><td>C</td><td>FC-8192,FC-8192,FC-1</td><td>76M</td><td>0.00007</td><td>0.982</td><td>0.00012</td><td>0.974</td></tr><tr><td>D</td><td>FC-4096,FC-4096,FC-4096,FC-1</td><td>38M</td><td>0.00009</td><td>0.978</td><td>0.00019</td><td>0.964</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "RMAC is a deep global image representation obtained from the last convolutional layer of a pretrained CNN on ImageNet classification task Russakovsky et al. (2015). When an image is fed into the network, the last convolutional layer outputs a $W \\times H \\times K$ response, where $K$ is the number of filters and $W$ and $H$ are the spatial width and height of the output, respectively, that depend on the network architecture as well as on the size of the input image. The response of the $k$ -th filter of the last convolutional layer can be represented by $\\Omega _ { k }$ , a 2D tensor of size $W \\times H$ . If $\\Omega _ { k } ( \\boldsymbol { p } )$ is the response at a particular position $p$ , and $R$ is a spatial region within the feature map, the regional feature vector $f _ { R }$ is defined as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/d78a9914e828505a1d8df51779c424d08674b92257e3c8c05ce94784424cc007.jpg",
+        "text": "$$\nf _ { R } = [ f _ { R , 1 } \\ldots f _ { R , k } \\ldots f _ { R , K } ] ^ { \\top }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $f _ { R , k } = \\operatorname* { m a x } _ { p \\in R } \\Omega _ { k } ( p )$ . Thus, $f _ { R }$ consists of the maximum activation of each filter inside the region $R$ . Several regional features are extracted at different multi-scale overlapping regions. Each of these regional vectors is independently post-processed with $\\ell 2$ -normalization, PCA-whitening and $\\ell 2$ -normalization, as suggested in Jegou & Chum (2012). Finally, regional vectors are summed ´ and $\\ell { 2 }$ -normalized once again to obtain the final compact vector. The size of the final vector is $K$ , which is independent of the size of the input image, its aspect ratio or the number of regions used. ",
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+    {
+        "type": "text",
+        "text": "3.3 SIMILARITY NETWORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To compare two images and obtain a visual similarity score we learn the similarity function $g$ by training a deep learning architecture. Given two input images $I _ { i }$ and $I _ { j }$ , we first extract their representations $x _ { i }$ and $x _ { j }$ , respectively, as explained in Section 3.2. The two $K$ -dimensional global vectors are concatenated and fed into the similarity network, as shown in Figure 1. This process is different to the standard siamese architecture Chopra et al. (2005) because the latter maps images into vector representations and updates the shared weights according to the learning protocol and our approach trains and updates the similarity network on top of high-quality vector representations. Moreover, in the similarity network architecture, weights in the image representation block are not necessarily shared. Figure 2 shows the difference between both approaches. ",
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+        "type": "text",
+        "text": "The similarity network is composed by a set of fully connected layers, each one of them followed by a non-linear function, such as ReLU Krizhevsky et al. (2012). The input of the network is fixed to be of $1 \\times K \\times 2$ size, so the size of the first layer is $1 \\times K \\times 2 \\times C h$ , where $C h$ is the number of channels. We consider hidden layers of size $1 \\times C h \\times 2 \\times C h$ . Finally, the output layer is of size $1 \\times C h \\times 2 \\times 1$ and it is not followed by a ReLU layer, as the output similarity score is expected to cover a full range of values, both positive and negative. The regression loss function, $L$ , penalizes when the predicted score of the network is far away from an annotated similarity score, such as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/710286d6b9817825e326f94f9f4d5afa636216a9ccd21f30a174827ce5eb9ef4.jpg",
+        "text": "$$\nL ( I _ { i } , I _ { j } ) = | s _ { i , j } - y _ { i , j } | = | g ( x _ { i } , x _ { j } , w _ { g } ) - y _ { i , j } |\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $s _ { i , j }$ is the network output and $y _ { i , j }$ is the annotated score. Four configurations A-D with different number of filters $C h$ and number of hidden layers are proposed and tested during our experiments, as shown in Table 1. ",
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+    {
+        "type": "text",
+        "text": "3.4 TRAINING SIMILARITY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The visual similarity network is trained in three stages. In each stage the weights are initialized by the trained weights of the previous stage while the learned task gets progressively more difficult. ",
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+    {
+        "type": "image",
+        "img_path": "images/f14d25dacad17f5f47fe3d267ac011ec55bc343bee77586d81f7d1f71f5a521d.jpg",
+        "image_caption": [
+            "Figure 3: Misclassified pairs. (Upper) Lower row: (dissimilar) similar images in which the network score is (lower) higher than the cosine similarity. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/38e7e36e2a240dc779a7a5dd10ddba811943cb7fd5980e4bba2bbff4f467c6a1.jpg",
+        "image_caption": [
+            "Figure 4: mAP versus $\\Delta$ . Rigid lines are DeepSimH scores, dashed lines are cosine similarity scores. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "STAGE 1: STANDARD METRIC ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In Stage 1, the network learns a standard similarity function based on the cosine similarity. We generate random pairs of vectors, $x _ { i }$ and $x _ { j }$ , and we assign the cosine similarity between them as the score label yi,j : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/445f3df3f580ff6ac08325efa6501793e3459e52f11b34cc022742ad1d2cb565.jpg",
+        "text": "$$\ny _ { i , j } = { \\frac { x _ { i } \\cdot x _ { j } } { \\| x _ { i } \\| \\| x _ { j } \\| } }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "In order to train the model in the full range of possible values, pairs are produced so that the cosine similarity is uniformly distributed within the training set. ",
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+    {
+        "type": "text",
+        "text": "STAGE 2: VISUAL SIMILARITY ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Stage 2, the basic similarity network learns to increase the similarity score when given two matching images and to decrease it when a pair of images is not a match. The weights in this training stage are initialized by the weights obtained during Stage 1. We now use pairs of image representation vectors $x _ { i }$ and $x _ { j }$ , randomly chosen from our training image dataset. The score label is set to: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/8c40564a466feb01d582506ead387537127c9f94512c3c31ccb096aa148cd34e.jpg",
+        "text": "$$\ny _ { i , j } = \\left\\{ \\begin{array} { l l } { \\frac { x _ { i } \\cdot x _ { j } } { \\| x _ { i } \\| \\| x _ { j } \\| } + \\Delta , } & { \\mathrm { i f } x _ { i } \\mathrm { a n d } x _ { j } \\mathrm { a r e } \\mathrm { s i m i l a r } } \\\\ { \\frac { x _ { i } \\cdot x _ { j } } { \\| x _ { i } \\| \\| x _ { j } \\| } - \\Delta , } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\Delta$ is the margin parameter. Thus, the model learns to discriminate when a pair of images are similar (dissimilar) and assigns it a higher (lower) value than the standard score. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this stage, the model learns how to compute a similarity score from examples of images that are known to be matching or non-matching. Therefore a relevant dataset to the final retrieval task should be used. Similarity between pairs might be decided using different techniques, such as image classes, score based on local features or manual labeling, among others. Without loss of generality, we consider two images as similar when they belong to the same class and as dissimilar when they belong to different classes. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "STAGE 3: HARD EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the Stage 3, the similarity network is refined by training it specifically by using difficult pairs of images. Previous works Gordo et al. (2016); Radenovic et al. (2016) have shown that fine-tunning ´ neural networks using difficult samples is very helpful in terms of performance. This is easy to understand: if the network is only trained by using easy pairs (e.g. a car and a dog), it will not be able to discriminate between difficult pairs (e.g. a car and a van). To choose the set of hard pairs we compute the scores of a random set of image pairs by using the network trained in Stage 2. Those pairs in which the network output is worse than the cosine similarity measure are selected as difficult pairs for retraining1. Examples of difficult image pairs can be seen in Figure 3. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 TESTING DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our approach is evaluated on the standard image retrieval datasets described below. ",
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+    },
+    {
+        "type": "text",
+        "text": "Oxford5k Philbin et al. (2007): a dataset that consists of 5,062 images of 11 different Oxford landmarks. The query set contains 55 annotated images, 5 per landmark. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Paris6k Philbin et al. (2008): a datasets that consists of 6,412 images of 11 different Paris landmarks. The query set contains 55 annotated images, 5 per landmark. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Land5k: a validation subset of the Landmarks database Babenko et al. (2014). It consists of the 4,915 validation images from 529 classes. A random selection of 45 images is used as queries. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Oxford105k, Paris106k: the large-scale versions of Oxford5k and Paris6k, respectively. They include 100,000 distractor images from Flickr Philbin et al. (2007). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In both the Oxford5k and the Paris6k collections query images are cropped according to the region of interest provided by the authors of the datasets. Evaluation is performed by computing the mean Average Precision (mAP), using the provided ground truth and algorithms. For Land5k we consider an image to be relevant to the query when it belongs to the same class. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 TRAINING DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            409,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For the purposes of this work, having a training dataset as similar as possible to the final similarity task is essential. We create several versions of the training dataset to evaluate the effect of using different samples in the training process. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Landmarks Gordo et al. (2016): an automatically cleaned subset of the full Landmarks Babenko et al. (2014) dataset which officially contains about 49,000 images from 586 landmarks. However, due to broken URLs, we could only download 33,119 training images and 4,915 validation images. This dataset does not contain images from classes that overlap with Oxford5k and Paris6k datasets as they were manually removed. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Landmarks-extra500: the Landmarks collection plus 250 random images from each of the Oxford5k and Paris6k datasets. In total, it contains 33,619 training images. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Landmarks-extra: the Landmarks collection in addition to about 500 images from Oxford5k and 1,700 images from Paris6k classes. In total, it contains 35,342 training images belonging to 605 different landmarks. Note that query images are not added in any case and they remain unseen by the system. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Image Representation. To compute RMAC representations we use the VGG16 network Simonyan & Zisserman (2014), which has been previously pre-trained on the ImageNet dataset Russakovsky et al. (2015). Unless otherwise stated, we use the default values proposed in Tolias et al. (2016) to obtain 512-dimensional RMAC vectors. VGG16 network is used off-the-shelf without any retraining or fine-tunning performed on top of it. Experimental results have shown that RMAC representations are very sensitive to the PCA matrices used in the post-processing step. As we are keeping query images unseen by the system and not using them in the PCA matrices computation as in Tolias et al. (2016), our results are slightly different to theirs. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Visual Similarity Learning. Similarity learning is trained using almost a million of random pairs, of which half of the pairs are visual matches and the other half are non-matches. PCA whitening is done using Paris5k images. As RMAC representation performs better in high resolution images, we re-scale all the images up to 1024 pixels, keeping the original aspect ratio of the pictures. For the similarity network, four different configurations A-D (Table 1) are explored during our experiments. The network is optimized using backpropagation and stochastic gradient descent. We use a learning rate of 0.001, a batch size of 100, a weight decay of 0.0005 and momentum of 0.9. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f0a6496cfe62f8006b173a5a2544900fd05f776c0b9a6e6b06b9b3a75d3b5de3.jpg",
+        "table_caption": [
+            "Table 2: mAP when using different training configurations and $\\Delta$ (in brackets) values. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">Landmarks</td><td colspan=\"3\">Landmarks-extra500</td><td colspan=\"3\">Landmarks-extra</td></tr><tr><td>Ox5k</td><td>Pa6k</td><td>La5k</td><td>Ox5k</td><td>Pa6k</td><td>La5k</td><td>Ox5k</td><td>Pa6k</td><td>La5k</td></tr><tr><td>Cosine</td><td>0.665</td><td>0.638</td><td>0.564</td><td>0.665</td><td>0.638</td><td>0.564</td><td>0.665</td><td>0.638</td><td>0.564</td></tr><tr><td>DeepCosine</td><td>0.638</td><td>0.596</td><td>0.549</td><td>0.638</td><td>0.596</td><td>0.549</td><td>0.638</td><td>0.596</td><td>0.549</td></tr><tr><td>OASIS</td><td>0.514</td><td>0.385</td><td>0.578</td><td>0.570</td><td>0.651</td><td>0.589</td><td>0.619</td><td>0.853</td><td>0.579</td></tr><tr><td>Linear (0.2)</td><td>0.598</td><td>0.660</td><td>0.508</td><td>0.611</td><td>0.632</td><td>0.514</td><td>0.602</td><td>0.581</td><td>0.502</td></tr><tr><td>DeepSim (0.2)</td><td>0.658</td><td>0.460</td><td>0.669</td><td>0.717</td><td>0.654</td><td>0.671</td><td>0.718</td><td>0.757</td><td>0.668</td></tr><tr><td>DeepSimH(0.2)</td><td>0.655</td><td>0.503</td><td>0.697</td><td>0.719</td><td>0.677</td><td>0.693</td><td>0.786</td><td>0.860</td><td>0.662</td></tr><tr><td>DeepSimH (0.4)</td><td>0.637</td><td>0.504</td><td>0.737</td><td>0.703</td><td>0.701</td><td>0.745</td><td>0.794</td><td>0.878</td><td>0.706</td></tr><tr><td>DeepSimH(0.6)</td><td>0.613</td><td>0.514</td><td>0.776</td><td>0.703</td><td>0.716</td><td>0.776</td><td>0.789</td><td>0.885</td><td>0.735</td></tr><tr><td>DeepSimH (0.8)</td><td>0.600</td><td>0.511</td><td>0.783</td><td>0.685</td><td>0.710</td><td>0.803</td><td>0.808</td><td>0.891</td><td>0.758</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Computational cost. Standard metrics are relatively fast and computationally cheap. Our visual similarity network involves the use of millions of parameters that inevitable increase the computational cost. However, it is still feasible to compute in a reasonable amount of time. In our experiments, training time is about 5 hours in a GeForce GTX 1080 GPU and testing time for a pair of images is $1 . 2 5 ~ \\mathrm { m s }$ on average (0.35 ms when using cosine similarity). ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 RESULTS ",
+        "text_level": 1,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 ARCHITECTURE DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Four different configurations A-D for the similarity neural network are proposed. We compare the performance of each one during Stage 1, when the network is trained with the standard cosine similarity measurement. If $s _ { l }$ is the network score and $y _ { l }$ is the cosine similarity of the $l$ -th pair with $l = 1 . . L$ , we evaluate each network by computing the mean squared error, MSE, and the correlation coefficient, $\\rho$ , as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/d8f58acf91d6fd5d24c57779e4843b25c806856561792d46ea8e28ee4bf1f260.jpg",
+        "text": "$$\nM S E = \\frac { 1 } { L } \\sum _ { l = 1 } ^ { L } ( s _ { l } - y _ { l } ) ^ { 2 } \\rho = \\frac { 1 } { L - 1 } \\sum _ { l = 1 } ^ { L } \\frac { s _ { l } - \\mu _ { s } } { \\sigma _ { s } } \\frac { y _ { l } - \\mu _ { y } } { \\sigma _ { y } }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mu _ { s }$ and $\\sigma _ { s }$ are the mean and standard deviation of the vector of network scores $s$ , and $\\mu _ { y }$ and $\\sigma _ { y }$ are the mean and standard deviation of the vectors of cosine similarities $y$ . ",
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+        "type": "text",
+        "text": "Results are shown in Table 1. Unsurprisingly, the configuration with bigger number of parameters, C, achieves the best MSE and $\\rho$ results, both in training and validation sets. However, the performance of networks B and $\\mathrm { D }$ is very close to the performance of network C. As network B requires only 21 million parameters and network C requires 76 million parameters, we keep configuration B as our default architecture for the rest of the experiments. ",
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+    {
+        "type": "text",
+        "text": "5.2 EVALUATION OF THE SIMILARITY NETWORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we study the benefits of using a non-metric distance function trained with neural networks. In order to isolate the contribution of the visual similarity computation and perform a fair comparison between different distance functions, we only train the similarity network part. However, an end-to-end training of the whole image retrieval pipeline is explored in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "To evaluate our similarity network, we compute the mAP at each stage of the training process (Section 4.2). Results when using different training datasets can be found in Table 2. Cosine similarity is computed as a baseline. We denote as DeepCosine the results obtained after the first stage, when the network is trained to mimic cosine similarity. Naturally, DeepCosine performs worse than the cosine similarity, as it is an estimation of the cosine metric. DeepSim refers to the results obtained after the second stage, when the network is fine-tunned to learn visual similarity with random pairs of images. DeepSimH are the results after the last stage, when the network is trained by using both random and hard pairs of images. We compare our approach against the standard similarity learning algorithm OASIS Chechik et al. (2010). Finally, we also conduct experiments on linear metric learning, which are denoted as Linear in Table 2, by training an affine transformation of the feature vectors using the same training protocol as described in Equation 5. ",
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+    {
+        "type": "table",
+        "img_path": "images/9dd4185fa3f5668994f2d7574074c065998b5dbf8794d87d31aaba00a6a75286.jpg",
+        "table_caption": [
+            "Table 3: mAP results for different state-of-the-art methods. Dim corresponds to the dimensionality of the feature representation. Similarity is the similarity function. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Method</td><td>Dim</td><td>Similarity</td><td>Ox5k</td><td>Ox105k</td><td>Pa6k</td><td>Pa106k</td></tr><tr><td rowspan=\"11\">Jltr-ertiti</td><td>Babenko et al. (2014)</td><td>512</td><td>L2</td><td>0.435</td><td>0.392</td><td>=</td><td>1</td></tr><tr><td>Sharif Razavian et al. (2014)</td><td>4096</td><td>Averaged L2</td><td>0.322</td><td>1</td><td>0.495</td><td>1</td></tr><tr><td>Wan et al. (2014)</td><td>4096</td><td>OASIS</td><td>0.466</td><td>1</td><td>0.867</td><td>=</td></tr><tr><td>Babenko &amp; Lempitsky (2015)</td><td>256</td><td>Cosine</td><td>0.657</td><td>0.642</td><td></td><td>=</td></tr><tr><td>Yue-Hei Ng et al. (2015)</td><td>128</td><td>L2</td><td>0.593</td><td>1</td><td>0.59</td><td>=</td></tr><tr><td>Kalantidis et al. (2016)</td><td>512</td><td>L2</td><td>0.708</td><td>0.653</td><td>0.797</td><td>0.722</td></tr><tr><td>Mohedano et al. (2016)</td><td>25k</td><td>Cosine</td><td>0.739</td><td>0.593</td><td>0.82</td><td>0.648</td></tr><tr><td>Salvador et al. (2016)</td><td>512</td><td>Cosine</td><td>0.588</td><td>1</td><td>0.656</td><td>1</td></tr><tr><td>Tolias et al. (2016)</td><td>512</td><td>Cosine</td><td>0.669</td><td>0.616</td><td>0.83</td><td>0.757</td></tr><tr><td>Jiménez et al. (2017)</td><td>512</td><td>Cosine</td><td>0.712</td><td>0.672</td><td>0.805</td><td>0.733</td></tr><tr><td>Ours (△ = 0.8)</td><td>512</td><td>DeepSimH</td><td>0.808</td><td>0.772</td><td>0.891</td><td>0.818</td></tr><tr><td rowspan=\"7\">Binunnau</td><td>Babenko et al. (2014)</td><td>512</td><td>L2</td><td>0.557</td><td>0.522</td><td>1</td><td>1</td></tr><tr><td>Gordo et al. (2016)</td><td>512</td><td>Cosine</td><td>0.831</td><td>0.786</td><td>0.871</td><td>0.797</td></tr><tr><td>Wan et al. (2014)</td><td>4096</td><td>OASIS</td><td>0.783</td><td>1</td><td>0.947</td><td>1</td></tr><tr><td>Radenovic et al. (2016)</td><td>512</td><td>Cosine</td><td>0.77</td><td>0.692</td><td>0.838</td><td>0.764</td></tr><tr><td>Salvador et al. (2016)</td><td>512</td><td>Cosine</td><td>0.71</td><td></td><td>0.798</td><td></td></tr><tr><td>Gordo et al. (2017)</td><td>2048</td><td>Cosine</td><td>0.861</td><td>0.828</td><td>0.945</td><td>0.906</td></tr><tr><td>Ours (△ = 0.8)</td><td>512</td><td>DeepSimH</td><td>0.882</td><td>0.821</td><td>0.882</td><td>0.829</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Our similarity networks outperform OASIS in all the testing datasets. Moreover when using Landmarks-clean-extra as training dataset, results are boosted with respect to the standard metric, achieving improvements ranging from $20 \\%$ (Oxford5k) to $40 \\%$ (Pairs6k). When using a small subset of images from Oxford5k and Paris6k classes, i.e. Landmarks-clean-extra-500 dataset, our similarity networks also improve mAP with respect to the cosine similarity in the three testing datasets. This indicates that visual similarity can be learnt even when using a reduced subset of the target image domain. Experiments on affine transformations show that, unlike our proposed methods, simple linear metrics are not able to properly fit Equation 5. However, visual similarity does not transfer well across domains when no images of the target domain are used during training. An extended discussion about the effects of the training dataset can be found in Appendix A. ",
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+    },
+    {
+        "type": "text",
+        "text": "Overall, these results suggests that our network is able to learn whether two images are similar or not and provide a similarity score accordingly. Figure 4 shows how the mAP is affected when using different values of $\\Delta$ . Except when $\\Delta = 0$ (i.e. visual similarity is not learned), DeepSimH always improves mAP with respect to the standard cosine similarity. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.3 COMPARISON WITH THE STATE OF THE ART. ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Finally, we compare our method against several state-of-the-art techniques (Table 3). As standard practice, works are split into two main groups: off-the-shelf and fine-tunning approaches. Off-theshelf are techniques that extract visual representations by using CNNs trained on ImageNet dataset Russakovsky et al. (2015) without modifying the network. On the other hand, fine-tunning methods retrain the network to compute more accurate visual representation. For a fair comparison, we only consider methods that represent each image with a single compact vector and do not apply query expansion or image re-ranking. When using off-the-shelf RMAC features, our DeepSimH approach outperforms previous methods in every dataset. To compare against fine-tunned methods, we compute RMAC vectors using the fine-tunned version of VGG16 proposed in Radenovic et al. (2016) ´ and training our DeepSimH exactly in the same way as in the off-the-shelf version. Accuracy is significantly improved when using our similarity network instead of the analogous cosine similarity method Radenovic et al. (2016). DeepSimH achieves the best mAP precision in´ $_ { \\mathrm { O X } 5 \\mathrm { k } }$ dataset and comes second in $_ { \\mathrm { O X 1 0 5 k } }$ and $\\mathrm { P a l 0 6 k }$ after Gordo et al. (2017), which uses the more complex and higher-dimensional ResNet He et al. (2016) instead of a VGG16 network for image representation. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We have presented a method for learning visual similarity directly from visual data. Instead of using a rigid metric distance, such as the standard cosine similarity, we propose to train a neural network model to learn a similarity estimation between a pair of visual representations previously extracted from input images. Our method outperforms state-of-the-art approaches based on rigid distances in standard image retrieval collection of images and experimental results showed that learning a nonmetric visual similarity function is beneficial in image retrieval tasks provided that a small subset of images of the same domain are available during training. Standard image retrieval techniques that are commonly applied after cosine similarity computation, such as query expansion or image re-ranking, might also be applied on top of the similarity network. Finally, we end with an open question, which is the subject of planned future work, concerning efficient computation of exact or approximate K-nearest neighbours based on the learned network similarity function. ",
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+    {
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+        "text": "Sergey Zagoruyko and Nikos Komodakis. Learning to compare image patches via convolutional neural networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2015. ",
+        "bbox": [
+            174,
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+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/4319978d225706a0e00ce9bc4ebe0fc5c78e6a37ce74916c2747867da9e0e095.jpg",
+        "image_caption": [
+            "Figure 5: mAP when using different number of target samples in the training set. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            140,
+            127,
+            812,
+            351
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX A TRAINING ON TARGET DATASET ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            434,
+            568,
+            449
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this appendix, a further discussion about the influence of the dataset used to train the similarity network and estimate the visual similarity between a pair of images is carried out. ",
+        "bbox": [
+            171,
+            464,
+            821,
+            493
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "As we already noted in Section 5.2, visual similarity does not transfer well across domains. A subset of samples from the target dataset is required during training to learn a meaningful similarity function. This is a well-known problem in the field of metric learning (Kulis et al. (2013)). In Figure 5, we explore the effect on performance when we use different subsets of samples from the target collection in addition to the Landmarks dataset (Gordo et al. (2016)) during the second stage of our training (Section 3.4). ",
+        "bbox": [
+            173,
+            500,
+            825,
+            583
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 5 shows that there is a clear correlation between the similarity network performance and the number of samples from the target dataset used during training. Indeed, in agreement with previous work in metric learning (Kulis et al. (2013)), we observe that not considering samples from the target dataset to train a similarity function might be harmful. The similarity network, however, outperforms standard metric results even when a small number of samples from the target collection is used during training: only 100 images from $_ { \\mathrm { O X } 5 \\mathrm { k } }$ and 250 images from Pa6k are required to outperform cosine similarity in $_ { \\mathrm { O X } 5 \\mathrm { k } }$ and Pa6k datasets, respectively. This fact suggests that the similarity network is able to generalize from a small subset of target samples and is not memorizing the distances in the training collection. ",
+        "bbox": [
+            173,
+            590,
+            825,
+            715
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Finally, we present some visual results of our findings. Figure 6 and Figure 7 show the t-Distributed Stochastic Neighbor Embedding (t-SNE) (Van Der Maaten, 2014) representation of $_ { \\mathrm { O X } 5 \\mathrm { k } }$ images when using RMAC as image representation, and cosine similarity or our similarity network as similarity function, respectively. Although RMAC descriptor with a standard metric is already performing well in terms of visual similarity (e.g., in Figure 6 images from Radcliffe camera are grouped together in the right bottom corner), performance can be pushed even more when our similarity network is used instead (Figure 7. In summary, these results indicate the benefit of training a similarity network over a standard metric function such as cosine similarity for the image retrieval task. ",
+        "bbox": [
+            173,
+            722,
+            825,
+            834
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/3b75d7f71a158f658f55774c1226cc9c9123f97cee7be8c809a112f09444badc.jpg",
+        "image_caption": [
+            "Figure 6: t-SNE plot for a subset of $5 0 0 \\mathrm { O x } 5 \\mathrm { k }$ images when using RMAC and cosine similarity. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            119,
+            802,
+            439
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/7e8787416ced5649f289dc8099d01f497114584fbe5615b98704a60c25ec1ac3.jpg",
+        "image_caption": [
+            "Figure 7: t-SNE plot for a subset of $5 0 0 \\mathrm { O x } 5 \\mathrm { k }$ images when using RMAC and DeepSim. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            194,
+            530,
+            802,
+            859
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX B END-TO-END TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            504,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "So far, we have isolated the similarity computation part in the image retrieval pipeline by only training the similarity network. In this way, it is easy to see that the improvement in the testing datasets compare to when using other similarity methods (Section 5.2) is, in fact, due to the visual similarity network function. In this appendix, however, we explore a real end-to-end approach for image retrieval. The end-to-end approach consists on feeding the system with pixels to obtain a visual similarity score between a pair of images. The whole pipeline is presented in Figure 8. For the feature extraction part, we adopt the MAC compact image representation, following Radenovic´ et al. (2016) work. For the visual similarity part, we use our visual similarity network DeepSim. The whole approach is end-to-end differentiable so backpropagation can be applied during training. ",
+        "bbox": [
+            173,
+            133,
+            825,
+            258
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/1a82b4116c061c1069df5e09b436350cd8a0fa681c672ea9e6ac9a5ddd6e8703.jpg",
+        "image_caption": [
+            "Figure 8: End-to-End architecture. The feature extraction part consists on a VGG16 network followed by a max-pooling and a l2-normalization layers. In the visual similarity part, two compact vectors are concatenated and forwarded to the DeepSim network to obtain a similarity score. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            275,
+            818,
+            429
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In this case, we use MAC Tolias et al. (2016) as compact image representation. After feeding a VGG16 network Simonyan & Zisserman (2014) with a pre-processed image, the feature maps from the last convolutional layer are obtained. These feature maps are then max-pooled over the whole region to obtain a compact vector, which is l2-normalized. The final dimensionality of the MAC vector does not depend on the input image size, but in the number of filters in the last convolutional layer. Image pre-processing includes resizing the image to 720 pixels on its largest side (maintaining aspect ratio) and mean subtraction. ",
+        "bbox": [
+            174,
+            501,
+            825,
+            598
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We initialize the VGG16 network with the weights trained on ImageNet dataset. We then learn the weights of the similarity network by freezing VGG16 weights and applying Stage 1 and Stage 2, as described in Section 3.4. Finally, for the end-to-end training, we unfreeze all the weights of the architecture and fine-tune all the layers one last time. As all the layers have been already pre-trained, the final end-to-end fine-tunning is performed in about 200,000 pairs of images from Landarmarksextra dataset (Section 4.2) for just 5,000 iterations. Note that we adopt MAC Tolias et al. (2016) instead of RMAC as it is easier to train and thus, the results are slightly worst. From Table 4 we note, firstly, a boost in performance when using DeepSim instead of the cosine similarity and finally, a significant improvement when the architecture is trained end-to-end with respect to both the baseline and when only training the visual similarity part. ",
+        "bbox": [
+            174,
+            604,
+            825,
+            744
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The results are unsurprising as fine-tuning the entire architecture allows us to fit better to a particular dataset. However the key message of the paper is that fine-tuning the final similarity computation, instead on relying on cosines as researchers have been doing so far, may be a worthwhile step that can push accuracy results higher irrespective of the feature vector computation. ",
+        "bbox": [
+            174,
+            751,
+            825,
+            808
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/935c4eca4983802bd27443169378bfb5f4cb7b0e8294f2b20ac0072995c5a13b.jpg",
+        "table_caption": [
+            "Table 4: mAP when training different parts of the image retrieval pipeline. In blue, the modules that are fine-tunned in every experiment. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Features</td><td>Similarity</td><td>Oxford5k</td><td>Paris6k</td><td>Landmarks5k</td></tr><tr><td>MAC</td><td>Cosine</td><td>0.481</td><td>0.539</td><td>0.494</td></tr><tr><td>MAC</td><td>DeepSim</td><td>0.509</td><td>0.683</td><td>0.589</td></tr><tr><td>MAC</td><td>DeepSim</td><td>0.555</td><td>0.710</td><td>0.685</td></tr></table>",
+        "bbox": [
+            271,
+            853,
+            718,
+            912
+        ],
+        "page_idx": 13
+    }
+]

parse/train/Syg-YfWCW/Syg-YfWCW.md

@@ -0,0 +1,359 @@
+# GO FOR A WALK AND ARRIVE AT THE ANSWER: REASONING OVER PATHS IN KNOWLEDGE BASES USING REINFORCEMENT LEARNING
+
+Rajarshi Das?,1, Shehzaad Dhuliawala?,1, Manzil Zaheer?,2   
+Luke Vilnis1, Ishan Durugkar3, Akshay Krishnamurthy1, Alex Smola4, Andrew McCallum1   
+{rajarshi, sdhuliawala, luke, akshay, mccallum}@cs.umass.edu   
+manzil@cmu.edu, ishand@cs.utexas.edu, alex@smola.org   
+1University of Massachusetts, Amherst, 2Carnegie Mellon University   
+3University of Texas at Austin, 4Amazon Web Services
+
+# ABSTRACT
+
+Knowledge bases (KB), both automatically and manually constructed, are often incomplete — many valid facts can be inferred from the KB by synthesizing existing information. A popular approach to KB completion is to infer new relations by combinatory reasoning over the information found along other paths connecting a pair of entities. Given the enormous size of KBs and the exponential number of paths, previous path-based models have considered only the problem of predicting a missing relation given two entities, or evaluating the truth of a proposed triple. Additionally, these methods have traditionally used random paths between fixed entity pairs or more recently learned to pick paths between them. We propose a new algorithm, MINERVA, which addresses the much more difficult and practical task of answering questions where the relation is known, but only one entity. Since random walks are impractical in a setting with unknown destination and combinatorially many paths from a start node, we present a neural reinforcement learning approach which learns how to navigate the graph conditioned on the input query to find predictive paths. On a comprehensive evaluation on seven knowledge base datasets, we found MINERVA to be competitive with many current state-of-the-art methods.
+
+# 1 INTRODUCTION
+
+Automated reasoning, the ability of computing systems to make new inferences from observed evidence, has been a long-standing goal of artificial intelligence. We are interested in automated reasoning on large knowledge bases (KB) with rich and diverse semantics (Suchanek et al., 2007; Bollacker et al., 2008; Carlson et al., 2010). KBs are highly incomplete (Min et al., 2013), and facts not directly stored in a KB can often be inferred from those that are, creating exciting opportunities and challenges for automated reasoning. For example, consider the small knowledge graph in Figure 1. We can answer the question “Who did Malala Yousafzai share her Nobel Peace prize with?” from the following reasoning path: Malala Yousafzai WonAward Nobel Peace Prize $2 0 1 4 $ AwardedTo Kailash Satyarthi. Our goal is to automatically learn such reasoning paths in KBs. We frame the learning problem as one of query answering, that is to say, answering questions of the form (Malala Yousafzai, SharesNobelPrizeWith, ?).
+
+From its early days, the focus of automated reasoning approaches has been to build systems that can learn crisp symbolic logical rules (McCarthy, 1960; Nilsson, 1991). Symbolic representations have also been integrated with machine learning especially in statistical relational learning (Muggleton et al., 1992; Getoor & Taskar, 2007; Kok & Domingos, 2007; Lao et al., 2011), but due to poor generalization performance, these approaches have largely been superceded by distributed vector representations. Learning embedding of entities and relations using tensor factorization or neural methods has been a popular approach (Nickel et al., 2011; Bordes et al., 2013; Socher et al., 2013, inter alia), but these methods cannot capture chains of reasoning expressed by KB paths. Neural multi-hop models (Neelakantan et al., 2015; Guu et al., 2015; Toutanova et al., 2016) address the aforementioned problems to some extent by operating on KB paths embedded in vector space. However, these models take as input a set of paths which are gathered by performing random walks independent of the query relation. Additionally, models such as those developed in Neelakantan et al. (2015); Das et al. (2017) use the same set of initially collected paths to answer a diverse set of query types (e.g. MarriedTo, Nationality, WorksIn etc.).
+
+![](images/a69f2413af7626b10b3a41ec0a77ddfd181895d6947d7b00aa9e15fcf6bc7516.jpg)  
+Figure 1: A small fragment of a knowledge base represented as a knowledge graph. Solid edges are observed and dashed edges are part of queries. Note how each query relation (e.g. SharesNobelPrizeWith, Nationality, etc.) can be answered by traversing the graph via “logical” paths between entity ‘Malala Yousafzai’ and the corresponding answer.
+
+This paper presents a method for efficiently searching the graph for answer-providing paths using reinforcement learning (RL) conditioned on the input question, eliminating any need for precomputed paths. Given a massive knowledge graph, we learn a policy, which, given the query (entity1, relation, ?), starts from entity1 and learns to walk to the answer node by choosing to take a labeled relation edge at each step, conditioning on the query relation and entire path history. This formulates the query-answering task as a reinforcement learning (RL) problem where the goal is to take an optimal sequence of decisions (choices of relation edges) to maximize the expected reward (reaching the correct answer node). We call the RL agent MINERVA for ”Meandering In Networks of Entities to Reach Verisimilar Answers.”
+
+Our RL-based formulation has many desirable properties. First, MINERVA has the built-in flexibility to take paths of variable length, which is important for answering harder questions that require complex chains of reasoning (Shen et al., 2017). Secondly, MINERVA needs no pretraining and trains on the knowledge graph from scratch with reinforcement learning; no other supervision or fine-tuning is required representing a significant advance over prior applications of RL in NLP. Third, our path-based approach is computationally efficient, since by searching in a small neighborhood around the query entity it avoids ranking all entities in the KB as in prior work. Finally, the reasoning paths found by our agent automatically form an interpretable provenance for its predictions.
+
+The main contributions of the paper are: (a) We present agent MINERVA, which learns to do query answering by walking on a knowledge graph conditioned on an input query, stopping when it reaches the answer node. The agent is trained using reinforcement learning, specifically policy gradients $( \ S 2 )$ . (b) We evaluate MINERVA on several benchmark datasets and compare favorably to Neural Theorem Provers (NTP) (Rocktaschel & Riedel, 2017) and Neural LP (Yang et al., 2017), which do logical ¨ rule learning in KBs, and also state-of-the-art embedding based methods such as DistMult (Yang et al., 2015) and ComplEx (Trouillon et al., 2016) and ConvE (Dettmers et al., 2018). (c) We also extend MINERVA to handle partially structured natural language queries and test it on the WikiMovies dataset $( \ S \ 3 . 3 )$ (Miller et al., 2016).
+
+We also compare to DeepPath (Xiong et al., 2017) which uses reinforcement learning to pick paths between entity pairs. The main difference is that the state of their RL agent includes the answer entity since it is designed for the simpler task of predicting if a fact is true or not. As such their method cannot be applied directly to our more challenging query answering task where the second entity is unknown and must be inferred. Nevertheless, MINERVA outperforms DeepPath on their benchmark NELL-995 dataset when compared in their experimental setting $( \ S \ 3 . 2 . 2 )$ .
+
+# 2 TASK AND MODEL
+
+We formally define the task of query answering in a KB. Let $\mathcal { E }$ denote the set of entities and $\mathcal { R }$ denote the set of binary relations. A KB is a collection of facts stored as triplets $( \mathbf { e } _ { 1 } , \mathbf { r } , \mathbf { e } _ { 2 } )$ where $\mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \in \mathcal { E }$ and $\mathbf { r } \in \mathcal { R }$ . From the KB, a knowledge graph $\mathcal { G }$ can be constructed where the entities $\mathrm { e } _ { 1 } , \mathrm { e } _ { 2 }$ are represented as the nodes and relation $r$ as labeled edge between them. Formally, a knowledge graph is a directed labeled multigraph $\mathcal { G } = ( V , E , \mathcal { R } )$ , where $V$ and $E$ denote the vertices and edges of the graph respectively. Note that $V = \mathcal { E }$ and $E \subseteq V \times \mathcal { R } \times V$ . Also, following previous approaches (Bordes et al., 2013; Neelakantan et al., 2015; Xiong et al., 2017), we add the inverse relation of every edge, i.e. for an edge $( \mathbf { e } _ { 1 } , \mathbf { r } , \mathbf { e } _ { 2 } ) \in E$ , we add the edge $\left( \mathbf { e } _ { 2 } , \mathbf { r } ^ { - 1 } , \mathbf { e } _ { 1 } \right)$ to the graph. (If the set of binary relations $\mathcal { R }$ does not contain the inverse relation $\mathrm { r } ^ { - 1 }$ , it is added to $\mathcal { R }$ as well.)
+
+Since KBs have a lot of missing information, two natural tasks have emerged in the information extraction community - fact prediction and query answering. Query answering seeks to answer questions of the form $\left( \mathbf { e } _ { 1 } , \mathbf { r } , ? \right)$ , e.g. Toronto, locatedIn, ?, whereas fact prediction involves predicting if a fact is true or not, e.g. (Toronto, locatedIn, Canada)?. Algorithms for fact prediction can be used for query answering, but with significant computation overhead, since all candidate answer entities must be evaluated, making it prohibitively expensive for large KBs with millions of entities. In this work, we present a query answering model, that learns to efficiently traverse the knowledge graph to find the correct answer to a query, eliminating the need to evaluate all entities.
+
+Query answering reduces naturally to a finite horizon sequential decision making problem as follows: We begin by representing the environment as a deterministic partially observed Markov decision process on a knowledge graph $\mathcal { G }$ derived from the KB $( \ S 2 . 1 )$ . Our RL agent is given an input query of the form $\left( \mathrm { { e } _ { l q } , \mathrm { { r } _ { q } , \mathrm { { ? } } } } \right)$ . Starting from vertex corresponding to $e _ { 1 q }$ in $\mathcal { G }$ , the agent follows a path in the graph stopping at a node that it predicts as the answer $( \ S 2 . 2 )$ . Using a training set of known facts, we train the agent using policy gradients more specifically by REINFORCE (Williams, 1992) with control variates $( \ S 2 . 3 )$ . Let us begin by describing the environment.
+
+2.1 ENVIRONMENT - STATES, ACTIONS, TRANSITIONS AND REWARDS
+
+Our environment is a finite horizon, deterministic partially observed Markov decision process that lies on the knowledge graph $\mathcal { G }$ derived from the KB. On this graph we will now specify a deterministic partially observed Markov decision process, which is a 5-tuple $( \mathcal { S } , \mathcal { O } , \mathcal { A } , \mathring { \delta } , \bar { R } )$ , each of which we elaborate below.
+
+States. The state space $s$ consists of all valid combinations in $\mathcal { E } \times \mathcal { E } \times \mathcal { R } \times \mathcal { E }$ . Intuitively, we want a state to encode the query $( \mathrm { e } _ { \mathrm { l q } } , \mathrm { r } _ { \mathrm { q } } )$ , the answer $( { \bf e } _ { 2 { \bf q } } )$ , and a location of exploration $\mathrm { e _ { t } }$ (current location of the RL agent). Thus overall a state $S \in S$ is represented by $S = ( { \bf e } _ { \mathrm { t } } , { \bf e } _ { 1 \mathrm { q } } , { \bf r } _ { \mathrm { q } } , { \bf e } _ { 2 \mathrm { q } } )$ and the state space consists of all valid combinations.
+
+Observations. The complete state of the environment is not observed. Intuitively, the agent knows its current location $\displaystyle \left( \mathrm { e _ { t } } \right)$ and $( { \bf e } _ { \mathrm { l q } } , { \bf r } _ { \mathrm { q } } )$ , but not the answer $\left( \mathrm { e } _ { 2 \mathrm { q } } \right)$ , which remains hidden. Formally, the observation function $\mathcal { O } : \mathcal { S }  \mathcal { \bar { E } } \times \mathbf { \bar { \mathcal { E } } } \times \mathcal { R }$ is defined as $\mathcal { O } ( s = ( { \bf e } _ { \mathrm { t } } , { \bf e } _ { \mathrm { l q } } , { \bf r } _ { \mathrm { q } } , { \bf e } _ { 2 \mathrm { q } } ) ) = ( { \bf e } _ { \mathrm { t } } , { \bf e } _ { \mathrm { l q } } , { \bf r } _ { \mathrm { q } } )$ .
+
+Actions. The set of possible actions $\mathcal { A } _ { S }$ from a state $S = ( { \bf e } _ { \mathrm { t } } , { \bf e } _ { 1 \mathrm { q } } , { \bf r } _ { \mathrm { q } } , { \bf e } _ { 2 \mathrm { q } } )$ consists of all outgoing edges of the vertex $\mathrm { e _ { t } }$ in $\mathcal { G }$ . Formally $\begin{array} { r } { A _ { S } = \{ ( \mathbf { e } _ { \mathrm { t } } , r , \nu ) \in E : S = ( \mathbf { e } _ { \mathrm { t } } , \overset { \cdot } { \mathbf { e } _ { \mathrm { l q } } } , \overset { \cdot } { \mathbf { r } } _ { \mathrm { q } } , \overset { \cdot } { \mathbf { e } _ { \mathrm { 2 q } } } ) , r \in \mathcal { R } , \nu \in V \} \bigcup \{ ( s , \varpi , s ) \} } \end{array}$ Basically, this means an agent at each state has option to select which outgoing edge it wishes to take having the knowledge of the label of the edge $r$ and destination vertex $\nu$ .
+
+During implementation, we unroll the computation graph up to a fixed number of time steps T. We augment each node with a special action called ‘NO OP’ which goes from a node to itself. Some questions are easier to answer and needs fewer steps of reasoning than others. This design decision allows the agent to remain at a node for any number of time steps. This is especially helpful when the agent has managed to reach a correct answer at a time step $t < \mathrm { T }$ and can continue to stay at the ‘answer node’ for the rest of the time steps. Alternatively, we could have allowed the agent to take a special ‘STOP’ action, but we found the current setup to work sufficiently well. As mentioned before, we also add the inverse relation of a triple, i.e. for the triple $( e _ { 1 } , r , e _ { 2 } )$ , we add the triple $( e _ { 2 } , r ^ { - 1 } , e _ { 1 } )$ to the graph. We found this important because this actually allows our agent to undo a potentially wrong decision.
+
+Transition. The environment evolves deterministically by just updating the state to the new vertex incident to the edge selected by the agent. The query and answer remains the same. Formally, the transition function is $\delta : { \mathcal { S } } \times { \mathcal { A } } \to { \mathcal { S } }$ defined by $\bar { \boldsymbol { \delta } } ( S , \bar { A } ) = ( \nu , \mathbf { e } _ { 1 \mathrm { q } } , \mathbf { r } _ { \mathrm { q } } , \mathbf { e } _ { 2 \mathrm { q } } )$ , where $S = ( { \bf e } _ { \mathrm { t } } , { \bf e } _ { 1 \mathrm { q } } , { \bf r } _ { \mathrm { q } } , { \bf e } _ { 2 \mathrm { q } } )$ and $A = \left( \mathbf { e } _ { \mathrm { t } } , r , \nu \right)$ ).
+
+Rewards. We only have a terminal reward of $+ 1$ if the current location is the correct answer at the end and 0 otherwise. To elaborate, if $S _ { T } = ( { \bf e } _ { \mathrm { t } } , { \bf e } _ { \mathrm { l q } } , { \bf r } _ { \mathrm { q } } , { \bf e } _ { 2 \mathrm { q } } )$ is the final state, then we receive a reward of $+ 1$ if $\mathbf { e } _ { \mathrm { t } } = \mathbf { e } _ { 2 \mathrm { q } }$ else 0., i.e. $R ( S _ { T } ) = \mathbb { I } \{ \mathbf { e } _ { \mathrm { t } } = \mathbf { e } _ { 2 \mathrm { q } } \}$ .
+
+# 2.2 POLICY NETWORK
+
+To solve the finite horizon deterministic partially observable Markov decision process described above, we design a randomized non-stationary history-dependent policy $\boldsymbol { \pi } = ( \mathbf { d _ { 1 } } , \mathbf { d } _ { 2 } , . . . , \mathbf { d _ { T - 1 } } )$ , where $\mathbf { d _ { t } } : H _ { t } \to \mathcal { P } ( \bar { \mathcal { A } _ { S _ { t } } } )$ and history $H _ { t } = ( H _ { t - 1 } , A _ { t - 1 } , O _ { t } )$ is just the sequence of observations and actions taken. We restrict ourselves to policies parameterized by long short-term memory network (LSTM) (Hochreiter & Schmidhuber, 1997).
+
+An agent based on LSTM encodes the history $H _ { t }$ as a continuous vector $\mathbf { h } _ { \mathbf { t } } \in \mathbb { R } ^ { 2 d }$ . We also have embedding matrix $\mathbf { r } \in \mathbb { R } ^ { | \mathcal { R } | \times d }$ and $\mathbf { e } \in \mathbb { R } ^ { | \mathcal { E } | \times d }$ for the binary relations and entities respectively. The history embedding for $H _ { t } = ( H _ { t - 1 } , A _ { t - 1 } , O _ { t } )$ is updated according to LSTM dynamics:
+
+$$
+\mathbf { h } _ { \mathrm { t } } = \mathrm { L S T M } \left( \mathbf { h } _ { \mathrm { t - 1 } } , \left[ \mathbf { a } _ { \mathrm { t - 1 } } ; \mathbf { o } _ { \mathrm { t } } \right] \right)
+$$
+
+where $\mathbf { a _ { t - 1 } } \in \mathbb { R } ^ { d }$ and $\mathbf { o _ { t } } \in \mathbb { R } ^ { d }$ denote the vector representation for action/relation at time $t - 1$ and observation/entity at time $t$ respectively and $[ ; ]$ denote vector concatenation. To elucidate, $\mathbf { a _ { t - 1 } } = \mathbf { r } _ { A _ { t - 1 } }$ , i.e. the embedding of the relation corresponding to label of the edge the agent chose at time $t - 1$ and $\mathbf { o } _ { \mathbf { t } } = \mathbf { e } _ { \mathrm { e } _ { t } }$ if $O _ { t } = ( { \bf e } _ { \mathrm { t } } , { \bf e } _ { \mathrm { 1 q } } , { \bf r } _ { \mathrm { q } } )$ i.e. the embedding of the entity corresponding to vertex the agent is at time $t$ .
+
+Based on the history embedding $\mathbf { h _ { t } }$ , the policy network makes the decision to choose an action from all available actions $( \boldsymbol { \mathcal { A } } _ { S _ { t } } )$ conditioned on the query relation. Recall that each possible action represents an outgoing edge with information of the edge relation label $l$ and destination vertex/entity $d$ . So embedding for each $A \in { \mathcal { A } } _ { S _ { t } }$ is $[ \mathbf { r } _ { \mathrm { I } } ; \mathbf { e _ { d } } ]$ , and stacking embeddings for all the outgoing edges we obtain the matrix $\mathbf { A _ { t } }$ . The network taking these as inputs is parameterized as a two-layer feedforward network with ReLU nonlinearity which takes in the current history representation $\mathbf { h _ { t } }$ and the embedding for the query relation $\mathbf { r _ { q } }$ and outputs a probability distribution over the possible actions from which a discrete action is sampled. In other words,
+
+$$
+\begin{array} { r l } & { \mathbf { d } _ { \mathbf { t } } = \mathrm { s o f t m a x } ( \mathbf { A } _ { \mathbf { t } } ( \mathbf { W } _ { 2 } \mathrm { R e L U } ( \mathbf { W } _ { 1 } [ \mathbf { h } _ { \mathbf { t } } ; \mathbf { o } _ { \mathbf { t } } ; \mathbf { r } _ { \mathbf { q } } ] ) ) ) , } \\ & { A _ { t } \sim \mathrm { C a t e g o r i c a l } ( \mathbf { d } _ { \mathbf { t } } ) . } \end{array}
+$$
+
+Note that the nodes in $\mathcal { G }$ do not have a fixed ordering or number of edges coming out from them. The size of matrix $\mathbf { A _ { t } }$ is $| { \mathcal { A } } _ { S _ { t } } | \times 2 d$ , so the decision probabilities $d _ { t }$ lies on simplex of size $\lvert A _ { S _ { t } } \rvert$ . Also the procedure above is invariant to order in which edges are presented as desired and falls in purview of neural networks designed to be permutation invariant (Zaheer et al., 2017). Finally, to summarize, the parameters of the LSTM, the weights $\mathbf { W _ { 1 } } , \mathbf { W _ { 2 } }$ , the corresponding biases (not shown above for brevity), and the embedding matrices form the parameters θ of the policy network.
+
+# 2.3 TRAINING
+
+For the policy network $\scriptstyle ( \pi _ { \mathsf { \boldsymbol { \theta } } } )$ described above, we want to find parameters θ that maximize the expected reward:
+
+$$
+J ( \mathbf { \theta } ) = \mathbb { E } _ { ( e _ { 1 } , r , e _ { 2 } ) \sim D } \mathbb { E } _ { A _ { 1 } , \ldots , A _ { T - 1 } \sim \pi _ { \Theta } } [ R ( S _ { T } ) | S _ { 1 } = ( e _ { 1 } , e _ { 1 } , r , e _ { 2 } ) ] ,
+$$
+
+where we assume there is a true underlying distribution $( \mathbf { e } _ { 1 } , \mathbf { r } , \mathbf { e } _ { 2 } ) \sim D$ . To solve this optimization problem, we employ REINFORCE (Williams, 1992) as follows:
+
+• The first expectation is replaced with empirical average over the training dataset.   
+• For the second expectation, we approximate by running multiple rollouts for each training example. The number of rollouts is fixed and for all our experiments we set this number to 20. For variance reduction, a common strategy is to use an additive control variate baseline (Hammersley, 2013; Fishman, 2013; Evans & Swartz, 2000). We use a moving average of the cumulative discounted reward as the baseline. We tune the weight of this moving average as a hyperparameter. Note that in our experiments we found that using a learned baseline performed similarly, but we finally settled for cumulative discounted reward as the baseline owing to its simplicity.   
+• To encourage diversity in the paths sampled by the policy at training time, we add an entropy regularization term to our cost function scaled by a constant $( \beta )$ .
+
+Table 1: Statistics of various datasets used in experiments.   
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">#entities</td><td rowspan="2">#relations</td><td rowspan="2">#facts</td><td rowspan="2">#queries</td><td colspan="2">#degree</td></tr><tr><td>avg.</td><td>median</td></tr><tr><td>COUNTRIES</td><td>272</td><td>2</td><td>1158</td><td>24</td><td>4.35</td><td>4</td></tr><tr><td>UMLS</td><td>135</td><td>49</td><td>5,216</td><td>661</td><td>38.63</td><td>28</td></tr><tr><td>KINSHIP</td><td>104</td><td>26</td><td>10686</td><td>1074</td><td>82.15</td><td>82</td></tr><tr><td>WN18RR</td><td>40,945</td><td>11</td><td>86,835</td><td>3134</td><td>2.19</td><td>2</td></tr><tr><td>NELL-995</td><td>75,492</td><td>200</td><td>154,213</td><td>3992</td><td>4.07</td><td>1</td></tr><tr><td>FB15K-237</td><td>14,505</td><td>237</td><td>272,115</td><td>20,466</td><td>19.74</td><td>14</td></tr><tr><td>WikiMovies</td><td>43,230</td><td>9</td><td>196,453</td><td>9952</td><td>6.65</td><td>4</td></tr></table>
+
+<table><tr><td></td><td>ComplEx</td><td>ConvE</td><td>DistMult</td><td>NTP</td><td>NTP-λ</td><td>NeuralLP</td><td>MINERVA</td></tr><tr><td>S1</td><td>99.37±0.4</td><td>100.0±0.00</td><td>97.91±0.01</td><td>90.83±15.4</td><td>100.0±0.00</td><td>100.0±0.0</td><td>100.0±0.00</td></tr><tr><td>S2</td><td>87.95±2.8</td><td>99.0±1.00</td><td>69.18±2.38</td><td>87.40±11.7</td><td>93.04±0.40</td><td>75.1 ± 0.3</td><td>92.36±2.41</td></tr><tr><td>S3</td><td>48.44±6.3</td><td>86.0±5.00</td><td>15.79±0.64</td><td>56.68±17.6</td><td>77.26±17.0</td><td>92.2 ± 0.2</td><td>95.10±1.20</td></tr></table>
+
+Table 2: Performance on three tasks of COUNTRIES dataset with AUC-PR metric. MINERVA significantly outperforms all other methods on the hardest task (S3). Also variance across runs for MINERVA is lower compared to other methods.
+
+# 3 EXPERIMENTS
+
+We now present empirical studies for MINERVA in order to establish that (i) MINERVA is competitive for query answering on small (Sec. 3.1.1) as well as large KBs (Sec. 3.1.2), (ii) MINERVA is superior to a path based models that do not search the KB efficiently or train query specific models (Sec. 3.2), (iii) MINERVA can not only be used for well formed queries, but can also easily handle partially structured natural language queries (Sec 3.3), (iv) MINERVA is highly capable of reasoning over long chains, and (v) MINERVA is robust to train and has much faster inference time (Sec. 3.5).
+
+# 3.1 KNOWLEDGE BASE QUERY ANSWERING
+
+To gauge the reasoning capability of MINERVA, we begin with task of query answering on KB, i.e. we want to answer queries of the form $( e _ { 1 } , r , ? )$ . Note that, as mentioned in Sec. 2, this task is subtly different from fact checking in a KB. Also, as most of the previous literature works in the regime of fact checking, their ranking includes variations of both $( e _ { 1 } , r , x )$ and $( x , r , e _ { 2 } )$ . However, since we do not have access to $e _ { 2 }$ in case of question answering scenario the same ranking procedure does not hold for us – we only need to rank on $( e _ { 1 } , r , x )$ . This difference in ranking made it necessary for us to re-run all the implementations of previous work. We used the implementation or the best pre-trained models (whenever available) of Rocktaschel & Riedel (2017); Yang et al. (2017) and Dettmers et al. ¨ (2018). For MINERVA to produce a ranking of answer entities during inference, we do a beam search with a beam width of 50 and rank entities by the probability of the trajectory the model took to reach the entity and remaining entities are given a rank of $\infty$ .
+
+Method We compare MINERVA with various state-of-the-art models using $\mathrm { H I T S } @ 1 , 3 , 1 0$ and mean reciprocal rank (MRR), which are standard metrics for KB completion tasks. In particular we compare against embedding based models - DistMult (Yang et al., 2015), ComplEx (Trouillon et al., 2016) and ConvE (Dettmers et al., 2018). For ConvE and ComplEx, we used the implementation released by Dettmers et al. $( 2 0 1 8 ) ^ { 1 }$ on the best hyperparameter settings reported by them. For DistMult, we use our highly tuned implementation (e.g. which performs better than the state-of-the-art results of Toutanova et al. (2015)). We also compare with two recent work in learning logical rules in KB namely Neural Theorem Provers (NTP) (Rocktaschel & Riedel, 2017) and NeuralLP (Yang et al., ¨ 2017). Rocktaschel & Riedel (2017) also reports a NTP model which is trained with an additional ¨ objective function of ComplEx (NTP- $\mathcal { \lambda }$ ). For these models, we used the implementation released by corresponding authors 2 3, again on the best hyperparameter settings reported by them.
+
+<table><tr><td>Data</td><td>Metric</td><td>ComplEx</td><td>ConvE</td><td>DistMult</td><td>NTP</td><td>NTP-λ</td><td>NeuralLP</td><td>MINERVA</td></tr><tr><td rowspan="4">KINSHIP</td><td>HITS@1</td><td>0.754</td><td>0.697</td><td>0.808</td><td>0.500</td><td>0.759</td><td>0.475</td><td>0.605</td></tr><tr><td>HITs@3</td><td>0.910</td><td>0.886</td><td>0.942</td><td>0.700</td><td>0.798</td><td>0.707</td><td>0.812</td></tr><tr><td>HITs@10</td><td>0.980</td><td>0.974</td><td>0.979</td><td>0.777</td><td>0.878</td><td>0.912</td><td>0.924</td></tr><tr><td>MRR</td><td>0.838</td><td>0.797</td><td>0.878</td><td>0.612</td><td>0.793</td><td>0.619</td><td>0.720</td></tr><tr><td rowspan="4">UMLS</td><td>HITs@1</td><td>0.823</td><td>0.894</td><td>0.916</td><td>0.817</td><td>0.843</td><td>0.643</td><td>0.728</td></tr><tr><td>HITs@3</td><td>0.962</td><td>0.964</td><td>0.967</td><td>0.906</td><td>0.983</td><td>0.869</td><td>0.900</td></tr><tr><td>HITs@10</td><td>0.995</td><td>0.992</td><td>0.992</td><td>0.970</td><td>1.000</td><td>0.962</td><td>0.968</td></tr><tr><td>MRR</td><td>0.894</td><td>0.933</td><td>0.944</td><td>0.872</td><td>0.912</td><td>0.778</td><td>0.825</td></tr></table>
+
+Table 3: Query answering results on KINSHIP and UMLS datasets.
+
+# 3.1.1 SMALLER DATASETS
+
+Dataset We use three standard datasets: COUNTRIES (Bouchard et al., 2015), KINSHIP, and UMLS (Kok & Domingos, 2007). The COUNTRIES dataset ontains countries, regions, and subregions as entities and is carefully designed to explicitly test the logical rule learning and reasoning capabilities of link prediction models. The queries are of the form LocatedIn(c, ?) and the answer is a region (e.g. LocatedIn(Egypt, ?) with the answer as Africa). The dataset has 3 tasks (S1-3 in table 2) each requiring reasoning steps of increasing length and difficulty (see Rocktaschel & Riedel (2017) for ¨ more details about the tasks). Following the design of the COUNTRIES dataset, for task S1 and S2, we set the maximum path length $T = 2$ and for S3, we set $T = 3$ . The Unified Medical Language System (UMLS) dataset, is from biomedicine. The entities are biomedical concepts (e.g. disease, antibiotic) and relations are like treats and diagnoses. The KINSHIP dataset contains kinship relationships among members of the Alyawarra tribe from Central Australia. For these two task we use maximum path length $T = 2$ . Also, for MINERVA we turn off entity in (1) in these experiments.
+
+Observations For the COUNTRIES dataset, in Table 2 we report a stronger metric - the area under the precision-recall curve - as is common in the literature. We can see that MINERVA compares favorably or outperforms all the baseline models except on the task S2 of COUNTRIES, where the ensemble model NTP- $\mathscr { \lambda }$ and ConvE outperforms it, albeit with a higher variance across runs. Our gains are much more prominent in task S3, which is the hardest among all the tasks.
+
+The Kinship and UMLS datasets are small KB datasets with around 100 entities each and as we see from Table 3, embedding based methods (ConvE, ComplEx and DistMult) perform much better than methods which aim to learn logical rules (NTP, NeuralLP and MINERVA). On Kinship, MINERVA outperforms both NeuralLP and NTP and matches the HITS $@ 1 0$ performance of NTP on UMLS. Unlike COUNTRIES, these datasets were not designed to test the logical rule learning ability of models and given the small size, embedding based models are able to get really high performance. Combination of both methods gives a slight increase in performance as can be seen from the results of NTP- $\mathcal { \Lambda }$ . However, when we initialized MINERVA with pre-trained embeddings of ComplEx, we did not find a significant increase in performance.
+
+# 3.1.2 LARGER DATASETS
+
+Dataset Next we evaluate MINERVA on three large KG datasets - WN18RR, FB15K-237 and NELL995. The WN18RR (Dettmers et al., 2018) and FB15K-237 (Toutanova et al., 2015) datasets are created from the original WN18 and FB15K datasets respectively by removing various sources of test leakage, making the datasets more realistic and challenging. The NELL-995 dataset released by Xiong et al. (2017) has separate graphs for each query relation, where a graph for a query relation can have triples from the test set of another query relation. For the query answering experiment, we combine all the graphs and removed all test triples (and the corresponding triples with inverse relations) from the graph. We also noticed that several triples in the test set had an entity (source or target) that never appeared in the graph. Since, there will be no trained embeddings for those entities, we removed them from the test set. This reduced the size of test set from 3992 queries to 2818 queries.4
+
+<table><tr><td>Data</td><td>Metric</td><td>ComplEx</td><td>ConvE</td><td>DistMult</td><td>NeuralLP</td><td>Path-Baseline</td><td>MINERVA</td></tr><tr><td rowspan="4">WN18RR</td><td>HITs@1</td><td>0.382</td><td>0.403</td><td>0.410</td><td>0.376</td><td>0.017</td><td>0.413</td></tr><tr><td>HITS@3</td><td>0.433</td><td>0.452</td><td>0.441</td><td>0.468</td><td>0.025</td><td>0.456</td></tr><tr><td>HITs@10</td><td>0.480</td><td>0.519</td><td>0.475</td><td>0.657</td><td>0.046</td><td>0.513</td></tr><tr><td>MRR</td><td>0.415</td><td>0.438</td><td>0.433</td><td>0.463</td><td>0.027</td><td>0.448</td></tr><tr><td rowspan="4">FB15K-237</td><td>HITs@1</td><td>0.303</td><td>0.313</td><td>0.275</td><td>0.166</td><td>0.169</td><td>0.217</td></tr><tr><td>HITS@3</td><td>0.434</td><td>0.457</td><td>0.417</td><td>0.248</td><td>0.248</td><td>0.329</td></tr><tr><td>HITs@10</td><td>0.572</td><td>0.600</td><td>0.568</td><td>0.348</td><td>0.357</td><td>0.456</td></tr><tr><td>MRR</td><td>0.394</td><td>0.410</td><td>0.370</td><td>0.227</td><td>0.227</td><td>0.293</td></tr><tr><td rowspan="4">NELL-995</td><td>HITs@1</td><td>0.612</td><td>0.672</td><td>0.610</td><td>-</td><td>0.300</td><td>0.663</td></tr><tr><td>HITS@3</td><td>0.761</td><td>0.808</td><td>0.733</td><td>=</td><td>0.417</td><td>0.773</td></tr><tr><td>HITs@10</td><td>0.827</td><td>0.864</td><td>0.795</td><td></td><td>0.497</td><td>0.831</td></tr><tr><td>MRR</td><td>0.694</td><td>0.747</td><td>0.680</td><td>=</td><td>0.371</td><td>0.725</td></tr></table>
+
+Table 4: Query answering results on WN18RR, FB15K-237 and NELL-995 datasets. NeuralLP does not scale to NELL-995 and hence the entries are kept blank.
+
+Observations Table 4 reports the query answering results on the larger WN18RR, FB15K-237 and NELL-995 datasets. We could not include the results of NeuralLP on NELL-995 since it didn’t scale to that size. Similarly NTP did not scale to any of the larger datasets. Apart from these, we are the first to report a comprehensive summary of performance of all baseline methods on these datasets.
+
+On NELL-995, MINERVA performs comparably to embedding based methods such as DistMult and ComplEx and performs comparably with ConvE on the stricter HITS $@ 1$ metric. ConvE, however outperforms us on HITS $@ 1 0$ on NELL-995. On WN18RR, logic based based methods (NeuralLP, MINERVA) generally outperform embedding based methods, with MINERVA achieving the highest score on HITS $@ 1$ metric and NeuralLP significantly outperforming on HITS $@ 1 0$ .
+
+We observe that on FB15K-237, however, embedding based methods dominate over MINERVA and NeuralLP. Upon deeper inspection, we found that the query relation types of FB15K-237 knowledge graph differs significantly from others.
+
+Analysis of query relations of FB15k-237: We analyzed the type of query relation types on the FB15K-237 dataset. Following Bordes et al. (2013), we categorized the query relations into (M)any to 1, 1 to M or 1 to 1 relations. An example of a M to 1 relation would be ‘/people/profession’ (What is the profession of person $\mathbf { \bar { \Sigma } } ^ { \mathbf { \Sigma } } ( \mathbf { X } ^ { \prime } ?$ ). An example of 1 to M relation would be /music/instrument/instrumentalists (‘Who plays the music instrument X?’) or ‘/people/ethnicity/people’ (‘Who are people with ethnicity X?’). From a query answering point of view, the answer to these questions is a list of entities. However, during evaluation time, the model is evaluated based on whether it is able to predict the one target entity which is in the query triple. Also, since MINERVA outputs the end points of the paths as target entities, it is sometimes possible that the particular target entity of the triple does not have a path from the source entity (however there are paths to other ‘correct’ answer entities). Table 9 (in appendix) shows few other examples of relations belonging to different classes.
+
+Following Bordes et al. (2013), we classify a relation as 1-to-M if the ratio of cardinality of tail to head entities is greater than 1.5 and as M-to-1 if it is lesser than 0.67. In the validation set of FB15K-237, $54 \%$ of the queries are 1-to-M, whereas only $26 \%$ are M-to-1. Contrasting it with NELL-995, $2 7 \%$ are 1-to-M and $36 \%$ are M-to-1 or UMLS where only $18 \%$ are 1-to-M. Table 10 (in appendix) shows few relations from FB15K-237 dataset which have high tail-to-head ratio. The average ratio for 1-TO-M relations in FB15K-237 is 13.39 (substantially higher than 1.5). As explained before, the current evaluation scheme is not suited when it comes to 1-to-M relations and the high percentage of 1-to-M relations in FB $1 5 \mathrm { K } \cdot 2 3 7$ also explains the sub optimal performance of MINERVA.
+
+We also check the frequency of occurrence of various unique path types. We define a path type as the sequence of relation types (ignoring the entities) in a path. Intuitively, a predictive path which generalizes across queries will occur many number of times in the graph. Figure 2 shows the plot. As we can see, the characteristics of FB15K-237 is quite different from other datasets. For example, in NELL-995, more than 1000 different path types occur more than 1000 times. WN18RR has only 11 different relation types which means there are only $1 1 ^ { 3 }$ possible path types of length 3 and even fewer number of them would be predictive. As can be seen, there are few path types which occur more than $1 0 ^ { 4 }$ times and around 50 of them occur more than 1000 times. However in FB15K-237, which has the highest number of relation types, we observe a sharp decrease in the number of path types which occur a significant number of times. Since MINERVA cannot find path types which repeat often, it finds it hard to learn path types that generalize.
+
+![](images/553c62e7ed024a8406667f586884f02ed4deb83ab5b92d6c3f8461d12a706365.jpg)  
+Figure 2: Count of number of unique path types of length 3 which occur more than $\mathbf { \epsilon } \cdot \mathbf { \gamma } _ { \mathbf { X } } \mathbf { \epsilon } )$ times in various datasets. For example, in NELL-995 there are more than $1 0 ^ { 3 }$ path types which occur more than $1 0 ^ { 3 }$ times. However, for FB15k-237, we see a sharp decrease as $\mathbf { \epsilon } _ { \mathbf { X } } \mathbf { \epsilon } _ { \mathbf { X } } ,$ becomes higher, suggesting that path types do not repeat often.
+
+# 3.2 COMPARISON WITH PATH BASED MODELS
+
+# 3.2.1 WITH RANDOM WALK MODELS
+
+In this experiment, we compare to a model which gathers path based on random walks and tries to predict the answer entity. Neural multi-hop models (Neelakantan et al., 2015; Toutanova et al., 2016), operate on paths between entity pairs in a KB. However these methods need to know the target entity in order to pre-compute paths between entity pairs. (Guu et al., 2015) is an exception in this regard as they do random walks starting from a source entity $\cdot _ { e 1 } \cdot$ and then using the path, they train a classifier to predict the target answer entity. However, they only consider one path starting from a source entity. In contrast, Neelakantan et al. (2015); Toutanova et al. (2016) use information from multiple paths between the source and target entity. We design a baseline model which combines the strength of both these approaches. Starting from $\cdot _ { e _ { 1 } } ,$ , the model samples $k = 1 0 0$ ) random paths of up to a maximum length of $T = 3$ . Following Neelakantan et al. (2015), we encode each paths with an LSTM followed by a max-pooling operation to featurize the paths. This feature is concatenated with the source entity and query relation vector which is then passed through a feed forward network which scores all possible target entities. The network is trained with a multi-class cross entropy objective based on observed triples and during inference we rank target entities according to the model score.
+
+The PATH-BASELINE column of table 4 shows the performance of this model on the three datasets. As we can see MINERVA outperforms this baseline significantly. This shows that a model which predicts based on a set of randomly sampled paths does not do as well as MINERVA because it either loses important paths during random walking or it fails to aggregate predictive features from all the $k$ paths, many of which would be irrelevant to answer the given query. The latter is akin to the problem with distant supervision (Mintz et al., 2009), where important evidence gets lost amidst a plethora of irrelevant information. However, by taking each step conditioned on the query relation, MINERVA can effectively reduce the search space and focus on paths relevant to answer the query.
+
+# 3.2.2 WITH DEEPPATH
+
+We also compare MINERVA with DeepPath which uses RL to pick paths between entity pairs. For a fair comparison, we only rank the answer entities against the negative examples in the dataset used in their experiments5 and report the mean average precision (MAP) scores for each query relation. DeepPath feeds the paths its agent gathers as input features to the path ranking algorithm (PRA) (Lao et al., 2011), which trains a per-relation classifier. But unlike them, we train one model which learns for all query relations so as to enable our agent to leverage from correlations and more data. If our agent is not able to reach the correct entity or one of the negative entities, the corresponding entities gets a score of negative infinity. If MINERVA fails to reach any of the entities in the set of correct and negative entities. then we fall back to a random ordering of the entities. As show in table 5, we outperform them or achieve comparable performance for all the query relations For this experiment, we set the maximum length $T = 3$ . Although training per-relation models is cumbersome and does not scale to massive KBs with thousands of relation types, we also train per-relation models of MINERVA replicating the settings of DeepPath (MINERVAa in table 5). MINERVAa outperforms DeepPath and performs similarly to MINERVA which is an encouraging result since training one model which performs well for all relation is highly desirable.
+
+Table 5: MAP scores for different query relations on the NELL-995 dataset. Note that in this comparison, MINERVA refers to only a single learnt model for all query relations which is competitive with individual DeepPath models trained separately for each query relation. We also trained MINERVA in the setting of DeepPath, i.e. training per-relation models (MINERVAa)   
+
+<table><tr><td>Task</td><td>DeepPath</td><td>MINERVA</td><td>MINERVAa</td></tr><tr><td>athleteplaysinleague</td><td>0.960</td><td>0.970</td><td>0.940</td></tr><tr><td>worksfor</td><td>0.711</td><td>0.825</td><td>0.810</td></tr><tr><td>organizationhiredperson</td><td>0.742</td><td>0.851</td><td>0.856</td></tr><tr><td>athleteplayssport</td><td>0.957</td><td>0.985</td><td>0.980</td></tr><tr><td>teamplayssport</td><td>0.738</td><td>0.846</td><td>0.880</td></tr><tr><td>personborninlocation</td><td>0.757</td><td>0.793</td><td>0.780</td></tr><tr><td>personleadsorganization</td><td>0.795</td><td>0.851</td><td>0.877</td></tr><tr><td>athletehomestadium</td><td>0.890</td><td>0.895</td><td>0.898</td></tr><tr><td>organizationheadquarteredincity</td><td>0.790</td><td>0.946</td><td>0.940</td></tr><tr><td>athleteplaysforteam</td><td>0.750</td><td>0.824</td><td>0.800</td></tr></table>
+
+# 3.3 PARTIALLY STRUCTURED QUERIES
+
+Queries in KBs are structured in the form of triples. However, this is unsatisfactory since for most real applications, the queries appear in natural language. As a first step in this direction, we extend MINERVA to take in “partially structured” queries. We use the WikiMovies dataset (Miller et al., 2016) which contains questions in natural language albeit generated by templates created by human annotators. An example question is “Which is a film written by Herb
+
+<table><tr><td>Model</td><td>Accuracy</td></tr><tr><td>Memory Network</td><td>78.5</td></tr><tr><td rowspan="3">QA system Key-Value Memory Network</td><td>93.5</td></tr><tr><td>93.9</td></tr><tr><td>94.6</td></tr><tr><td>Neural LP MINERVA</td><td>96.7</td></tr></table>
+
+Table 6: Performance on WikiMovies
+
+Freed?”. WikiMovies also has an accompanying KB which can be used to answer all the questions.
+
+We link the entity occurring in the question to the KB via simple string matching. To form the vector representation of the query relation, we design a simple question encoder which computes the average of the embeddings of the question words. The word embeddings are learned from scratch and we do not use any pretrained embeddings. We compare our results with those reported in Yang et al. (2017) (table 6). For this experiment, we found that $T = 1$ sufficed, suggesting that WikiMovies is not the best testbed for multihop reasoning, but this experiment is a promising first step towards the realistic setup of using KBs to answer natural language question.
+
+# 3.4 GRID WORLD PATH FINDING
+
+While chains in KB need not be very long to get good empirical results (Neelakantan et al., 2015; Das et al., 2017; Yang et al., 2017), in principle MINERVA can be used to learn long reasoning chains. To evaluate the same, we test our model on a synthetic 16-by-16 grid world dataset created by Yang et al. (2017), where the task is to navigate to a particular cell (answer entity) starting from a random cell (start entity) by following a set of directions (query relation). The KB consists of atomic triples of the form ((2,1), North, (1,1)) – entity (1,1) is north of entity (2,1). The queries consists of a sequence of directions (e.g. North, SouthWest, East). The queries are classified into classes based on the path lengths. Figure 3 shows the accuracy on varying path lengths. Compared to Neural LP, MINERVA is much more robust to queries, which require longer path, showing minimal degradation in performance for even the longest path in the dataset.
+
+![](images/faf82edb7f06d44d51171d677bd4f2a3e34f3ddba99872d5109fc677e3f06c2e.jpg)  
+Figure 3: Grid world experiment: We significantly outperform NeuralLP for longer path lengths.
+
+![](images/8c641037b54000c547c9115d252d6c9bdf5580cbd9346d4ee179e44f36d5da7a.jpg)  
+Figure 4: Based on the query relation our agent assigns different probabilities to different actions. The dashed edges in the top row denote query relation. Examples in the bottom row are from the WikiMovies dataset and hence the questions are partially structured.
+
+# 3.5 FURTHER ANALYSIS
+
+Training time. Figure 5 plots the HITS $@ 1 0$ scores on the development set against the training time comparing MINERVA with DistMult. It can be seen that MINERVA converges to a higher score much faster than DistMult. It is also interesting to note that even during the early stages of the training, MINERVA has much higher performance than that of DistMult, as during these initial stages, MINERVA would just be doing random walks in the neighborhood of the source
+
+![](images/5c8daad0a603b56d7b6005e2d0a4bc85799c640df9c714ff454c59a0971848a6.jpg)  
+Figure 5: HITS@10 on the development set versus training time.
+
+entity $( e _ { 1 } )$ . This implies that MINERVA’s approach of searching for an answer in the neighborhood of $e _ { 1 }$ is a much more efficient and smarter strategy than ranking all entities in the knowledge graph (as done by DistMult and other related methods).
+
+Inference Time. At test time, embedding based methods such as ConvE, ComplEx and DistMult rank all entities in the graph. Hence, for a test-time query, the running time is always $\mathcal { O } ( | \mathcal { E } | )$ where $\mathcal { R }$ denotes the set of entities $\left( = \right.$ nodes) in the graph. MINERVA, on the other hand is efficient at inference time since it has to essentially search for answer entities in its local neighborhood. The many cost at inference time for MINERVA is to compute probabilities for all outgoing edges along the path. Thus inference time of MINERVA only depends on degree distribution of the graph. If we assume the knowledge graph to obey a power law degree distribution, like many natural graphs, then for MINERVA the average inference time can be shown to be $O \big ( \frac { \alpha } { \alpha - 1 } \big )$ , when the coefficient of the power law $\alpha > 1$ . The median inference time for MINERVA is $O ( 1 )$ for all values of $\alpha$ . Note that these quantities are independent of size of entities $| \mathcal { E } |$ . For instance, on the test dataset of WN18RR, the wall clock inference time of MINERVA is 63s whereas that of a GPU implementation of DistMult, which is the simplest among the lot, is 211s. Similarly the wall-clock inference time on the test set of NELL-995 for a GPU implementation of DistMult is 115s whereas that of MINERVA is 35s.
+
+Query based Decision Making. At each step before making a decision, our agent conditions on the query relation. Figure 4 shows examples, where based on the query relation, the probabilities are peaked on different actions. For example, when the query relation is WorksFor, MINERVA assigns a much higher probability of taking the edge CoachesTeam than AthletePlaysInLeague. We also see similar behavior on the WikiMovies dataset where the query consists of words instead of fixed schema relation.
+
+Model Robustness. Table 7 also reports the mean and standard deviation across three independent runs of MINERVA. We found it easy to obtain/reproduce the highest scores across several runs as can be seen from the low deviations in scores.
+
+Table 7: Mean and Standard deviation across runs for various datasets.
+
+Effectiveness of Remembering Path History. MINERVA encodes the history of decisions it has taken in the past using LSTMs. To test the importance of remembering the sequence of decisions, we did an ablation study in which the agent chose the next action based on only local information i.e. current entity and query and did not have access to the history $\mathrm { h } _ { \mathrm { t } }$ . For the KINSHIP dataset, we observe a $27 \%$ points decrease in HITS $@ 1$ and $13 \%$ decrease in HITS $@ 1 0$ . For grid-world, it is also not surprising that we see a big drop in performance. The final accuracy is 0.23 for path lengths 2-4 and 0.04 for lengths 8-10. For FB15K-237 the HITS $@ 1 0$ performance dropped from 0.456 to 0.408.
+
+NO-OP and Inverse Relations. At each step, MINERVA can choose to take a NO-OP edge and remain at the same node. This gives the agent the flexibility of taking paths of variable lengths. Some questions are easier to answer than others and require fewer steps of reasoning and if the agent reaches the answer early, it can choose to remain there. Example (i) in table 8 shows such an example. Similarly inverse relation gives the agent the ability to recover from a potentially wrong decision it has taken before. Example (ii) shows such an example, where the agent took a incorrect decision at the first step but was able to revert the decision because of the presence of inverted edges.
+
+Table 8: A few example of paths found by MINERVA on the COUNTRIES and NELL. MINERVA can learn general rules as required by the COUNTRIES dataset (example (i)). It can learn shorter paths if necessary (example (ii)) and has the ability to correct a previously taken decision (example (iii))   
+
+<table><tr><td>Dataset</td><td>HITS@1</td><td>HITS@3</td><td>HITS@10</td></tr><tr><td>NELL-995</td><td>0.66±0.029</td><td>0.77±0.0016</td><td>0.83±0.0030</td></tr><tr><td>FB15K-237</td><td>0.22±0.002</td><td>0.33 ±0.0008</td><td>0.46±0.0006</td></tr><tr><td>WN18RR</td><td>0.41±0.030</td><td>0.45±0.0180</td><td>0.51±0.0005</td></tr></table>
+
+# 4 RELATED WORK
+
+Learning vector representations of entities and relations using tensor factorization (Nickel et al., 2011; 2012; Bordes et al., 2013; Riedel et al., 2013; Nickel et al., 2014; Yang et al., 2015) or neural methods (Socher et al., 2013; Toutanova et al., 2015; Verga et al., 2016) has been a popular approach to reasoning with a knowledge base. However, these methods cannot capture more complex reasoning patterns such as those found by following inference paths in KBs. Multi-hop link prediction approaches (Lao et al., 2011; Neelakantan et al., 2015; Guu et al., 2015; Toutanova et al., 2016; Das et al., 2017) address the problems above, but the reasoning paths that they operate on are gathered by performing random walks independent of the type of query relation. Lao et al. (2011) further filters paths from the set of sampled paths based on the restriction that the path must end at one of the target entities in the training set and are within a maximum length. These constraints make them query dependent but they are heuristic in nature. Our approach eliminates any necessity to pre-compute paths and learns to efficiently search the graph conditioned on the input query relation.
+
+Inductive Logic Programming (ILP) (Muggleton et al., 1992) aims to learn general purpose predicate rules from examples and background knowledge. Early work in ILP such as FOIL (Quinlan, 1990), PROGOL (Muggleton, 1995) are either rule-based or require negative examples which is often hard to find in KBs (by design, KBs store true facts). Statistical relational learning methods (Getoor & Taskar, 2007; Kok & Domingos, 2007; Schoenmackers et al., 2010) along with probabilistic logic (Richardson & Domingos, 2006; Broecheler et al., 2010; Wang et al., 2013) combine machine learning and logic but these approaches operate on symbols rather than vectors and hence do not enjoy the generalization properties of embedding based approaches.
+
+There are few prior work which treat inference as search over the space of natural language. Nogueira & Cho (2016) propose a task (WikiNav) in which each the nodes in the graph are Wikipedia pages and the edges are hyperlinks to other wiki pages. The entity is to be represented by the text in the page and hence the agent is required to reason over natural language space to navigate through the graph. Similar to WikiNav is Wikispeedia (West et al., 2009) in which an agent needs to learn to traverse to a given target entity node (wiki page) as quickly as possible. Angeli & Manning (2014) propose natural logic inference in which they cast the inference as a search from a query to any valid premise. At each step, the actions are one of the seven lexical relations introduced by MacCartney & Manning (2007).
+
+Neural Theorem Provers (NTP) (Rocktaschel & Riedel, 2017) and Neural LP (Yang et al., 2017) are ¨ methods to learn logical rules that can be trained end-to-end with gradient based learning. NTPs are constructed by Prolog’s backward chaining inference method. It operates on vectors rather than symbols, thereby providing a success score for each proof path. However, since a score can be computed between any two vectors, the computation graph becomes quite large because of such soft-matching during substitution step of backward chaining. For tractability, it resorts to heuristics such as only keeping the top-K scoring proof paths trading-off guarantees for exact gradients. Also the efficacy of NTPs has yet to be shown on large KBs. Neural LP introduces a differential rule learning system using operators defined in TensorLog (Cohen, 2016). It has a LSTM based controller with a differentiable memory component (Graves et al., 2014; Sukhbaatar et al., 2015) and the rule scores are calculated via attention. Even though, differentiable memory allows end to end training, it necessitates accessing the entire memory, which can be computationally expensive. RL approaches capable of hard selection of memory (Zaremba & Sutskever, 2015) are computationally attractive. MINERVA uses a similar hard selection of relation edges to walk on the graph. More importantly, MINERVA outperforms both these methods on their respective benchmark datasets.
+
+DeepPath (Xiong et al., 2017) uses RL based approaches to find paths in KBs. However, the state of their MDP requires the target entity to be known in advance and hence their path finding strategy is dependent on knowing the answer entity. MINERVA does not need any knowledge of the target entity and instead learns to find the answer entity among all entities. DeepPath, additionally feeds its gathered paths to Path Ranking Algorithm (Lao et al., 2011), whereas MINERVA is a complete system trained to do query answering. DeepPath also uses fixed pretrained embeddings for its entity and relations. Lastly, on comparing MINERVA with DeepPath in their experimental setting on the NELL dataset, we match their performance or outperform them. MINERVA is also similar to methods for learning to search for structured prediction (Collins & Roark, 2004; Daume III & Marcu, 2005; ´ Daume III et al., 2009; Ross et al., 2011; Chang et al., 2015). These methods are based on imitating a ´ reference policy (oracle) which make near-optimal decision at every step. In our problem setting, it is unclear what a good reference policy would be. For example, a shortest path oracle between two entities would be unideal, since the answer providing path should depend on the query relation.
+
+# 5 CONCLUSION
+
+We explored a new way of automated reasoning on large knowledge bases in which we use the knowledge graphs representation of the knowledge base and train an agent to walk to the answer node conditioned on the input query. We achieve state-of-the-art results on multiple benchmark knowledge base completion tasks and we also show that our model is robust and can learn long chains-ofreasoning. Moreover it needs no pretraining or initial supervision. Future research directions include applying more sophisticated RL techniques and working directly on textual queries and documents.
+
+# ACKNOWLEDGEMENTS
+
+We are grateful to Patrick Verga for letting us use his implementation of DistMult. This work was supported in part by the Center for Data Science and the Center for Intelligent Information Retrieval, in part by DARPA under agreement number FA8750-13-2-0020, in part by Defense Advanced Research Agency (DARPA) contract number HR0011-15-2-0036, in part by the National Science Foundation (NSF) grant numbers DMR-1534431 and IIS-1514053 and in part by the Chan Zuckerberg Initiative under the project Scientific Knowledge Base Construction. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the sponsor.
+
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+Patrick Verga, David Belanger, Emma Strubell, Benjamin Roth, and Andrew McCallum. Multilingual relation extraction using compositional universal schema. In NAACL, 2016.   
+William Yang Wang, Kathryn Mazaitis, and William W Cohen. Programming with personalized pagerank: a locally groundable first-order probabilistic logic. In CIKM, 2013.   
+Robert West, Joelle Pineau, and Doina Precup. Wikispeedia: An online game for inferring semantic distances between concepts. In IJCAI, 2009.   
+Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 1992.   
+Wenhan Xiong, Thien Hoang, and William Yang Wang. Deeppath: A reinforcement learning method for knowledge graph reasoning. In EMNLP, 2017.   
+Bishan Yang, Wen-tau Yih, Xiaodong He, Jianfeng Gao, and Li Deng. Embedding entities and relations for learning and inference in knowledge bases. In ICLR, 2015.   
+Fan Yang, Zhilin Yang, and William W Cohen. Differentiable learning of logical rules for knowledge base reasoning. In NIPS, 2017.   
+Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Ruslan Salakhutdinov, and Alexander Smola. Deep sets. In NIPS, 2017.   
+Wojciech Zaremba and Ilya Sutskever. Reinforcement learning neural turing machines. arXiv:1505.00521, 2015.
+
+Table 9: Few example facts belonging to m to 1, 1 to m relations in FB15K-237   
+
+<table><tr><td rowspan=1 colspan=1>Relation</td><td rowspan=1 colspan=1>tail/head</td></tr><tr><td rowspan=1 colspan=1>/people/marriage_union_type/unions_of-this_type./people/marriage/location_of_ceremony</td><td rowspan=1 colspan=1>129.75</td></tr><tr><td rowspan=1 colspan=1>/organization/role/leaders./organization/leadership/organization</td><td rowspan=1 colspan=1>65.15</td></tr><tr><td rowspan=1 colspan=1>/location/country/second_level_divisions</td><td rowspan=1 colspan=1>49.18</td></tr><tr><td rowspan=1 colspan=1>/user/ktrueman/default_domain/international_organization/member_states</td><td rowspan=1 colspan=1>36.5</td></tr><tr><td rowspan=1 colspan=1>/base/marchmadness/ncaa_basketball_tournament/seeds./base/marchmadness/ncaa_tournament_seed/team</td><td rowspan=1 colspan=1>33.6</td></tr></table>
+
+Table 10: Few example 1-to-M relations from FB15K-237 with high cardinality ratio of tail to head.
+
+# 6 APPENDIX
+
+# 6.1 HYPERPARAMETERS
+
+Experimental Details We choose the relation and embedding dimension size as 200. The action embedding is formed by concatenating the entity and relation embedding. We use a 3 layer LSTM with hidden size of 400. The hidden layer size of MLP (weights $\mathbf { W _ { 1 } }$ and $\mathbf { W } _ { 2 }$ ) is set to 400. We use Adam (Kingma & Ba, 2014) with the default parameters in REINFORCE for the update.
+
+In our experiments, we tune our model over two hyper parameters, viz., $\beta$ which is the entropy regularization constant and $\lambda$ which is the moving average constant for the REINFORCE baseline. The table 11 lists the best hyper parameters for all the datasets.
+
+Table 11: Best hyper parameters   
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>β</td><td rowspan=1 colspan=1>入</td><td rowspan=1 colspan=1>Path Length</td></tr><tr><td rowspan=1 colspan=1>UMLS</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>KINSHIP</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>Countries S1</td><td rowspan=1 colspan=1>0.01</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>Countries S2</td><td rowspan=1 colspan=1>0.02</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>Countries S3</td><td rowspan=1 colspan=1>0.01</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>3</td></tr><tr><td rowspan=1 colspan=1>WN18RR</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>3</td></tr><tr><td rowspan=1 colspan=1>NELL-995</td><td rowspan=1 colspan=1>0.06</td><td rowspan=1 colspan=1>0.0</td><td rowspan=1 colspan=1>3</td></tr><tr><td rowspan=1 colspan=1>FB15K-237</td><td rowspan=1 colspan=1>0.02</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>3</td></tr><tr><td rowspan=1 colspan=1>WIKIMOVIES</td><td rowspan=1 colspan=1>0.15</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td></tr></table>
+
+# 6.2 ADDENDUM TO NELL RESULTS
+
+The NELL dataset released by Xiong et al. (2017) includes two additional tasks for which the scores were not reported in the paper and so we were unable to compare them against DeepPath. Nevertheless, we ran MINERVA on these tasks and report our results in table 12 for completeness.
+
+Table 12: NELL results for the remaining tasks   
+
+<table><tr><td>Task</td><td>Single Model</td><td>DeepPath setup</td></tr><tr><td>agentbelongstoorganization</td><td>0.86</td><td>0.87</td></tr><tr><td>teamplaysinleague</td><td>0.97</td><td>0.95</td></tr></table>

parse/train/SyxrxR4KPS/SyxrxR4KPS.md

@@ -0,0 +1,320 @@
+# DEEP NEUROETHOLOGY OF A VIRTUAL RODENT
+
+Josh Merel?1, Diego Aldarondo $^ { \star 2 , 3 }$ , Jesse Marshal $^ { \star 3 , 4 }$ , Yuval Tassa1, Greg Wayne1, Bence Olveczky ¨ 3,4
+
+1DeepMind, London, UK.   
+2Program in Neuroscience, 3Center for Brain Science, 4Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA.   
+jsmerel@google.com, diegoaldarondo@g.harvard.edu,   
+jesse d marshall@fas.harvard.edu
+
+# ABSTRACT
+
+Parallel developments in neuroscience and deep learning have led to mutually productive exchanges, pushing our understanding of real and artificial neural networks in sensory and cognitive systems. However, this interaction between fields is less developed in the study of motor control. In this work, we develop a virtual rodent as a platform for the grounded study of motor activity in artificial models of embodied control. We then use this platform to study motor activity across contexts by training a model to solve four complex tasks. Using methods familiar to neuroscientists, we describe the behavioral representations and algorithms employed by different layers of the network using a neuroethological approach to characterize motor activity relative to the rodent’s behavior and goals. We find that the model uses two classes of representations which respectively encode the task-specific behavioral strategies and task-invariant behavioral kinematics. These representations are reflected in the sequential activity and population dynamics of neural subpopulations. Overall, the virtual rodent facilitates grounded collaborations between deep reinforcement learning and motor neuroscience.
+
+# 1 INTRODUCTION
+
+Animals have nervous systems that allow them to coordinate their movement and perform a diverse set of complex behaviors. Mammals, in particular, are generalists in that they use the same general neural network to solve a wide variety of tasks. This flexibility in adapting behaviors towards many different goals far surpasses that of robots or artificial motor control systems. Hence, studies of the neural underpinnings of flexible behavior in mammals could yield important insights into the classes of algorithms capable of complex control across contexts and inspire algorithms for flexible control in artificial systems (Merel et al., 2019b).
+
+Recent efforts at the interface of neuroscience and machine learning have sparked renewed interest in constructive approaches in which artificial models that solve tasks similar to those solved by animals serve as normative models of biological intelligence. Researchers have attempted to leverage these models to gain insights into the functional transformations implemented by neurobiological circuits, prominently in vision (Khaligh-Razavi & Kriegeskorte, 2014; Yamins et al., 2014; Kar et al., 2019), but also increasingly in other areas, including audition (Kell et al., 2018) and navigation (Banino et al., 2018; Cueva & Wei, 2018). Efforts to construct models of biological locomotion systems have informed our understanding of the mechanisms and evolutionary history of bodies and behavior (Grillner et al., 2007; Ijspeert et al., 2007; Ramdya et al., 2017; Nyakatura et al., 2019). Neural control approaches have also been applied to the study of reaching movements, though often in constrained behavioral paradigms (Lillicrap & Scott, 2013), where supervised training is possible (Sussillo et al., 2015; Michaels et al., 2019).
+
+While these approaches model parts of the interactions between animals and their environments (Chiel & Beer, 1997), none attempt to capture the full complexity of embodied control, involving how an animal uses its senses, body and behaviors to solve challenges in a physical environment.
+
+The development of models of embodied control is valuable to the field of motor neuroscience, which typically focuses on restricted behaviors in controlled experimental settings. It is also valuable for AI research, where flexible models of embodied control could be applicable to robotics.
+
+Here, we introduce a virtual model of a rodent to facilitate grounded investigation of embodied motor systems. The virtual rodent affords a new opportunity to directly compare principles of artificial control to biological data from real-world rodents, which are more experimentally accessible than humans. We draw inspiration from emerging deep reinforcement learning algorithms which now allow artificial agents to perform complex and adaptive movement in physical environments with sensory information that is increasingly similar to that available to animals (Peng et al., 2016; 2017; Heess et al., 2017; Merel et al., 2019a;c). Similarly, our virtual rodent exists in a physical world, equipped with a set of actuators that must be coordinated for it to behave effectively. It also possesses a sensory system that allows it to use visual input from an egocentric camera located on its head and proprioceptive input to sense the configuration of its body in space.
+
+There are several questions one could answer using the virtual rodent platform. Here we focus on the problem of embodied control across multiple tasks. While some efforts have been made to analyze neural activity in reduced systems trained to solve multiple tasks (Song et al., 2017; Yang et al., 2019), those studies lacked the important element of motor control in a physical environment. Our rodent platform presents the opportunity to study how representations of movements as well as sequences of movements change as a function of goals and task contexts.
+
+To address these questions, we trained our virtual rodent to solve four complex tasks within a physical environment, all requiring the coordinated control of its body. We then ask “Can a neuroscientist understand a virtual rodent?” – a more grounded take on the originally satirical “Can a biologist fix a radio?” (Lazebnik, 2002) or the more recent “Could a neuroscientist understand a microprocessor?” (Jonas & Kording, 2017). We take a more sanguine view of the tremendous advances that have been made in computational neuroscience in the past decade, and posit that the supposed ‘failure’ of these approaches in synthetic systems is partly a misdirection. Analysis approaches in neuroscience were developed with the explicit purpose of understanding sensation and action in real brains, and often implicitly rooted in the types of architectures and processing that are thought relevant in biological control systems. With this philosophy, we use analysis approaches common in neuroscience to explore the types of representations and dynamics that the virtual rodent’s neural network employs to coordinate multiple complex movements in the service of solving motor and cognitive tasks.
+
+# 2 APPROACH
+
+# 2.1 VIRTUAL RODENT BODY
+
+![](images/8639b3c7d5c3c3b4825ff38bb186f6c5cf1b61335864c27ebc9506840809e1f3.jpg)  
+Figure 1: (A) Anatomical skeleton of a rodent (as reference; not part of physical simulation). (B) A body designed around the skeleton to match the anatomy and model collisions with the environment. (C) Purely cosmetic skin to cover the body. (D) Semi-transparent visualization of (A)-(C) overlain.
+
+We implemented a virtual rodent body (Figure 1) in MuJoCo (Todorov et al., 2012), based on measurements of laboratory rats (see Appendix A.1). The rodent body has 38 controllable degrees of freedom. The tail, spine, and neck consist of multiple segments with joints, but are controlled by tendons that co-activate multiple joints (spatial tendons in MuJoCo). The rodent will be released as part of dm control/locomotion.
+
+The virtual rodent has access to proprioceptive information as well as “raw” egocentric RGB-camera $6 4 \times 6 4$ pixels) input from a head-mounted camera. The proprioceptive inputs include internal joint angles and angular velocities, the positions and velocities of the tendons that provide actuation, egocentric vectors from the root (pelvis) of the body to the positions of the head and paws, a vestibular-like upright orientation vector, touch or contact sensors in the paws, as well as egocentric acceleration, velocity, and 3D angular velocity of the root.
+
+# 2.2 VIRTUAL RODENT TASKS
+
+![](images/023d3ca2fd213021246063387e1a45996b8d869e35ac665ccb43f84c5fc03c9d.jpg)  
+Figure 2: Visualizations of four tasks the virtual rodent was trained to solve: (A) jumping over gaps (“gaps run”), (B) foraging in a maze (“maze forage”), (C) escaping from a hilly region (“bowl escape”), and (D) touching a ball twice with a forepaw with a precise timing interval between touches (“two-tap”).
+
+We implemented four tasks adapted from previous work in deep reinforcement learning and motor neuroscience (Merel et al., 2019a; Tassa et al., 2018; Kawai et al., 2015) to encourage diverse motor behaviors in the rodent. The tasks are as follows: (1) Run along a corridor, over “gaps”, with a reward for traveling along the corridor at a target velocity (Figure 2A). (2) Collect all the blue orbs in a maze, with a sparse reward for each orb collected (Figure 2B). (3) Escape a bowl-shaped region by traversing hilly terrain, with a reward proportional to distance from the center of the bowl (Figure 2C). (4) Approach orbs in an open field, activate them by touching them with a forepaw, and touch them a second time after a precise interval of $8 0 0 \mathrm { m s }$ with a tolerance of $\pm 1 0 0 \mathrm { m s }$ ; there is a time-out period if the touch is not within the tolerated window and rewards are provided sparsely on the first and second touch (Figure 2D). We did not provide the agent with a cue or context indicating its task. Rather, the agent had to infer the task from the visual input and behave appropriately.
+
+# 2.3 TRAINING A MULTI-TASK POLICY
+
+![](images/11a13f9194d634d2f4aab607177c6bdcb2c975788a5ea199ecef4324e4e17d02.jpg)  
+Figure 3: The virtual rodent agent architecture. Egocentric visual image inputs are encoded into features via a small residual network (He et al., 2016) and proprioceptive state observations are encoded via a small multi-layer perceptron. The features are passed into a recurrent LSTM module (Hochreiter & Schmidhuber, 1997). The core module is trained by backpropogation during training of the value function. The outputs of the core are also passed as features to the policy module (with the dashed arrow indicating no backpropogation along this path during training) along with shortcut paths from the proprioceptive observations as well as encoded features. The policy module consists of one or more stacked LSTMs (with or without skip connections) which then produce the actions via a stochastic policy.
+
+Emboldened by recent results in which end-to-end RL produces a single terrain-adaptive policy (Peng et al., 2016; 2017; Heess et al., 2017), we trained a single architecture on the multiple motorcontrol-reliant tasks (see Figure 3). To train a single policy to perform all four tasks, we used an IMPALA-style setup for actor-critic DeepRL (Espeholt et al., 2018); parallel workers collected rollouts, logged them to a replay, from which a central learner sampled data to perform updates. The value-function critic was trained using off-policy correction via V-trace. To update the actor, we used a variant of MPO (Abdolmaleki et al., 2018) where the E-step is performed using advantages determined from the empirical returns and the value-function, instead of the Q-function (Song et al., 2019). Empirically, we found that the “escape” task was more challenging to learn during interleaved training relative to the other tasks. Consequently, we present results arising from training a singletask expert on the escape task and training the multi-task policies using kickstarting for that task (Schmitt et al., 2018), with a weak coefficient (.001 or .005). Kickstarting on this task made the seeds more reliably solve all four tasks, facilitating comparison of the multi-task policies with different architectures (i.e. the policy having 1, 2, or 3 layers, with or without skip connections across those layers). The procedure yields a single neural network that uses visual inputs to determine how to behave and coordinates its body to move in ways required to solve the tasks. See video examples of a single policy solving episodes of each task: gaps, forage, escape, and two-tap.
+
+![](images/91b42c963300f5a2b7162b067eb9426bb53d1881f5c829d0623f855e6e285fe6.jpg)  
+Figure 4: Ethology of the virtual rodent. (A) Example jumping sequence in gaps run task with a representative subset of recorded behavioral features. Dashed lines denote the time of the corresponding frames (top). (B) tSNE embedding of 60 behavioral features describing the pose and kinematics of the virtual rodent allows identification of rodent behaviors. Points are colored by hand-labeling of behavioral clusters identified by watershed clustering. (C) The first two principal components of different behavioral features reveals that behaviors are more shared across tasks at short, $5 { - } 2 5 \ \mathrm { H z }$ timescales (fast kinematics), but no longer $0 . 3 – 5 \mathrm { H z }$ timescales (slow kinematics).
+
+We analyzed the virtual rodent’s neural network activity in conjunction with its behavior to characterize how it solves multiple tasks (Figure 4A). We used analyses and perturbation techniques adapted from neuroscience, where a range of techniques have been developed to highlight the properties of real neural networks. Biological neural networks have been hypothesized to control, select, and modulate movement through a variety of debated mechanisms, ranging from explicit neural representations of muscle forces and behavioral primitives, to more abstract production of neural dynamics that could underly movement (Graziano, 2006; Kalaska, 2009; Churchland et al., 2012). A challenge with nearly all of these models however is that they have largely been inspired by findings from individual behavioral tasks, making it unclear how to generalize them to a broader range of naturalistic behaviors. To provide insight into mechanisms underlying movement in the virtual rodent, and to potentially give insight by proxy into the mechanisms underlying behavior in real rats, we thus systematically tested how the different network layers encoded and generated different aspects of movement.
+
+For all analyses we logged the virtual rodent’s kinematics, joint angles, computed forces, sensory inputs, and the cell unit activity of the LSTMs in core and policy layers during 25 trials per task from each network architecture.
+
+# 3.1 VIRTUAL RODENTS EXHIBIT BEHAVIORAL FLEXIBILITY.
+
+We began our analysis by quantitatively describing the behavioral repertoire of the virtual rodent. A challenge in understanding the neural mechanisms underlying behavior is that it can be described at many timescales. On short timescales, one could describe rodent locomotion using a set of actuators that produce joint-specific patterns of forces and kinematics. However on longer timescales, these force patterns are organized into coordinated, re-used movements, such as running, jumping, and turning. These movements can be further combined to form behavioral strategies or goal-directed behaviors. Relating neural representations to motor behaviors therefore requires analysis methods that span multiple timescales of behavioral description. To systematically examine the classes of behaviors these networks learn to generate and how they are differentially deployed across tasks, we developed sets of behavioral features that describe the kinematics of the animal on fast (5-25 $\mathrm { H z }$ ), intermediate $\mathrm { 1 - 2 5 \ : H z ) }$ or slow $( 0 . 3 { - } 5 \mathrm { H z } )$ timescales (Appendix A.2, A.3 ). As validation that these features reflected meaningful differences across behaviors, embedding these features using tSNE (Maaten & Hinton, 2008) produced a behavioral map in which virtual rodent behaviors, were segregated to different regions of the map (Figure 4B)(see video). This behavioral repertoire of the virtual rodent consisted of many behaviors observed in rodents, such as rearing, jumping, running, climbing and spinning. While the exact kinematics of the virtual rodent’s behaviors did not exactly match those observed in real rats, they did reproduce unexpected features. For instance the stride frequency of the virtual rodent during galloping matches that observed in rats (Appendix A.3).
+
+We next investigated how these behaviors were used by the virtual rodent across tasks. On short timescales, low-level motor features like joint speed and actuator forces occupied similar regions in principal component space (Figure 4C). In contrast, behavioral kinematics, especially on long, $0 . 3 – 5 \ \mathrm { H z }$ timescales, were more differentiated across tasks. Similar results held when examining overlap in other dimensions using multidimensional scaling. Overall this suggests that the network learned to adapt similar movements in a selective manner for different tasks, suggesting that the agent exhibited a form of behavioral flexibility.
+
+![](images/67bcf88435a27a7365450b59b3b78035122bee8d5d50547b9b995054a5efcee3.jpg)  
+3.2 NETWORKS PRIMARILY REFLECT BEHAVIORS, NOT FORCES   
+Figure 5: Representational structure of the rodent’s neural network. (A) Example similarity matrices of neural networks and behavioral descriptors. We grouped behavioral descriptors into 50 clusters that and we computed the average neural population vector during each cluster (AppendixA.4). Similarity was assessed by computing the dot product of either the neural population vector or the behavioral feature vector within each cluster. (B) Centered Kernel Alignment (CKA) index of neural and behavioral feature similarity matrices for 3 and 1 policy layer architectures. (C) CKA index of feature similarity matrices across all pairs of network layers. (D) Average CKA index between core and policy layers and behavioral features, compared across architectures. Points show values from individual network seeds. Policy values are averaged across layers.
+
+We next examined the neural activity patterns underlying the virtual rodent’s behavior to test if networks produced behaviors through explicit representations of forces, kinematics or behaviors. As expected, core and policy units operate on distinct timescales (See Appendix A.3, Figure 9). Units in the core typically fluctuated over timescales of 1-10 seconds, likely representing variables associated with context and reward. In contrast, units in policy layers were more active over subsecond timescales, potentially encoding motor and behavioral features.
+
+To quantify which aspects of behavior were encoded in the core and policy layers, and how these patterns varied across layers, we used representational similarity analysis (RSA) (Kriegeskorte et al., 2008; Kriegeskorte & Diedrichsen, 2019). RSA provides a global measure of how well different features are encoded in layers of a neural network by analyzing the geometries of network activity upon exposure to several stimuli, such as objects. To apply RSA, first a representational similarity (or equivalently, dissimilarity) matrix is computed that quantifies the similarity of neural population responses to a set of stimuli. To test if different neural populations show similar stimulus encodings, these similarity matricies can then be directly compared across different network layers. Multiple metrics, such as the matrix correlation or dot product can be used to compare these neural representational similarity matricies. Here we used the linear centered kernel alignment (CKA) index, which shows invariance to orthonormal rotations of population activity (Kornblith et al., 2019).
+
+RSA can also be used to directly test how well a particular stimulus feature is encoded in a population. If each stimuli can be quantitively described by one or more feature vectors, a similarity matrix can also be computed across the set of stimuli themselves. The strength of encoding of a particular set of features can by measured by comparing the correlation of the stimulus feature similarity matrix and the neuronal similarity matrix. The correlation strength directly reflects the ability of a linear decoder trained on the neuronal population vector to distinguish different stimuli (Kriegeskorte & Diedrichsen, 2019). Unlike previous applications of RSA in the analysis of discrete stimuli such as objects, (Khaligh-Razavi & Kriegeskorte, 2014; Yamins et al., 2014) behavior evolves continuously. To adapt RSA to behavioral analysis, we partitioned time by discretizing each behavioral feature into 50 clusters (Appendix A.4).
+
+As expected, RSA revealed that core and policy layers encoded somewhat distinct behavioral features. Policy layers contained greater information about fast timescale kinematics in a manner that was largely conserved across layers, while core layers showed more moderate encoding of kinematics that was stronger for slow behavioral features (Figure 5B,C). This difference in encoding was largely consistent across all architectures tested (Figure 5D).
+
+The feature encoding of policy networks was somewhat consistent with the emergence of a hierarchy of behavioral abstraction. In networks trained with three policy layers, representations were distributed in timescales across layers, with the last layer (policy 2) showing stronger encoding of fast behavioral features, and the first layer (policy 0) instead showing stronger encoding of slow behavioral features. However, policy layer activity, even close to the motor periphery, did not show strong explicit encoding of behavioral kinematics or forces.
+
+# 3.3 BEHAVIORAL REPRESENTATIONS ARE SHARED ACROSS TASKS
+
+We then investigated the degree to which the rodent’s neural networks used the same neural representations to produce behaviors, such as running or spinning, that were shared across tasks. Embedding population activity into two-dimensions using multidimensional scaling revealed that core neuron representations were highly distinct across all tasks, while policy layers contained more overlap (Figure 6A), suggesting that some behavioral representations were re-used. Comparison of representational similarity matricies for behaviors that were shared across tasks revealed that policy layers tended to possess a relatively similar encoding of behavioral features, especially fast behavioral features, over tasks (Figure 6C; Appendix A.4). This was validated by inspection of neural activity during individual behaviors shared across tasks (Appendix A.5, Figure 10). Core layer representations across almost all behavioral categories were more variable across tasks, consistent with encoding behavioral sequences or task variables.
+
+Interestingly, when comparing this cross-task encoding similarity across architectures, we found that one layer networks showed a marked increase in the similarity of behavioral encoding across tasks (Figure 6D). This suggests that in networks with lower computational capacity, animals must rely on a smaller, shared behavioral representation across tasks.
+
+![](images/7cfb7930dde41aa8ab9af0c64233ad336dd435af1695a82c86c9ca1d187677b6.jpg)  
+Figure 6: Policy representations are shared across tasks. (A) Two-dimensional multidimensional scaling embeddings of core and policy activity shows that while policy representations overlap across some tasks, core representations are largely distinct. (B) CKA index of the policy 2 and core network representations of behavioral features during behaviors shared across different tasks (Appendix A.4). Policy 2, but not core networks show similar encoding patterns across the across the maze forage and two-tap tasks, as well as the gaps run and maze forage tasks, consistent with the shared behaviors used across these tasks. (C) The similarity of behavioral feature encoding (CKA index) across different architectures demonstrates that networks with fewer layers show greater similarity across tasks. Points show values from individual seeds.
+
+![](images/19e9873163a055927997db4616b4e2828e139ce2c7387fb4a81445bd3f1d088d.jpg)  
+Figure 7: Neurons in core and policy networks show sequential activity during stereotyped behavior. (A) Example video stills showing the virtual rodent engaged in the two-tap task (B) Average absolute $\mathbf { Z }$ -scored activity traces of all 128 neurons in each layer during performance of the two-tap sequence. Traces are sorted by the time of peak average firing rate. Dashed lines indicate the times of first and second taps. Sequential neural activity is present during the two-tap sequence.
+
+While RSA described which behavioral features were represented in core and policy activity, we were also interested in describing how neural activity changes over time to produce different behaviors. We began by analyzing neural activity during the production of stereotyped behaviors. Activity patterns in the two-tap task showed peak activity in core and policy units that was sequentially organized (Figure 7), uniformly tiling time between both taps of the two-tap sequence. This sequential activation was observed across tasks and behaviors in the policy network, including during running (see video) where, consistent with policy networks encoding short-timescale kinematic features in a task-invariant manner, neural activity sequences were largely conserved across tasks (See Appendix A.5, Figure 10). These sequences were reliably repeated across instances of the respective behaviors, and in the case of the two-tap sequence, showed reduced neural variability relative to surrounding timepoints (See Appendix A.6, Figure 11).
+
+![](images/3c7fff60bb112e65a8254add79376b65378c141f6c3b1c79f39d6dd1ec12d779.jpg)  
+Figure 8: Latent network dynamics within tasks reflect rodent behavior on different timescales. (A) Vector field representation of the first two principal components of neural activity in the core and final policy layers during the two-tap task. PC spaces show signatures of rotational dynamics. (B) Vector field representation of first two jPC planes for the core and final policy layers during the twotap task. Apparent rotations within the different planes are associated with behaviors and behavioral features of different timescales, labeled above. Columns denote layer (as in (A)), while rows denote jPC plane. (C) Characteristic frequency of rotations within each jPC plane. Groups of three points respectively indicate the first, second, and third jPC planes for a given layer. Rotations in the core are slower than those in the policy. (D) Variance explained by each jPC plane.
+
+The finding of sequential activity hints at a putative mechanism for the rodent’s behavioral production. We next hoped to systematically quantify the types of sequential and dynamical activity present in core and policy networks without presupposing the behaviors of interest. To describe population dynamics in relation to behavior, we first applied principal components analysis (PCA) to the activity during the performance of single tasks, and visualized the gradient of the population vector as a vector field. Figure 8A shows such a vector field representation of the first two principal components of the core and final policy layer during the two-tap task. We generated vector fields by discretizing the PC space into a two-dimensional grid and calculating the average neural activity gradient with respect to time for each bin.
+
+The vector fields showed strong signatures of rotational dynamics across all layers, likely a signature of previously described sequential activity. To extract rotational patterns, we used jPCA, a dimensionality reduction method that extracts latent rotational dynamics in neural activity (Churchland et al., 2012). The resulting jPCs form an orthonormal basis that spans the same space as the first six traditional PCs, while maximally emphasizing rotational dynamics. Figure 8B shows the vector fields of the first two jPC planes for the core and final policy layers along with their characteristic frequency. Consistent with our previous findings, jPC planes in the core have lower characteristic frequencies than those in policy layers across tasks (Figure 8C). The jPC planes also individually explained a large percentage of total neural variability (Figure 8D).
+
+These rotational dynamics in the policy and core jPC planes were respectively associated with the production of behaviors and the reward structure of the task. For example, in the two-tap task, rotations in the fastest jPC plane in the core were concurrent with the approach to reward, while rotations in the second fastest jPC were concurrent with long timescale transitions between running to the orb and performing the two-tap sequence. Similarly, the fastest jPC in policy layers was correlated with the phase of running, while the second fastest was correlated with the phase of the two-tap sequence (video). This trend of core and policy neural dynamics respectively reflecting behavioral and task-related features was also present in other tasks. For example, in the maze forage task, the first two jPC planes in the core respectively correlated with reaching the target orb and discovering the location of new orbs, while those in the policy were correlated with low-level locomotor features such as running phase (video). Along with RSA, these findings support a model in which the core layer transforms sensory information into a contextual signal in a task-specific manner. This signal then modulates activity in the policy toward different trajectories that generate appropriate behaviors in a more task-independent fashion. For a more complete set of behaviors with neural dynamics visualizations overlaid, see Appendix A.7.
+
+# 3.5 NEURAL PERTURBATIONS CORROBORATE DISTINCT ROLES ACROSS LAYERS
+
+To causally demonstrate the differing roles of core and policy units in respectively encoding taskrelevant features and movement, we performed silencing and activation of different neuronal subsets in the two-tap task. We identified two stereotyped behaviors (rears and spinning jumps) that were reliably used in two different seeds of the agent to reach the orb in the task. We ranked neurons according to the degree of modulation of their z-scored activity during the performance of these behaviors. We then inactivated subsets of neurons by clamping activity to the mean values between the first and second taps and observed the effects of inactivation on trial success and behavior.
+
+In both seeds analyzed, inactivation of policy units had a stronger effect on motor behavior than the inactivation of core units. For instance, in the two-tap task, ablation of 64 neurons in the final policy layer disrupts the performance of the spinning jump (Appendix A.8 Figure 12B video). In contrast, ablation of behavior-modulated core units did not prevent the production of the behavior, but mildly affected the way in which the behavior is directed toward objects in the environment. For example, ablation of a subset of core units during the performance of a spinning jump had a limited effect, but sometimes resulted in jumps that missed the target orbs (video; See Appendix A.8, Figure 12C).
+
+We also performed a complementary perturbation aimed to elicit behaviors by overwriting the cell state of neurons in each layer with the average time-varying trajectory of neural activity measured during natural performance of a target behavior. The efficacy of stimulation was found to depend on the gross body posture and behavioral state of an animal, but was nevertheless successful in some cases. For example, during the two-tap sequence, we were able to elicit spinning movements common to searching behaviors in the forage task (video; See Appendix A.8, Figure 12D, E). The efficacy of this activation was more reliable in layers closer to the motor output (Figure 12D). In fact, activation of core units rarely elicited spins, but rather elicited sporadic dashes reminiscent of the searching strategy of many models during the forage task (video).
+
+# 4 DISCUSSION
+
+For many computational neuroscientists and artificial intelligence researchers, an aim is to reverseengineer the nervous system at an appropriate level of abstraction. In the motor system, such an effort requires that we build embodied models of animals equipped with artificial nervous systems capable of controlling their synthetic bodies across a range of behavior. Here we introduced a virtual rodent capable of performing a variety of complex locomotor behaviors to solve multiple tasks using a single policy. We then used this virtual nervous system to study principles of the neural control of movement across contexts and described several commonalities between the neural activity of artificial control and previous descriptions of biological control.
+
+A key advantage of this approach relative to experimental approaches in neuroscience is that we can fully observe sensory inputs, neural activity, and behavior, facilitating more comprehensive testing of theories related to how behavior can be generated. Furthermore, we have complete knowledge of the connectivity, sources of variance, and training objectives of each component of the model, providing a rare ground truth to test the validity of our neural analyses. With these advantages in mind, we evaluated our analyses based on their capacity to both describe the algorithms and representations employed by the virtual rodent and recapitulate the known functional objectives underlying its creation without prior knowledge.
+
+To this end, our description of core and policy as respectively representing value and motor production is consistent with the model’s actor-critic training objectives. But beyond validation, our analyses provide several insights into how these objectives are reached. RSA revealed that the cell activity of core and policy layers had greater similarity with behavioral and postural features than with short-timescale actuators. This suggests that the representation of behavior is useful in the moment-to-moment production of motor actions in artificial control, a model that has been previously proposed in biological action selection and motor control (Mink, 1996; Graziano, 2006). These behavioral representations were more consistent across tasks in the policy than in the core, suggesting that task context and value activity in the core engaged task-specific behavioral strategies through the reuse of shared motor activity in the policy.
+
+Our analysis of neural dynamics suggests that reused motor activity patterns are often organized as sequences. Specifically, the activity of policy units uniformly tiles time in the production of several stereotyped behaviors like running, jumping, spinning, and the two-tap sequence. This finding is consistent with reports linking sequential neural activity to the production of stereotyped motor and task-oriented behavior in rodents (Berke et al., 2009; Rueda-Orozco & Robbe, 2015; Dhawale et al., 2019), including during task delay periods (Akhlaghpour et al., 2016), as well as in singing birds (Albert & Margoliash, 1996; Hahnloser et al., 2002). Similarly, by relating rotational dynamics to the virtual rodent’s behavior, we found that different behaviors were seemingly associated with distinct rotations in neural activity space that evolved at different timescales. These findings are consistent with a hierarchical control scheme in which policy layer dynamics that generate reused behaviors are activated and modulated by sensorimotor signals from the core.
+
+This work represents an early step toward the constructive modeling of embodied control for the purpose of understanding the neural mechanisms behind the generation of behavior. Incrementally and judiciously increasing the realism of the model’s embodiment, behavioral repertoire, and neural architecture is a natural path for future research. Our virtual rodent possesses far fewer actuators and touch sensors than a real rodent, uses a vastly different sense of vision, and lacks integration with olfactory, auditory, and whisker-based sensation (see Zhuang et al., 2017). While the virtual rodent is capable of locomotor behaviors, an increased diversity of tasks involving decision making, memory-based navigation, and working memory could give insight into “cognitive” behaviors of which rodents are capable. Furthermore, biologically-inspired design of neural architectures and training procedures should facilitate comparisons to real neural recordings and manipulations. We expect that this comparison will help isolate residual elements of animal behavior generation that are poorly captured by current models of motor control, and encourage the development of artificial neural architectures that can produce increasingly realistic behavior.
+
+# AUTHOR CONTRIBUTIONS
+
+Josh and Yuval built the rodent MuJoCo model, with measurements collected by Diego and Jesse. Josh trained the virtual rodent model. Jesse performed behavioral and neural representation analyses. Diego performed neural dynamics analyses. Josh, Jesse, and Diego drafted the manuscript. All authors contributed to the conception of the project.
+
+# ACKNOWLEDGMENTS
+
+The rodent skeleton reference model was purchased from leo3Dmodels on TurboSquid. Thanks to Max Cant for the rodent skin, and Marcus Wainwright for the skybox and ground textures. D.A. was supported by NSF GRFP DGE1745303. J.D.M was supported by a fellowship from the Helen Hay Whitney foundation sponsored by Vertex and a K99/R00 award from the NINDS.
+
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+
+# A APPENDIX
+
+# A.1 RAT MEASUREMENTS
+
+To construct the virtual rodent model, we obtained the mass and lengths of the largest body segments that influence the physical properties of the virtual rodent. First, we dissected cadavers of two female Long-Evans rats, and measured the mass of relevant limb segments and organs. Next, we measured the lengths of body segments over the skin of animals anesthetized with $2 \%$ v/v isoflurane anesthesia in oxygen. We confirmed that these skin based measurements approximated bone lengths by measuring bone lengths in a third cadaver. The care and experimental manipulation of all animals were reviewed and approved by the appropriate Institutional Animal Care and Use Committee.
+
+Table 1: Before weighing, limb segments were divided at their respective joints. Mass of all segments includes all bones, skin, muscle, fascia and adipose layers. L and R refer to the left and right sides of the animal. Precision of measurements listed without decimal places is $\pm 0 . 5 \mathrm { g }$   
+
+<table><tr><td colspan="3">Animal (#)</td><td rowspan="2"></td></tr><tr><td></td><td>63</td><td>64</td></tr><tr><td>Body part</td><td></td><td>Mass (g)</td><td>Average mass (g)</td></tr><tr><td>Hindlimb L</td><td>21</td><td>26</td><td>23.5</td></tr><tr><td>Hindlimb R</td><td>21</td><td>26</td><td>23.5</td></tr><tr><td>Tail</td><td>8</td><td>10</td><td>9</td></tr><tr><td>Forelimb R</td><td>11</td><td>14</td><td>12.5</td></tr><tr><td>Forelimb L</td><td>12</td><td>13</td><td>12.5</td></tr><tr><td>Full torso</td><td>176</td><td>187</td><td>181.5</td></tr><tr><td>Head</td><td>26</td><td>26</td><td>26</td></tr><tr><td>Upper torso</td><td>78</td><td>71</td><td>74.5</td></tr><tr><td>Lower torso</td><td>98</td><td>114</td><td>106</td></tr><tr><td>Torso without organs</td><td>54</td><td>58</td><td>56</td></tr><tr><td>Intestines and stomach</td><td>22</td><td>32</td><td>27</td></tr><tr><td>Liver</td><td>26</td><td>17</td><td>21.5</td></tr><tr><td>Pelvis and kidneys</td><td>74</td><td>80</td><td>77</td></tr><tr><td>Jaw</td><td>2.43</td><td>4.70</td><td>3.57</td></tr><tr><td>Skull</td><td>23</td><td>21</td><td>22</td></tr><tr><td>Tail (base to mid)</td><td>5.92</td><td>7.20</td><td>6.56</td></tr><tr><td>Tail (mid to tip)</td><td>1.78</td><td>2.30</td><td>2.04</td></tr><tr><td>Scapula L</td><td>3.19</td><td>4.70</td><td>3.94</td></tr><tr><td>Humerus L</td><td>6.25</td><td>4.70</td><td>5.48</td></tr><tr><td>Radius/ulna L</td><td>2.61</td><td>2.8</td><td>2.70</td></tr><tr><td>Forepaw L</td><td>0.53</td><td>0.5</td><td>0.52</td></tr><tr><td>Scapula R</td><td>2.23</td><td>3.9</td><td>3.07</td></tr><tr><td>Humerus R</td><td>6.08</td><td>6.7</td><td>6.39</td></tr><tr><td>Radius/ulna R</td><td>2.17</td><td>3.3</td><td>2.74</td></tr><tr><td>Forepaw R</td><td>0.53</td><td>0.5</td><td>0.52</td></tr><tr><td>Hindpaw L</td><td>1.66</td><td>1.7</td><td>1.68</td></tr><tr><td>TibiaL</td><td>9</td><td>9</td><td>9</td></tr><tr><td>Femur L</td><td>13</td><td>16</td><td>14.5</td></tr><tr><td>Hindpaw R</td><td>1.81</td><td>1.6</td><td>1.71</td></tr><tr><td>Tibia R</td><td>5</td><td>6</td><td>5.5</td></tr><tr><td>Femur R</td><td>13</td><td>18</td><td>15.5</td></tr><tr><td>Total</td><td>281</td><td>301</td><td>291</td></tr></table>
+
+Table 2: Length measurements of limb segments used to construct the virtual rodent model from 7 female Long-Evans rats. Measurements were performed using calipers either over the skin or over dissected bones $( ^ { * } )$ . Thoracic and sacral refer to vertebral segments. L and R refer to the left and right sides of the animal’s body.   
+
+<table><tr><td colspan="9">Animal (#)</td></tr><tr><td></td><td>48</td><td>62</td><td>55</td><td>56</td><td>64</td><td>63</td><td>62*</td><td>Average± std</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Age (days)</td><td>382 325</td><td>82 273</td><td>330 389</td><td>330 348</td><td>83 283</td><td>83 269</td><td>83 273</td><td>309 ± 47</td></tr><tr><td>Mass (g)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Body part</td><td colspan="8">Length (mm)</td></tr><tr><td>Ankle to claw L</td><td>40.2</td><td>39.5</td><td>39.7</td><td>37.8</td><td>39.9</td><td>41.5</td><td>39.8</td><td>39.8 ± 1.1</td></tr><tr><td>Ankle to toe L</td><td>38.4</td><td>38.12</td><td>37.7</td><td>35.6</td><td>36.6</td><td>39.3</td><td>38</td><td>37.7 ± 1.2</td></tr><tr><td>Ankle to pad L</td><td>23.4</td><td>22.2</td><td>23</td><td>22.12</td><td>22.5</td><td>23.3</td><td>6.4</td><td>20.4 ± 6.2</td></tr><tr><td>Ankle to claw R</td><td></td><td>38.2</td><td>40.4</td><td>38.3</td><td>39.3</td><td>39.6</td><td>38.3</td><td>39.0 ± 0.9</td></tr><tr><td>Ankle to toe R</td><td></td><td>37</td><td>38.7</td><td>36.3</td><td>37.7</td><td>38.6</td><td>36.2</td><td>37.4 ± 1.1</td></tr><tr><td>Ankle to pad R</td><td></td><td>22.4</td><td>23.3</td><td>21.9</td><td>21.8</td><td>23.1</td><td>24.1</td><td>22.8 ± 0.9</td></tr><tr><td>Tibia L</td><td>50</td><td>36.3</td><td>38.5</td><td>49.2</td><td>35.8</td><td></td><td></td><td></td></tr><tr><td>Femur L</td><td>44.5</td><td>31.6</td><td>32.1</td><td>37.9</td><td>33.4</td><td>38.7</td><td>34.1</td><td>40.4 ± 6.5</td></tr><tr><td>Tibia R</td><td></td><td>36.7</td><td>39.1</td><td>37.9</td><td></td><td>35.35</td><td>32.4</td><td>35.3 ± 4.6</td></tr><tr><td>Femur R</td><td></td><td>32.9</td><td>32.1</td><td>38.7</td><td>35.1</td><td>38.4</td><td>36.18</td><td>37.2 ± 1.5</td></tr><tr><td>Pelvis</td><td>25.8</td><td></td><td>31.7</td><td>30.2</td><td>31.9</td><td>32.1</td><td>32.6</td><td>33.4 ± 2.6</td></tr><tr><td>Wrist to claw L</td><td>15</td><td>32 18.8</td><td>17.6</td><td>18.6</td><td>26.7 16</td><td>27.2 19.02</td><td>19.2</td><td>28.9 ± 2.7</td></tr><tr><td>Wrist to finger L</td><td></td><td>16</td><td>15.8</td><td>17.4</td><td>15.5</td><td>17.07</td><td>17.6</td><td>17.7 ± 1.6 16.6 ± 0.9</td></tr><tr><td>Wrist to pad L</td><td></td><td>6</td><td>6.4</td><td>8.34</td><td>4.9</td><td>6.1</td><td>6.4</td><td>6.4 ± 1.1</td></tr><tr><td>Wrist to olecranon L</td><td>29.1</td><td>34</td><td>32.5</td><td>31.7</td><td>33.9</td><td>32.1</td><td>29.9</td><td>31.9 ± 1.9</td></tr><tr><td>Humerus L</td><td>31.9</td><td>29.52</td><td>31</td><td>28.2</td><td>27</td><td>31.2</td><td>25.4</td><td>29.2 ± 2.4</td></tr><tr><td>Scapula L</td><td>22.7</td><td>24</td><td>26.4</td><td>29.3</td><td>25.9</td><td>29.1</td><td>26.2</td><td>26.2 ± 2.4</td></tr><tr><td>Wrist to claw R</td><td></td><td>16.8</td><td>17</td><td>17.8</td><td>15.9</td><td>16.3</td><td>18.1</td><td>17.0 ± 0.8</td></tr><tr><td>Wrist to finger R</td><td></td><td>14.1</td><td>13</td><td>15.6</td><td>15.6</td><td>15.3</td><td>16.9</td><td>15.1 ± 1.4</td></tr><tr><td>Wrist to pad R</td><td></td><td>5.6</td><td>5.8</td><td>6.55</td><td>5.2</td><td>5</td><td>5.8</td><td>5.7 ± 0.5</td></tr><tr><td>Wrist to olecranon R</td><td></td><td>30.6</td><td>33.5</td><td>31.2</td><td>30.4</td><td>31.8</td><td>29.9</td><td>31.2 ± 1.3</td></tr><tr><td>Humerus R</td><td></td><td>28.2</td><td>33.5</td><td>28.8</td><td>25</td><td>28.2</td><td>25.2</td><td>28.1 ± 3.1</td></tr><tr><td>Scapula R</td><td></td><td>23.8</td><td>29.5</td><td>25.9</td><td></td><td></td><td></td><td></td></tr><tr><td>Headcap width</td><td>39</td><td></td><td></td><td></td><td>26.2</td><td>28.8</td><td>24.4</td><td>26.4± 2.3 39</td></tr><tr><td>Headcap length</td><td>30</td><td></td><td></td><td></td><td></td><td></td><td></td><td>30</td></tr><tr><td>Skull width</td><td>38.8</td><td>23.35</td><td>23</td><td>21.8</td><td>22.8</td><td>23.9</td><td>22.2</td><td>25.1 ± 6.1</td></tr><tr><td>Skull length</td><td>57</td><td>51.1</td><td>61</td><td>56.48</td><td>53.16</td><td>58.13</td><td>48</td><td>55.0 ± 4.5</td></tr><tr><td>Skull height</td><td></td><td></td><td></td><td></td><td>21.59</td><td>21.5</td><td>21</td><td>21.4 ± 0.3</td></tr><tr><td>Head to thoracic</td><td></td><td>48.6</td><td>71.4</td><td>68.68</td><td>65</td><td>60.4</td><td>71.2</td><td>64.2 ± 8.7</td></tr><tr><td>Thoracic to sacral</td><td></td><td></td><td>73.6</td><td>62.9</td><td>65.04</td><td>64.7</td><td>68.8</td><td>68.0 ± 4.6</td></tr><tr><td>Head to sacral</td><td>145</td><td>73.1</td><td>145.5</td><td>127.05</td><td>127.2</td><td>123.7</td><td>140.9</td><td>133.6 ± 9.7</td></tr><tr><td>Head width</td><td>53.4</td><td>126</td><td></td><td></td><td></td><td></td><td></td><td>53.4</td></tr><tr><td>Ear</td><td>18</td><td>17.55</td><td>19.3</td><td>17.9</td><td>19.2</td><td>18.8</td><td></td><td>18.5 ± 0.7</td></tr><tr><td>Eye</td><td>7.2</td><td>8.25</td><td>8.6</td><td>8.8</td><td>8.2</td><td>8.3</td><td></td><td>8.2 ± 0.6</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+# A.2 BEHAVIORAL ANALYSIS
+
+We generated features describing the whole-body pose and kinematics of the virtual rodent on fast, intermediate, and slow temporal scales. To describe the whole-body pose, we took the top 15 principal components of the virtual rodent’s joint angles and joint positions to yield two 15 dimensional sets of eigenpostures (Stephens et al., 2008). We combined these into a 30 dimensional set of postural features. To describe the animal’s whole-body kinematics, we computed the continuous wavelet transform of each eigenposture using a Morlet wavelet spanning 25 scales. For each set of eigenpostures this yielded a 375 dimensional time-frequency representation of the underlying kinematics. We then computed the top 15 principal components of each 375 dimensional time-frequency representation and combined them to yield a 30 dimensional representational description of the animal’s behavioral kinematics. To facilitate comparison of kinematics to neural representations on different timescales, we used three sets of wavelet frequencies on 1 to $2 5 \ \mathrm { H z }$ (intermediate), 0.3 to $5 \ : \mathrm { H z }$ (slow) or $5 { - } 2 5 \ \mathrm { H z }$ (fast) timescales. In separate work, we have found that combining postural and kinematic information improves separation of animal behaviors in behavioral embeddings. Therefore, we combined postural and dynamical features, the later on intermediate timescales, to yield a 60 dimensional set of ‘behavioral features’ that we used to map the animal’s behavior using tSNE (Figure 4C) (Berman et al., 2014). tSNEs were made using the Barnes-Hut approximation with a perplexity of 30.
+
+![](images/4ee230bc4a104c4cb785052cf9dc87e6078e6ba3b81777a90583d2471d1d32de.jpg)  
+A.3 POWER SPECTRAL DENSITY OF BEHAVIOR AND NETWORK ACTIVITY   
+Figure 9: (A) Power spectral density estimates of four different features describing animal behavior, computed by averaging the spectral density of the top ten principal components of each feature, weighted by the variance they explain. (B) Power spectral density estimates of four different network layers, computed by averaging the spectral density of the top ten principal components of each matrix of activations, weighted by the variance they explain. Notice that policy layers have more power in high frequency bands than core layers. Arrows mark peaks in the power spectra corresponding to locomotion. Notably, the $4 { - } 5 \ \mathrm { H z }$ frequency of galloping in the virtual rat matches that measured in laboratory rats (Heglund & Taylor, 1988). Power spectral density was computed using Welch’s method using a $1 0 \mathrm { { s } }$ window size and $5 \mathrm { ~ s ~ }$ overlap.
+
+# A.4 REPRESENTATIONAL SIMILARITY ANALYSIS
+
+We used representational similarity analysis to compare population representations across different network layers and to compute the encoding strength of different features describing animal behavior in the population. Representational similarity analysis has in the past been used to compare neural population responses in tasks where behavioral stimuli are discrete, for instance corpuses of objects or faces (Kriegeskorte et al., 2008; Kriegeskorte & Diedrichsen, 2019). A challenge in scaling such approaches to neural analysis in the context of behavior is that behavior unfolds continuously in time. It is thus a priori unclear how to discretize behavior into discrete chunks in which to compare representations.
+
+Formally, we defined eight sets of features $\mathit { B _ { i = 1 \dots 8 } }$ describing the behavior of the animal on different timescales. These included features such as joint angles, the angular speed of the joint angles, eigenposture coefficients, and actuator forces that vary on short timescales, as well as behavioral kinematics, which vary on longer timescales and ‘behavioral features’, which consisted of both kinematics and eigenpostures. Each feature set is a matrix $B _ { i } \in \mathbb { R } ^ { M x q _ { i } }$ where $M$ is the number of timepoints in the experiment and $q _ { i }$ is the number of features in the set. We discretized each set $B _ { i }$ using $\mathbf { k }$ -means clustering with $k = 5 0$ to yield a partition of the timepoints in the experiment $P _ { i }$ .
+
+Using the discretization defined in $P _ { i }$ , we can perform representational similarity analysis to compare the structure of population responses across neural network layers $L _ { m }$ and $L _ { n }$ or between a given network layer and features of the behavior $B _ { i }$ . Following notation in (Kornblith et al., 2019) we let $X \in \mathbb { R } ^ { k x \cdot p }$ be a matrix of population responses across $p$ neurons and the $k$ behavioral categories in $P _ { i }$ . We let $Y \in \mathbb { R } ^ { k x q }$ be either the matrix of population responses from $q$ neurons in a distinct network layer, or a set of $q$ features describing the behavior of the animal in the feature set $B _ { i }$ .
+
+After computing the response matricies in a given behavioral partition, we compared the representational structure of the matricies $X X ^ { T }$ and $\check { Y Y } ^ { T }$ . To do so, we compute the similarity between these matricies using the linear Centered Kernel Alignment index, which is invariant under orthonormal rotations of the population activity. Following (Kornblith et al., 2019), the CKA coeffient is:
+
+$$
+C K A ( X X ^ { T } , Y Y ^ { T } ) = \frac { \| X Y ^ { T } \| _ { F } } { \| X X ^ { T } \| _ { F } \| Y Y ^ { T } \| _ { F } }
+$$
+
+Where $\| \cdot \| _ { F }$ is the Frobenius norm. For centered $X$ and $Y$ , the numerator is equivalent to the dot-product between the vectorized responses $\| X Y ^ { T } \| _ { F } = \langle \mathrm { v e c } ( X X ^ { T } ) , \mathrm { v e c } ( Y Y ^ { T } ) \rangle$ .
+
+For a given network layer $L _ { m }$ , and a behavioral partition $P _ { i }$ , we can denote $X X ^ { T } = D _ { P _ { i } } ^ { L _ { m } } = D _ { i } ^ { m }$ Similarly, for a given feature set $B _ { i }$ , let $D _ { P _ { i } } ^ { B _ { i } } = D _ { i } ^ { i }$ . Thus we are interested in characterizing both
+
+$$
+C K A \left( D _ { i } ^ { m } , D _ { i } ^ { n } \right)
+$$
+
+and
+
+$$
+C K A \left( D _ { i } ^ { m } , D _ { i } ^ { i } \right) .
+$$
+
+The former equation describes the similarity across two layers of the network, and the later describes the similarity of the network activity to a set of behavioral descriptors.
+
+An additional challenge comes when restricting this analysis to comparing the neural representations of behavioral across different tasks $T _ { a }$ , $T _ { b }$ , where not all behaviors are necessarily used in each task. To make such a comparison, we denote $B _ { i } ( T _ { a } )$ to be the set of behavioral clusters observed in task $T _ { a }$ , and $B _ { i } ^ { T _ { a } T _ { b } } = B _ { i } ( T _ { a } ) \cap B _ { i } ( T _ { a } )$ to be the set of behaviors used in each of the two tasks. We can then define a restricted partition of timepoints for each task $P _ { i } ^ { T _ { a } , T _ { b } }$ or $P _ { i } ^ { T _ { b } , T _ { a } }$ that includes only these behaviors, and compute the representational similarity between the same layer across tasks:
+
+$$
+C K A \left( D _ { i , T _ { a } } ^ { m } , D _ { i , T _ { b } } ^ { m } \right) .
+$$
+
+We have presented a means of performing representational similarity analysis across continuous time domains, where the natural units of discretization are unclear and likely manifold. While we focused on analyzing responses on the population level, it is likely that different subspaces of the population may encode information about distinct behavioral features at different timescales, which is still an emerging domain in representational similarity analysis techniques.
+
+![](images/c2a6ddf23b4ed356dffaa5db6fff4e1709bf3694f0d9191d7ade5cd93679a072.jpg)  
+Figure 10: Average activity in the final policy layer (policy 2) during running cycles across different tasks. In each heatmap, rows correspond to the absolute averaged z-scored activity for individual neurons, while columns denote time relative to the mid stance of the running phase. Across heatmaps, neurons are sorted by the time of peak activity in the tasks denoted on the left, such that each column of heatmaps contains the same average activity information with rearranged rows. Aligned running bouts were acquired by manually segmenting the the principal component space of policy 2 activity to find instances of mid-stance running and analyzing the surrounding $2 0 0 \mathrm { m s }$ .
+
+# A.6 STEREOTYPED BEHAVIOR INITIATION AND NEURAL VARIABILITY
+
+During the execution of stereotyped behaviors, neural variability was reduced (Figure 11). Recall that in our setting, neurons have no intrinsic noise, but inherit motor noise through observations of the state (i.e. via sensory reafference). This effect loosely resembles, and perhaps informs one line of interpretation of the widely reported phenomenon of neural variability reducing with stimulus or task onset (Churchland et al., 2010). Our reproduction of this effect, which simply emerges from training, suggests that variance modulation may partly arise from moments in a task that benefit from increased behavioral precision (Renart & Machens, 2014).
+
+![](images/6b68a4c39694412686de128e665a7da68d6865eb2acac8a39ace635ab95fd84b.jpg)  
+Figure 11: Quantification of neural variability in inter-tap interval of two-tap task relative to the second tap. (A) Example normalized activity traces of ten randomly selected neurons in the final policy layer. Lines indicate mean normalized activity whiles shaded regions range from the 20th percentile to the 80th percentile. Dashed lines indicate the times of first and second taps. (B) Standard deviation of normalized activity across all neurons in the final policy layer as a function of time relative to the second tap. Lines indicate the mean standard deviation while shaded regions range from the 20th percentile to the 80th percentile. Observe that variability is reduced during the two-tap interval.
+
+# A.7 NEURAL DYNAMICS VISUALIZED DURING TASK BEHAVIOR
+
+For completeness, we provide links to videos of a few variants of neural dynamics for each task.
+
+Table 3: Links to representative visualizations of neural dynamics and behavior   
+
+<table><tr><td>Network</td><td>Visualization</td><td>Task (link)</td></tr><tr><td>1-layer policy</td><td>PCA</td><td>gaps</td></tr><tr><td></td><td>PCA</td><td>forage</td></tr><tr><td rowspan="6">3-layer policy</td><td>PCA</td><td></td></tr><tr><td></td><td>escape</td></tr><tr><td>PCA</td><td>two-tap</td></tr><tr><td>PCA</td><td>gaps</td></tr><tr><td>PCA</td><td>forage</td></tr><tr><td>PCA</td><td>escape</td></tr><tr><td></td><td>PCA</td><td>two-tap</td></tr><tr><td rowspan="3">3-layer policy</td><td>jPCA</td><td>gaps</td></tr><tr><td>jPCA</td><td>forage</td></tr><tr><td>jPCA</td><td>escape</td></tr><tr><td></td><td> jPCA</td><td>two-tap</td></tr></table>
+
+![](images/7e8deee90d5d49f7283a2ecc3d59466908cfda5655ff4272f850afdeeb79bb57.jpg)  
+Figure 12: Causal manipulations reveal distinct roles for core and policy layers in the production of behavior. (A) Two-tap accuracy during the inactivation of units modulated by idiosyncratic behaviors within the two-tap sequence. Core inactivation has a weaker negative effect on trial success than policy inactivation for several levels of inactivation. (B) Representative example of a failed trial during inactivation of the final policy layer in a model that performs a spinning jump during the two-tap sequence. The model is incapable of producing the spinning jump behavior while inactivated. (C) Representative example of a failed trial during core inactivation in a model that performs a spinning jump during the two-tap sequence. The model is still able to perform the spinning jump behavior, but misses the orb. (D) Proportion of attempts at stimulation that successfully elicited spin behavior during the two-tap task. The efficacy of this activation was more reliable in layers closer to the motor output. (E) Representative example of a single trial in which an extra spin occurs after policy 2 activation.

parse/train/oXSw7laxwUpln/oXSw7laxwUpln.md

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+# An Empirical Investigation of Catastrophic Forgetting in Gradient-Based Neural Networks
+
+Ian J. Goodfellow Mehdi Mirza Da Xiao Aaron Courville Yoshua Bengio
+
+goodfeli@iro.umontreal.ca mirzamom@iro.umontreal.ca xiaoda99@bupt.edu.cn aaron.courville@umontreal.ca yoshua.bengio@umontreal.ca
+
+# Abstract
+
+Catastrophic forgetting is a problem faced by many machine learning models and algorithms. When trained on one task, then trained on a second task, many machine learning models “forget” how to perform the first task. This is widely believed to be a serious problem for neural networks. Here, we investigate the extent to which the catastrophic forgetting problem occurs for modern neural networks, comparing both established and recent gradient-based training algorithms and activation functions. We also examine the effect of the relationship between the first task and the second task on catastrophic forgetting. We find that it is always best to train using the dropout algorithm– the dropout algorithm is consistently best at adapting to the new task, remembering the old task, and has the best tradeoff curve between these two extremes. We find that different tasks and relationships between tasks result in very different rankings of activation function performance. This suggests that the choice of activation function should always be cross-validated.
+
+# 1. Introduction
+
+Catastrophic forgetting(McCloskey & Cohen, 1989; Ratcliff, 1990) is a problem that affects neural networks, as well as other learning systems, including both biological and machine learning systems. When a learning system is first trained on one task, then trained on a second task, it may forget how to perform the first task. For example, a machine learning system trained with a convex objective will always reach the same configuration at the end of training on the second task, regardless of how it was initialized. This means that an SVM that is trained on two different tasks will completely forget how to perform the first task. Whenever the SVM is able to correctly classify an example from the original task, it is only due to chance similarities between the two tasks.
+
+A well-supported model of biological learning in human beings suggests that neocortical neurons learn using an algorithm that is prone to catastrophic forgetting, and that the neocortical learning algorithm is complemented by a virtual experience system that replays memories stored in the hippocampus in order to continually reinforce tasks that have not been recently performed (McClelland et al., 1995). As machine learning researchers, the lesson we can glean from this is that it is acceptable for our learning algorithms to suffer from forgetting, but they may need complementary algorithms to reduce the information loss. Designing such complementary algorithms depends on understanding the characteristics of the forgetting experienced by our contemporary primary learning algorithms.
+
+In this paper we investigate the extent to which catastrophic forgetting affects a variety of learning algorithms and neural network activation functions. Neuroscientific evidence suggests that the relationship between the old and new task strongly influences the outcome of the two successive learning experiences (McClelland). Consequently, we examine three different types of relationship between tasks: one in which the tasks are functionally identical but with different formats of the input, one in which the tasks are similar, and one in which the tasks are dissimilar.
+
+We find that dropout (Hinton et al., 2012) is consistently the best training algorithm for modern feedforward neural nets. The choice of activation function has a less consistent effect–different activation functions are preferable depending on the task and relationship between tasks, as well as whether one places greater emphasis on adapting to the new task or retaining performance on the old task. When training with dropout, maxout (Goodfellow et al., 2013b) is the only activation function to consistently appear somewhere on the frontier of performance tradeoffs for all tasks we considered. However, maxout is not the best function at all points along the tradeoff curve, and does not have as consistent performance when trained without dropout, so it is still advisable to cross-validate the choice of activation function, particularly when training without dropout.
+
+We find that in most cases, dropout increases the optimal size of the net, so the resistance to forgetting may be explained mostly by the larger nets having greater capacity. However, this effect is not consistent, and when using dissimilar task pairs, dropout usually decreases the size of the net. This suggests dropout may have other more subtle beneficial effects to characterize in the future.
+
+# 2. Related work
+
+Catastrophic forgetting has not been a well-studied property of neural networks in recent years. This property was well-studied in the past, but has not received much attention since the deep learning renaissance that began in 2006. Srivastava et al. (2013) repopularized the idea of studying this aspect of modern deep neural nets.
+
+However, the main focus of this work was not to study catastrophic forgetting, so the experiments were limited. Only one neural network was trained in each case. The networks all used the same hyperparameters, and the same heuristically chosen stopping point. Only one pair of tasks was employed, so it is not clear whether the findings apply only to pairs of tasks with the same kind and degree of similarity or whether the findings generalize to many kinds of pairs of tasks. Only one training algorithm, standard gradient descent was employed. We move beyond all of these limitations by training multiple nets with different hyperparameters, stopping using a validation set, evaluating using three task pairs with different task similarity profiles, and including the dropout algorithm in our set of experiments.
+
+# 3. Methods
+
+In this section, we describe the basic algorithms and techniques used in our experiments.
+
+# 3.1. Dropout
+
+Dropout (Hinton et al., 2012; Srivastava, 2013) is a recently introduced training algorithm for neural networks. Dropout is designed to regularize neural networks in order to improve their generalization performance.
+
+Dropout training is a modification to standard stochastic gradient descent training. When each example is presented to the network during learning, the input states and hidden unit states of the network are multiplied by a binary mask. The zeros in the mask cause some units to be removed from the network. This mask is generated randomly each time an example is presented. Each element of the mask is sampled independently of the others, using some fixed probability $p$ . At test time, no units are dropped, and the weights going out of each unit are multiplied by $p$ to compensate for that unit being present more often than it was during training.
+
+Dropout can be seen as an extremely efficient means of training exponentially many neural networks that share weights, then averaging together their predictions. This procedure resembles bagging, which helps to reduce the generalization error. The fact that the learned features must work well in the context of many different models also helps to regularize the model.
+
+Dropout is a very effective regularizer. Prior to the introduction of dropout, one of the main ways of reducing the generalization error of a neural network was simply to restrict its capacity by using a small number of hidden units. Dropout enables training of noticeably larger networks. As an example, we performed random hyperparameter search with 25 experiments in each case to find the best two-layer rectifier network (Glorot et al., 2011a) for classifying the MNIST dataset. When training with dropout, the best network according to the validation set had $5 6 . 4 8 \%$ more parameters than the best network trained without dropout.
+
+We hypothesize that the increased size of optimally functioning dropout nets means that they are less prone to the catastrophic forgetting problem than traditional neural nets, which were regularized by constraining the capacity to be just barely sufficient to perform the first task.
+
+# 3.2. Activation functions
+
+Each of the hidden layers of our neural networks transforms some input vector $x$ into an output vector $h$ . In all cases, this is done by first computing a presynaptic activation $z = W x + b$ where $W$ is a matrix of learnable parameters and $b$ is a vector of learnable parameters. The presynaptic activation $z$ is then transformed into a post-synaptic activation $h$ by an activation function: $h = f ( z )$ . $h$ is then provided as the input to the next layer.
+
+We studied the following activation functions:
+
+1. Logistic sigmoid:
+
+$$
+\forall i , f ( z ) _ { i } = { \frac { 1 } { 1 + \exp ( - z _ { i } ) } }
+$$
+
+2. Rectified linear (Jarrett et al., 2009; Glorot et al., 2011a):
+
+$$
+\forall i , f ( z ) _ { i } = \operatorname* { m a x } ( 0 , z _ { i } )
+$$
+
+3. Hard Local Winner Take All (LWTA) (Srivastava et al., 2013):
+
+$$
+\forall i , f ( z ) _ { i } = g ( i , z ) z _ { i } .
+$$
+
+Here $g$ is a gating function. $z$ is divided into disjoint blocks of size $k$ , and $g ( i , z )$ is $^ { 1 }$ if $z _ { i }$ is the maximal element of its group. If more than one element is tied for the maximum, we break the tie uniformly at random 1. Otherwise $g ( i , z )$ is $0$ .
+
+4. Maxout (Goodfellow et al., 2013b):
+
+$$
+\forall i , f ( z ) _ { i } = \operatorname* { m a x } _ { j } \left\{ z _ { k i } , \dots , z _ { k ( i + 1 ) - 1 } \right\}
+$$
+
+We trained each of these four activation functions with each of the two algorithms we considered, for a total of eight distinct methods.
+
+# 3.3. Random hyperparameter search
+
+Making fair comparisons between different deep learning methods is difficult. The performance of most deep learning methods is a complicated non-linear function of multiple hyperparameters. For many applications, the state of the art performance is obtained by a human practitioner selecting hyperparameters for some deep learning method. Human selection is problematic for comparing methods because the human practitioner may be more skillful at selecting hyperparameters for methods that he or she is familiar with. Human practitioners may also have a conflict of interest predisposing them to selecting better hyperparameters for methods that they prefer.
+
+Automated selection of hyperparameters allows more fair comparison of methods with a complicated dependence on hyperparameters. However, automated selection of hyperparameters is challenging. Grid search suffers from the curse of dimensionality, requiring exponentially many experiments to explore highdimensional hyperparameter spaces. In this work, we use random hyperparameter search (Bergstra $\&$ Bengio, 2012) instead. This method is simple to implement and obtains roughly state of the art results using only 25 experiments on simple datasets such as MNIST.
+
+Other more sophisticated methods of hyperparameter search, such as Bayesian optimization, may be able to obtain better results, but we found that random search was able to obtain state of the art performance on the tasks we consider, so we did not think that the greater complication of using these methods was justified. More sophisticated methods of hyperparameter feedback may also introduce some sort of bias into the experiment, if one of the methods we study satisfies more of the modeling assumptions of the hyperparameter selector.
+
+# 4. Experiments
+
+All of our experiments follow the same basic form. For each experiment, we define two tasks: the “old task” and the “new task.” We examine the behavior of neural networks that are trained on the old task, then trained on the new task.
+
+For each definition of the tasks, we run the same suite of experiments for two kinds of algorithms: stochastic gradient descent training, and dropout training. For each of these algorithms, we try four different activation functions: logistic sigmoid, rectifier, hard LWTA, and maxout.
+
+For each of these eight conditions, we randomly generate 25 random sets of hyperparameters. See the code accompanying the paper for details. In all cases, we use a model with two hidden layers followed by a softmax classification layer. The hyperparameters we search over include the magnitude of the maxnorm constraint (Srebro & Shraibman, 2005) for each layer, the method used to initialize the weights for each layer and any hyper-parameters associated with such method, the initial biases for each layer, the parameters controlling a saturating linear learning rate decay and momentum increase schedule, and the size of each layer.
+
+We did not search over some hyperparameters for which good values are reasonably well-known. For example, for dropout, the best probability of dropping a hidden unit is known to usually be around 0.5, and the best probability of dropping a visible unit is known to usually be around 0.2. We used these wellknown constants on all experiments. This may reduce the maximum possible performance we are able to obtain using our search, but it makes the search function much better with only 25 experiments since fewer of the experiments fail dramatically.
+
+We did our best to keep the hyperparameter searches comparable between different methods. We always used the same hyperparameter search for SGD as for dropout. For the different activation functions, there are some slight differences between the hyperameter searches. All of these differences are related to parameter initialization schemes. For LWTA and maxout, we always set the initial biases to 0, since randomly initializing a bias for each unit can make one unit within a group win the max too often, resulting in dead filters. For rectifiers and sigmoids, we randomly select the initial biases, but using different distributions. Sigmoid networks can benefit from significantly negative initial biases, since this encourages sparsity, but these initializations are fatal to rectifier networks, since a significantly negative initial bias can prevent a unit’s parameters from ever receiving non-zero gradient. Rectifier units can also benefit from slightly positive initial biases, because they help prevent rectifier units from getting stuck, but there is no known reason to believe this helps sigmoid units. We thus use a different range of initial biases for the rectifiers and the sigmoids. This was necessary to make sure that each method is able to achieve roughly state of the art performance with only 25 experiments in the random search. Likewise, there are some differences in the way we initialize the weights for each activation function. For all activation functions, we initialize the weights from a uniform distribution over small values, in at least some cases. For maxout and LWTA, this is always the method we use. For rectifiers and sigmoids, the hyperparameter search may also choose to use the initialization method advocated by Martens $\&$ Sutskever (2011). In this method, all but $k$ of the weights going into a unit are set to 0, while the remaining $k$ are set to relatively large random values. For maxout and LWTA, this method performs poorly because different filters within the same group can be initialized to have extremely dissimilar semantics.
+
+In all cases, we first train on the “old task” until the validation set error has not improved in the last 100 epochs. Then we restore the parameters corresponding to the best validation set error, and begin training on the “new task”. We train until the error on the union of the old validation set and new validation set has not improved for 100 epochs.
+
+After running all 25 randomly configured experiments for all 8 conditions, we make a possibilities frontier curve showing the minimum amount of test error on the new task obtaining for each amount of test error on the old task. Specifically, these plots are made by drawing a curve that traces out the lower left frontier of the cloud of points of all (old task test error, new task test error) pairs encountered by all 25 models during the course of training on the new task, with one point generated after each pass through the training set. Note that these test set errors are computed after training on only a subset of the training data, because we do not train on the validation set. It is possible to improve further by also training on the validation set, but we do not do so here because we only care about the relative performance of the different methods, not necessarily obtaining state of the art results.
+
+(Usually possibilities frontier curves are used in scenarios where higher values are better, and the curves trace out the higher edge of a convex hull of scatterplot. Here, we are plotting error rates, so the lower values are better and the curves trace out the lower edge of a convex hull of a scatterplot. We used error rather than accuracy so that log scale plots would compress regions of bad performance and expand regions of good performance, in order to highlight the differences between the best-performing methods. Note that the log scaling sometimes makes the convex regions apear non-convex)
+
+# 4.1. Input reformatting
+
+Many naturally occurring tasks are highly similar to each other in terms of the underlying structure that must be understood, but have the input presented in a different format.
+
+For example, consider learning to understand Italian after already learning to understand Spanish. Both tasks share the deeper underlying structure of being a natural language understanding problem, and furthermore, Italian and Spanish have similar grammar. However, the specific words in each language are different. A person learning Italian thus benefits from having a pre-existing representation of the general structure of the language. The challenge is to learn to map the new words into these structures (e.g., to attach the Italian word “sei” to the pre-existing concept of the second person conjugation of the verb “to be”) without damaging the ability to understand Spanish. The ability to understand Spanish could diminish if the learning algorithm inadvertently modifies the more abstract definition of language in general (i.e., if neurons that were used for verb conjugation before now get re-purposed for plurality agreement) rather than exploiting the pre-existing definition, or if the learning algorithm removes the associations between individual Spanish words and these pre-existing concepts (e.g., if the net retains the concept of there being a second person conjugation of the verb “to be” but forgets that the Spanish word “eres” corresponds to it).
+
+To test this kind of learning problem, we designed a simple pair of tasks, where the tasks are the same, but with different ways of formatting the input. Specifically, we used MNIST classification, but with a different permutation of the pixels for the old task and the new task. Both tasks thus benefit from having concepts like penstroke detectors, or the concept of penstrokes being combined to form digits. However, the meaning of any individual pixel is different. The net must learn to associate new collections of pixels to penstrokes, without significantly disrupting the old higher level concepts, or erasing the old connections between pixels and penstrokes.
+
+The classification performance results are presented in Fig. 1. Using dropout improved the two-task validation set performance for all models on this task pair. We show the effect of dropout on the optimal model size in Fig. 2. While the nets were able to basically succeed at this task, we don’t believe that they did so by mapping different sets of pixels into pre-existing concepts. We visualized the first layer weights of the best net (in terms of combined validation set error) and their apparent semantics do not noticeably change between when training on the old task concludes and training on the new task begins. This suggests that the higher layers of the net changed to be able to accomodate a relatively arbitrary projection of the input, rather than remaining the same while the lower layers adapted to the new input format.
+
+# 4.2. Similar tasks
+
+We next considered what happens when the two tasks are not exactly the same, but semantically similar, and using the same input format. To test this case, we used sentiment analysis of two product categories of Amazon reviews (Blitzer et al., 2007) as the two tasks.
+
+![](images/439876043b347125563b4302f22b1573a3769eb0dc365574e3ccd131b6543e32.jpg)  
+Figure 2. Optimal model size with and without dropout on the input reformatting tasks.
+
+![](images/1a891c3439c2f88cc6860282836db0ce0c400f5a86513b0c6e0c7a28c3b8fe5d.jpg)  
+Figure 4. Optimal model size with and without dropout on the similar tasks experiment.
+
+The task is just to classify the text of a product review as positive or negative in sentiment. We used the same preprocessing as (Glorot et al., 2011b).
+
+The classification performance results are presented in Fig. 3. Using dropout improved the two-task validation set performance for all models on this task pair. We show the effect of dropout on the optimal model size in Fig. 6.
+
+# 4.3. Dissimilar tasks
+
+We next considered what happens when the two tasks are semantically similar. To test this case, we used Amazon reviews as one task, and MNIST classification as another. In order to give both tasks the same output size, we used only two classes of the MNIST dataset. To give them the same validation set size, we randomly subsampled the remaining examples of the MNIST validation set (since the MNIST validation set was originally larger than the Amazon validation set, and we don’t want the estimate of the performance on the Amazon dataset to have higher variance than the MNIST one). The Amazon dataset as we preprocessed it earlier has 5,000 input features, while MNIST has only 784. To give the two tasks the same input size, we reduced the dimensionality of the Amazon data with PCA.
+
+Classification performance results are presented in
+
+![](images/d8623b848906d291104729dc1978e8515a386bd637b64b936abd9433ac89725f.jpg)  
+Figure 1. Possibilities frontiers for the input reformatting experiment.
+
+![](images/ca28351bde9517c0e6ed2b65896dd91ab510b89828faa915af1f3da1aecb6479.jpg)  
+Figure 3. Possibilities frontiers for the similar tasks experiment.
+
+![](images/a55adeed4d54a90c359f1fd3e22a4eb47abbcc1d53858ba394e840616270b07e.jpg)  
+Figure 5. Possibilities frontiers for the dissimilar tasks experiment.
+
+![](images/1c8ac5562c915153a85593c6c6554ae9820ab36e59998b20d37bb95f71e3cd08.jpg)  
+Figure 6. Optimal model size with and without dropout on the disimilar tasks experiment.   
+Fig. 5. Using dropout improved the two-task validation set performance for all models on this task pair. We show the effect of dropout on the optimal model size in Fig. 6.
+
+# 5. Discussion
+
+Our experiments have shown that training with dropout is always beneficial, at least on the relatively small datasets we used in this paper. Dropout improved performance for all eight methods on all three task pairs. Dropout works the best in terms of performance on the new task, performance on the old task, and points along the tradeoff curve balancing these two extremes, for all three task pairs. Dropout’s resistance to forgetting may be explained in part by the large model sizes that can be trained with dropout. On the input-reformatted task pair and the similar task pair, dropout never decreased the size of the optimal model for any of the four activation functions we tried. However, dropout seems to have additional properties that can help prevent forgetting that we do not yet have an explanation for. On the dissimilar tasks experiment, dropout improved performance but reduced the size of the optimal model for most of the activation functions, and on the other task pairs, it occasionally had no effect on the optimal model size.
+
+The only recent previous work on catastrophic forgetting(Srivastava et al., 2013) argued that the choice of activation function has a significant effect on the catastrophic forgetting properties of a net, and in particular that hard LWTA outperforms logistic sigmoid and rectified linear units in this respect when trained with stochastic gradient descent.
+
+In our more extensive experiments we found that the choice of activation function has a less consistent effect than the choice of training algorithm. When we performed experiments with different kinds of task pairs, we found that the ranking of the activation functions is very problem dependent. For example, logistic sigmoid is the worst under some conditions but the best under other conditions. This suggests that one should always cross-validate the choice of activation function, as long as it is computationally feasible. We also reject the idea that hard LWTA is particular resistant to catastrophic forgetting in general, or that it makes the standard SGD training algorithm more resistant to catastrophic forgetting. For example, when training with SGD on the input reformatting task pair, hard LWTA’s possibilities frontier is worse than all activation functions except sigmoid for most points along the curve. On the similar task pair, LWTA with SGD is the worst of all eight methods we considered, in terms of best performance on the new task, best performance on the old task, and in terms of attaining points close to the origin of the possibilities frontier plot. However, hard LWTA does perform the best in some circumstances (it has the best performance on the new task for the dissimilar task pair ). This suggests that it is worth including hard LWTA as one of many activation functions in a hyperparameter search. LWTA is however never the leftmost point in any of our three task pairs, so it is probably only useful in sequential task settings where forgetting is an issue.
+
+When computational resources are too limited to experiment with multiple activation functions, we recommend using the maxout activation function trained with dropout. This is the only method that appears on the lower-left frontier of the performance tradeoff plots for all three task pairs we considered.
+
+# Acknowledgments
+
+We would like to thank the developers of Theano (Bergstra et al., 2010; Bastien et al., 2012), Pylearn2 (Goodfellow et al., 2013a). We would also like to thank NSERC, Compute Canada, and Calcul Qu´ebec for providing computational resources. Ian Goodfellow is supported by the 2013 Google Fellowship in Deep Learning.
+
+# References
+
+Bastien, Fr´ed´eric, Lamblin, Pascal, Pascanu, Razvan, Bergstra, James, Goodfellow, Ian J., Bergeron, Arnaud, Bouchard, Nicolas, and Bengio, Yoshua. Theano: new features and speed improvements. Deep Learning and Unsupervised Feature Learning NIPS 2012 Workshop, 2012.
+
+Bergstra, James and Bengio, Yoshua. Random search for hyper-parameter optimization. Journal of Machine Learning Research, 13:281–305, February 2012.
+
+Bergstra, James, Breuleux, Olivier, Bastien, Fr´ed´eric,
+
+Lamblin, Pascal, Pascanu, Razvan, Desjardins, Guillaume, Turian, Joseph, Warde-Farley, David, and Bengio, Yoshua. Theano: a CPU and GPU math expression compiler. In Proceedings of the Python for Scientific Computing Conference (SciPy), June 2010. Oral Presentation.
+
+Blitzer, John, Dredze, Mark, and Pereira, Fernando. Biographies, bollywood, boom-boxes and blenders: Domain adaptation for sentiment classification. In ACL ’07, pp. 440–447, 2007.
+
+Glorot, Xavier, Bordes, Antoine, and Bengio, Yoshua. Deep sparse rectifier neural networks. In JMLR W&CP: Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics (AISTATS 2011), April 2011a.
+
+Glorot, Xavier, Bordes, Antoine, and Bengio, Yoshua. Domain adaptation for large-scale sentiment classification: A deep learning approach. In Proceedings of theTwenty-eight International Conference on Machine Learning (ICML’11), volume 27, pp. 97–110, June 2011b.
+
+Goodfellow, Ian J., Warde-Farley, David, Lamblin, Pascal, Dumoulin, Vincent, Mirza, Mehdi, Pascanu, Razvan, Bergstra, James, Bastien, Fr´ed´eric, and Bengio, Yoshua. Pylearn2: a machine learning research library. arXiv preprint arXiv:1308.4214, 2013a.
+
+Goodfellow, Ian J., Warde-Farley, David, Mirza, Mehdi, Courville, Aaron, and Bengio, Yoshua. Maxout networks. In Dasgupta, Sanjoy and McAllester, David (eds.), Proceedings of the 30th International Conference on Machine Learning (ICML’13), pp. 13191327. ACM, 2013b. URL http://icml.cc/ 2013/.
+
+Hinton, Geoffrey E., Srivastava, Nitish, Krizhevsky, Alex, Sutskever, Ilya, and Salakhutdinov, Ruslan. Improving neural networks by preventing coadaptation of feature detectors. Technical report, arXiv:1207.0580, 2012.
+
+Jarrett, Kevin, Kavukcuoglu, Koray, Ranzato, Marc’Aurelio, and LeCun, Yann. What is the best multi-stage architecture for object recognition? In Proc. International Conference on Computer Vision (ICCV’09), pp. 2146–2153. IEEE, 2009.
+
+Martens, James and Sutskever, Ilya. Learning recurrent neural networks with Hessian-free optimization. In Proc. ICML’2011. ACM, 2011.
+
+McClelland, J. L., McNaughton, B. L., and O’Reilly, R. C. Why there are complementary learning systems in the hippocampus and neocortex: Insights from the successes and failures of connectionist models of learning and memory. Psychological Review, 102:419–457, 1995.
+
+McClelland, James L.
+
+McCloskey, M. and Cohen, N. J. Catastrophic interference in connectionist networks: The sequential learning problem. In Bower, G. H. (ed.), The Psychology of Learning and Motivation, Vol. 24, pp. 109–164. Academic Press, San Diego, CA, 1989.
+
+Ratcliff, R. Connectionist models of recognition memory: constraints imposed by learning and forgetting functions. Psychological review, 97(2):285–308, April 1990. ISSN 0033-295X. URL http://view. ncbi.nlm.nih.gov/pubmed/2186426.
+
+Srebro, Nathan and Shraibman, Adi. Rank, tracenorm and max-norm. In Proceedings of the 18th Annual Conference on Learning Theory, pp. 545– 560. Springer-Verlag, 2005.
+
+Srivastava, Nitish. Improving neural networks with dropout. Master’s thesis, U. Toronto, 2013.
+
+Srivastava, Rupesh K, Masci, Jonathan, Kazerounian, Sohrob, Gomez, Faustino, and Schmidhuber, J¨urgen. Compete to compute. In Burges, C.J.C., Bottou, L., Welling, M., Ghahramani, Z., and Weinberger, K.Q. (eds.), Advances in Neural Information Processing Systems 26, pp. 2310–2318. 2013. URL http://media.nips.cc/nipsbooks/ nipspapers/paper_files/nips26/1109.pdf.

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+    {
+        "type": "text",
+        "text": "An Empirical Investigation of Catastrophic Forgetting in Gradient-Based Neural Networks ",
+        "text_level": 1,
+        "bbox": [
+            156,
+            109,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ian J. Goodfellow Mehdi Mirza Da Xiao Aaron Courville Yoshua Bengio ",
+        "bbox": [
+            88,
+            194,
+            235,
+            270
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "goodfeli@iro.umontreal.ca mirzamom@iro.umontreal.ca xiaoda99@bupt.edu.cn aaron.courville@umontreal.ca yoshua.bengio@umontreal.ca ",
+        "bbox": [
+            604,
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+            270
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            238,
+            295,
+            321,
+            313
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Catastrophic forgetting is a problem faced by many machine learning models and algorithms. When trained on one task, then trained on a second task, many machine learning models “forget” how to perform the first task. This is widely believed to be a serious problem for neural networks. Here, we investigate the extent to which the catastrophic forgetting problem occurs for modern neural networks, comparing both established and recent gradient-based training algorithms and activation functions. We also examine the effect of the relationship between the first task and the second task on catastrophic forgetting. We find that it is always best to train using the dropout algorithm– the dropout algorithm is consistently best at adapting to the new task, remembering the old task, and has the best tradeoff curve between these two extremes. We find that different tasks and relationships between tasks result in very different rankings of activation function performance. This suggests that the choice of activation function should always be cross-validated. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1. Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Catastrophic forgetting(McCloskey & Cohen, 1989; Ratcliff, 1990) is a problem that affects neural networks, as well as other learning systems, including both biological and machine learning systems. When a learning system is first trained on one task, then trained on a second task, it may forget how to perform the first task. For example, a machine learning system trained with a convex objective will always reach the same configuration at the end of training on the second task, regardless of how it was initialized. This means that an SVM that is trained on two different tasks will completely forget how to perform the first task. Whenever the SVM is able to correctly classify an example from the original task, it is only due to chance similarities between the two tasks. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "A well-supported model of biological learning in human beings suggests that neocortical neurons learn using an algorithm that is prone to catastrophic forgetting, and that the neocortical learning algorithm is complemented by a virtual experience system that replays memories stored in the hippocampus in order to continually reinforce tasks that have not been recently performed (McClelland et al., 1995). As machine learning researchers, the lesson we can glean from this is that it is acceptable for our learning algorithms to suffer from forgetting, but they may need complementary algorithms to reduce the information loss. Designing such complementary algorithms depends on understanding the characteristics of the forgetting experienced by our contemporary primary learning algorithms. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper we investigate the extent to which catastrophic forgetting affects a variety of learning algorithms and neural network activation functions. Neuroscientific evidence suggests that the relationship between the old and new task strongly influences the outcome of the two successive learning experiences (McClelland). Consequently, we examine three different types of relationship between tasks: one in which the tasks are functionally identical but with different formats of the input, one in which the tasks are similar, and one in which the tasks are dissimilar. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We find that dropout (Hinton et al., 2012) is consistently the best training algorithm for modern feedforward neural nets. The choice of activation function has a less consistent effect–different activation functions are preferable depending on the task and relationship between tasks, as well as whether one places greater emphasis on adapting to the new task or retaining performance on the old task. When training with dropout, maxout (Goodfellow et al., 2013b) is the only activation function to consistently appear somewhere on the frontier of performance tradeoffs for all tasks we considered. However, maxout is not the best function at all points along the tradeoff curve, and does not have as consistent performance when trained without dropout, so it is still advisable to cross-validate the choice of activation function, particularly when training without dropout. ",
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+    },
+    {
+        "type": "text",
+        "text": "We find that in most cases, dropout increases the optimal size of the net, so the resistance to forgetting may be explained mostly by the larger nets having greater capacity. However, this effect is not consistent, and when using dissimilar task pairs, dropout usually decreases the size of the net. This suggests dropout may have other more subtle beneficial effects to characterize in the future. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2. Related work ",
+        "text_level": 1,
+        "bbox": [
+            89,
+            458,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Catastrophic forgetting has not been a well-studied property of neural networks in recent years. This property was well-studied in the past, but has not received much attention since the deep learning renaissance that began in 2006. Srivastava et al. (2013) repopularized the idea of studying this aspect of modern deep neural nets. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "However, the main focus of this work was not to study catastrophic forgetting, so the experiments were limited. Only one neural network was trained in each case. The networks all used the same hyperparameters, and the same heuristically chosen stopping point. Only one pair of tasks was employed, so it is not clear whether the findings apply only to pairs of tasks with the same kind and degree of similarity or whether the findings generalize to many kinds of pairs of tasks. Only one training algorithm, standard gradient descent was employed. We move beyond all of these limitations by training multiple nets with different hyperparameters, stopping using a validation set, evaluating using three task pairs with different task similarity profiles, and including the dropout algorithm in our set of experiments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3. Methods ",
+        "text_level": 1,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we describe the basic algorithms and techniques used in our experiments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.1. Dropout ",
+        "text_level": 1,
+        "bbox": [
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+            156,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Dropout (Hinton et al., 2012; Srivastava, 2013) is a recently introduced training algorithm for neural networks. Dropout is designed to regularize neural networks in order to improve their generalization performance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Dropout training is a modification to standard stochastic gradient descent training. When each example is presented to the network during learning, the input states and hidden unit states of the network are multiplied by a binary mask. The zeros in the mask cause some units to be removed from the network. This mask is generated randomly each time an example is presented. Each element of the mask is sampled independently of the others, using some fixed probability $p$ . At test time, no units are dropped, and the weights going out of each unit are multiplied by $p$ to compensate for that unit being present more often than it was during training. ",
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+        "type": "text",
+        "text": "Dropout can be seen as an extremely efficient means of training exponentially many neural networks that share weights, then averaging together their predictions. This procedure resembles bagging, which helps to reduce the generalization error. The fact that the learned features must work well in the context of many different models also helps to regularize the model. ",
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+        "type": "text",
+        "text": "Dropout is a very effective regularizer. Prior to the introduction of dropout, one of the main ways of reducing the generalization error of a neural network was simply to restrict its capacity by using a small number of hidden units. Dropout enables training of noticeably larger networks. As an example, we performed random hyperparameter search with 25 experiments in each case to find the best two-layer rectifier network (Glorot et al., 2011a) for classifying the MNIST dataset. When training with dropout, the best network according to the validation set had $5 6 . 4 8 \\%$ more parameters than the best network trained without dropout. ",
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+        "text": "We hypothesize that the increased size of optimally functioning dropout nets means that they are less prone to the catastrophic forgetting problem than traditional neural nets, which were regularized by constraining the capacity to be just barely sufficient to perform the first task. ",
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+        "text": "3.2. Activation functions ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Each of the hidden layers of our neural networks transforms some input vector $x$ into an output vector $h$ . In all cases, this is done by first computing a presynaptic activation $z = W x + b$ where $W$ is a matrix of learnable parameters and $b$ is a vector of learnable parameters. The presynaptic activation $z$ is then transformed into a post-synaptic activation $h$ by an activation function: $h = f ( z )$ . $h$ is then provided as the input to the next layer. ",
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+        "text": "We studied the following activation functions: ",
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+    {
+        "type": "text",
+        "text": "1. Logistic sigmoid: ",
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+        "type": "equation",
+        "img_path": "images/05c1cbdec876a3eeeed4b971ae2c1c0049a20f276a5850e42ebde7f21afb8eb1.jpg",
+        "text": "$$\n\\forall i , f ( z ) _ { i } = { \\frac { 1 } { 1 + \\exp ( - z _ { i } ) } }\n$$",
+        "text_format": "latex",
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+        "text": "2. Rectified linear (Jarrett et al., 2009; Glorot et al., 2011a): ",
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+        "type": "equation",
+        "img_path": "images/f7e651ff4834977f83931a91bd6b98a11fe399b95a46033e8647b5f31d8c6dfb.jpg",
+        "text": "$$\n\\forall i , f ( z ) _ { i } = \\operatorname* { m a x } ( 0 , z _ { i } )\n$$",
+        "text_format": "latex",
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+        "text": "3. Hard Local Winner Take All (LWTA) (Srivastava et al., 2013): ",
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+    {
+        "type": "equation",
+        "img_path": "images/a88d4ee4ded98b20884611bb237f0ed1eafc36a5b657587e0305d4be6a7567c0.jpg",
+        "text": "$$\n\\forall i , f ( z ) _ { i } = g ( i , z ) z _ { i } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "text",
+        "text": "Here $g$ is a gating function. $z$ is divided into disjoint blocks of size $k$ , and $g ( i , z )$ is $^ { 1 }$ if $z _ { i }$ is the maximal element of its group. If more than one element is tied for the maximum, we break the tie uniformly at random 1. Otherwise $g ( i , z )$ is $0$ . ",
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+        "text": "4. Maxout (Goodfellow et al., 2013b): ",
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+    },
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+        "type": "equation",
+        "img_path": "images/fc3b9a61a6ea7980b30b8f05887282c9c6f056416c110537a0f82f9d94163570.jpg",
+        "text": "$$\n\\forall i , f ( z ) _ { i } = \\operatorname* { m a x } _ { j } \\left\\{ z _ { k i } , \\dots , z _ { k ( i + 1 ) - 1 } \\right\\}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We trained each of these four activation functions with each of the two algorithms we considered, for a total of eight distinct methods. ",
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+        "type": "text",
+        "text": "3.3. Random hyperparameter search ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Making fair comparisons between different deep learning methods is difficult. The performance of most deep learning methods is a complicated non-linear function of multiple hyperparameters. For many applications, the state of the art performance is obtained by a human practitioner selecting hyperparameters for some deep learning method. Human selection is problematic for comparing methods because the human practitioner may be more skillful at selecting hyperparameters for methods that he or she is familiar with. Human practitioners may also have a conflict of interest predisposing them to selecting better hyperparameters for methods that they prefer. ",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "Automated selection of hyperparameters allows more fair comparison of methods with a complicated dependence on hyperparameters. However, automated selection of hyperparameters is challenging. Grid search suffers from the curse of dimensionality, requiring exponentially many experiments to explore highdimensional hyperparameter spaces. In this work, we use random hyperparameter search (Bergstra $\\&$ Bengio, 2012) instead. This method is simple to implement and obtains roughly state of the art results using only 25 experiments on simple datasets such as MNIST. ",
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+        "type": "text",
+        "text": "Other more sophisticated methods of hyperparameter search, such as Bayesian optimization, may be able to obtain better results, but we found that random search was able to obtain state of the art performance on the tasks we consider, so we did not think that the greater complication of using these methods was justified. More sophisticated methods of hyperparameter feedback may also introduce some sort of bias into the experiment, if one of the methods we study satisfies more of the modeling assumptions of the hyperparameter selector. ",
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+        "type": "text",
+        "text": "4. Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "All of our experiments follow the same basic form. For each experiment, we define two tasks: the “old task” and the “new task.” We examine the behavior of neural networks that are trained on the old task, then trained on the new task. ",
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+    {
+        "type": "text",
+        "text": "For each definition of the tasks, we run the same suite of experiments for two kinds of algorithms: stochastic gradient descent training, and dropout training. For each of these algorithms, we try four different activation functions: logistic sigmoid, rectifier, hard LWTA, and maxout. ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "For each of these eight conditions, we randomly generate 25 random sets of hyperparameters. See the code accompanying the paper for details. In all cases, we use a model with two hidden layers followed by a softmax classification layer. The hyperparameters we search over include the magnitude of the maxnorm constraint (Srebro & Shraibman, 2005) for each layer, the method used to initialize the weights for each layer and any hyper-parameters associated with such method, the initial biases for each layer, the parameters controlling a saturating linear learning rate decay and momentum increase schedule, and the size of each layer. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "We did not search over some hyperparameters for which good values are reasonably well-known. For example, for dropout, the best probability of dropping a hidden unit is known to usually be around 0.5, and the best probability of dropping a visible unit is known to usually be around 0.2. We used these wellknown constants on all experiments. This may reduce the maximum possible performance we are able to obtain using our search, but it makes the search function much better with only 25 experiments since fewer of the experiments fail dramatically. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "We did our best to keep the hyperparameter searches comparable between different methods. We always used the same hyperparameter search for SGD as for dropout. For the different activation functions, there are some slight differences between the hyperameter searches. All of these differences are related to parameter initialization schemes. For LWTA and maxout, we always set the initial biases to 0, since randomly initializing a bias for each unit can make one unit within a group win the max too often, resulting in dead filters. For rectifiers and sigmoids, we randomly select the initial biases, but using different distributions. Sigmoid networks can benefit from significantly negative initial biases, since this encourages sparsity, but these initializations are fatal to rectifier networks, since a significantly negative initial bias can prevent a unit’s parameters from ever receiving non-zero gradient. Rectifier units can also benefit from slightly positive initial biases, because they help prevent rectifier units from getting stuck, but there is no known reason to believe this helps sigmoid units. We thus use a different range of initial biases for the rectifiers and the sigmoids. This was necessary to make sure that each method is able to achieve roughly state of the art performance with only 25 experiments in the random search. Likewise, there are some differences in the way we initialize the weights for each activation function. For all activation functions, we initialize the weights from a uniform distribution over small values, in at least some cases. For maxout and LWTA, this is always the method we use. For rectifiers and sigmoids, the hyperparameter search may also choose to use the initialization method advocated by Martens $\\&$ Sutskever (2011). In this method, all but $k$ of the weights going into a unit are set to 0, while the remaining $k$ are set to relatively large random values. For maxout and LWTA, this method performs poorly because different filters within the same group can be initialized to have extremely dissimilar semantics. ",
+        "bbox": [
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "In all cases, we first train on the “old task” until the validation set error has not improved in the last 100 epochs. Then we restore the parameters corresponding to the best validation set error, and begin training on the “new task”. We train until the error on the union of the old validation set and new validation set has not improved for 100 epochs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "After running all 25 randomly configured experiments for all 8 conditions, we make a possibilities frontier curve showing the minimum amount of test error on the new task obtaining for each amount of test error on the old task. Specifically, these plots are made by drawing a curve that traces out the lower left frontier of the cloud of points of all (old task test error, new task test error) pairs encountered by all 25 models during the course of training on the new task, with one point generated after each pass through the training set. Note that these test set errors are computed after training on only a subset of the training data, because we do not train on the validation set. It is possible to improve further by also training on the validation set, but we do not do so here because we only care about the relative performance of the different methods, not necessarily obtaining state of the art results. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "(Usually possibilities frontier curves are used in scenarios where higher values are better, and the curves trace out the higher edge of a convex hull of scatterplot. Here, we are plotting error rates, so the lower values are better and the curves trace out the lower edge of a convex hull of a scatterplot. We used error rather than accuracy so that log scale plots would compress regions of bad performance and expand regions of good performance, in order to highlight the differences between the best-performing methods. Note that the log scaling sometimes makes the convex regions apear non-convex) ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "4.1. Input reformatting ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Many naturally occurring tasks are highly similar to each other in terms of the underlying structure that must be understood, but have the input presented in a different format. ",
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+    },
+    {
+        "type": "text",
+        "text": "For example, consider learning to understand Italian after already learning to understand Spanish. Both tasks share the deeper underlying structure of being a natural language understanding problem, and furthermore, Italian and Spanish have similar grammar. However, the specific words in each language are different. A person learning Italian thus benefits from having a pre-existing representation of the general structure of the language. The challenge is to learn to map the new words into these structures (e.g., to attach the Italian word “sei” to the pre-existing concept of the second person conjugation of the verb “to be”) without damaging the ability to understand Spanish. The ability to understand Spanish could diminish if the learning algorithm inadvertently modifies the more abstract definition of language in general (i.e., if neurons that were used for verb conjugation before now get re-purposed for plurality agreement) rather than exploiting the pre-existing definition, or if the learning algorithm removes the associations between individual Spanish words and these pre-existing concepts (e.g., if the net retains the concept of there being a second person conjugation of the verb “to be” but forgets that the Spanish word “eres” corresponds to it). ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "To test this kind of learning problem, we designed a simple pair of tasks, where the tasks are the same, but with different ways of formatting the input. Specifically, we used MNIST classification, but with a different permutation of the pixels for the old task and the new task. Both tasks thus benefit from having concepts like penstroke detectors, or the concept of penstrokes being combined to form digits. However, the meaning of any individual pixel is different. The net must learn to associate new collections of pixels to penstrokes, without significantly disrupting the old higher level concepts, or erasing the old connections between pixels and penstrokes. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The classification performance results are presented in Fig. 1. Using dropout improved the two-task validation set performance for all models on this task pair. We show the effect of dropout on the optimal model size in Fig. 2. While the nets were able to basically succeed at this task, we don’t believe that they did so by mapping different sets of pixels into pre-existing concepts. We visualized the first layer weights of the best net (in terms of combined validation set error) and their apparent semantics do not noticeably change between when training on the old task concludes and training on the new task begins. This suggests that the higher layers of the net changed to be able to accomodate a relatively arbitrary projection of the input, rather than remaining the same while the lower layers adapted to the new input format. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.2. Similar tasks ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We next considered what happens when the two tasks are not exactly the same, but semantically similar, and using the same input format. To test this case, we used sentiment analysis of two product categories of Amazon reviews (Blitzer et al., 2007) as the two tasks. ",
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+    {
+        "type": "image",
+        "img_path": "images/439876043b347125563b4302f22b1573a3769eb0dc365574e3ccd131b6543e32.jpg",
+        "image_caption": [
+            "Figure 2. Optimal model size with and without dropout on the input reformatting tasks. "
+        ],
+        "image_footnote": [],
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+        "image_caption": [
+            "Figure 4. Optimal model size with and without dropout on the similar tasks experiment. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "The task is just to classify the text of a product review as positive or negative in sentiment. We used the same preprocessing as (Glorot et al., 2011b). ",
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+    {
+        "type": "text",
+        "text": "The classification performance results are presented in Fig. 3. Using dropout improved the two-task validation set performance for all models on this task pair. We show the effect of dropout on the optimal model size in Fig. 6. ",
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+    {
+        "type": "text",
+        "text": "4.3. Dissimilar tasks ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We next considered what happens when the two tasks are semantically similar. To test this case, we used Amazon reviews as one task, and MNIST classification as another. In order to give both tasks the same output size, we used only two classes of the MNIST dataset. To give them the same validation set size, we randomly subsampled the remaining examples of the MNIST validation set (since the MNIST validation set was originally larger than the Amazon validation set, and we don’t want the estimate of the performance on the Amazon dataset to have higher variance than the MNIST one). The Amazon dataset as we preprocessed it earlier has 5,000 input features, while MNIST has only 784. To give the two tasks the same input size, we reduced the dimensionality of the Amazon data with PCA. ",
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+    {
+        "type": "text",
+        "text": "Classification performance results are presented in ",
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+    {
+        "type": "image",
+        "img_path": "images/d8623b848906d291104729dc1978e8515a386bd637b64b936abd9433ac89725f.jpg",
+        "image_caption": [
+            "Figure 1. Possibilities frontiers for the input reformatting experiment. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 5
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+        "image_caption": [
+            "Figure 3. Possibilities frontiers for the similar tasks experiment. "
+        ],
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+        "image_caption": [
+            "Figure 5. Possibilities frontiers for the dissimilar tasks experiment. "
+        ],
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+        "image_caption": [
+            "Figure 6. Optimal model size with and without dropout on the disimilar tasks experiment. ",
+            "Fig. 5. Using dropout improved the two-task validation set performance for all models on this task pair. We show the effect of dropout on the optimal model size in Fig. 6. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "5. Discussion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our experiments have shown that training with dropout is always beneficial, at least on the relatively small datasets we used in this paper. Dropout improved performance for all eight methods on all three task pairs. Dropout works the best in terms of performance on the new task, performance on the old task, and points along the tradeoff curve balancing these two extremes, for all three task pairs. Dropout’s resistance to forgetting may be explained in part by the large model sizes that can be trained with dropout. On the input-reformatted task pair and the similar task pair, dropout never decreased the size of the optimal model for any of the four activation functions we tried. However, dropout seems to have additional properties that can help prevent forgetting that we do not yet have an explanation for. On the dissimilar tasks experiment, dropout improved performance but reduced the size of the optimal model for most of the activation functions, and on the other task pairs, it occasionally had no effect on the optimal model size. ",
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+    {
+        "type": "text",
+        "text": "The only recent previous work on catastrophic forgetting(Srivastava et al., 2013) argued that the choice of activation function has a significant effect on the catastrophic forgetting properties of a net, and in particular that hard LWTA outperforms logistic sigmoid and rectified linear units in this respect when trained with stochastic gradient descent. ",
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+    {
+        "type": "text",
+        "text": "In our more extensive experiments we found that the choice of activation function has a less consistent effect than the choice of training algorithm. When we performed experiments with different kinds of task pairs, we found that the ranking of the activation functions is very problem dependent. For example, logistic sigmoid is the worst under some conditions but the best under other conditions. This suggests that one should always cross-validate the choice of activation function, as long as it is computationally feasible. We also reject the idea that hard LWTA is particular resistant to catastrophic forgetting in general, or that it makes the standard SGD training algorithm more resistant to catastrophic forgetting. For example, when training with SGD on the input reformatting task pair, hard LWTA’s possibilities frontier is worse than all activation functions except sigmoid for most points along the curve. On the similar task pair, LWTA with SGD is the worst of all eight methods we considered, in terms of best performance on the new task, best performance on the old task, and in terms of attaining points close to the origin of the possibilities frontier plot. However, hard LWTA does perform the best in some circumstances (it has the best performance on the new task for the dissimilar task pair ). This suggests that it is worth including hard LWTA as one of many activation functions in a hyperparameter search. LWTA is however never the leftmost point in any of our three task pairs, so it is probably only useful in sequential task settings where forgetting is an issue. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "When computational resources are too limited to experiment with multiple activation functions, we recommend using the maxout activation function trained with dropout. This is the only method that appears on the lower-left frontier of the performance tradeoff plots for all three task pairs we considered. ",
+        "bbox": [
+            89,
+            440,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
+            91,
+            546,
+            243,
+            559
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank the developers of Theano (Bergstra et al., 2010; Bastien et al., 2012), Pylearn2 (Goodfellow et al., 2013a). We would also like to thank NSERC, Compute Canada, and Calcul Qu´ebec for providing computational resources. Ian Goodfellow is supported by the 2013 Google Fellowship in Deep Learning. ",
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+        "page_idx": 7
+    },
+    {
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parse/train/r1ZdKJ-0W/r1ZdKJ-0W.md

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+# DEEP GAUSSIAN EMBEDDING OF GRAPHS: UNSUPERVISED INDUCTIVE LEARNING VIA RANKING
+
+Aleksandar Bojchevski, Stephan Gunnemann ¨ Technical University of Munich, Germany {a.bojchevski,guennemann}@in.tum.de
+
+# ABSTRACT
+
+Methods that learn representations of nodes in a graph play a critical role in network analysis since they enable many downstream learning tasks. We propose Graph2Gauss – an approach that can efficiently learn versatile node embeddings on large scale (attributed) graphs that show strong performance on tasks such as link prediction and node classification. Unlike most approaches that represent nodes as point vectors in a low-dimensional continuous space, we embed each node as a Gaussian distribution, allowing us to capture uncertainty about the representation. Furthermore, we propose an unsupervised method that handles inductive learning scenarios and is applicable to different types of graphs: plain/attributed, directed/undirected. By leveraging both the network structure and the associated node attributes, we are able to generalize to unseen nodes without additional training. To learn the embeddings we adopt a personalized ranking formulation w.r.t. the node distances that exploits the natural ordering of the nodes imposed by the network structure. Experiments on real world networks demonstrate the high performance of our approach, outperforming state-of-the-art network embedding methods on several different tasks. Additionally, we demonstrate the benefits of modeling uncertainty – by analyzing it we can estimate neighborhood diversity and detect the intrinsic latent dimensionality of a graph.
+
+# 1 INTRODUCTION
+
+Graphs are a natural representation for a wide variety of real-life data, from social and rating networks (Facebook, Amazon), to gene interactions and citation networks (BioGRID, arXiv). Node embeddings are a powerful and increasingly popular approach to analyze such data (Cai et al., 2017). By operating in the embedding space, one can employ proved learning techniques and bypass the difficulty of incorporating the complex node interactions. Tasks such as link prediction, node classification, community detection, and visualization all greatly benefit from these latent node representations. Furthermore, for attributed graphs by leveraging both sources of information (network structure and attributes) one is able to learn more useful representations compared to approaches that only consider the graph (Yang et al., 2015; Pan et al., 2016; Ganguly & Pudi, 2017).
+
+All existing (attributed) graph embedding approaches represent each node by a single point in a low-dimensional continuous vector space. Representing the nodes simply as points, however, has a crucial limitation: we do not have information about the uncertainty of that representation. Yet uncertainty is inherent when describing a node in a complex graph by a single point only. Imagine a node for which the different sources of information are conflicting with each other, e.g. pointing to different communities or even revealing contradicting underlying patterns. Such discrepancy should be reflected in the uncertainty of its embedding. As a solution to this problem, we introduce a novel embedding approach that represents nodes as Gaussian distributions: each node becomes a full distribution rather than a single point. Thereby, we capture uncertainty about its representation.
+
+To effectively capture the non-i.i.d. nature of the data arising from the complex interactions between the nodes, we further propose a novel unsupervised personalized ranking formulation to learn the embeddings. Intuitively, from the point of view of a single node, we want nodes in its immediate neighborhood to be closest in the embedding space, while nodes multiple hops away should become increasingly more distant. This ordering between the nodes imposed by the network structure w.r.t the distances between their embeddings naturally leads to our ranking formulation. Taking into account this natural ranking from each node’s point of view, we learn more powerful embeddings since we incorporate information about the network structure beyond first and second order proximity.
+
+Furthermore, when node attributes (e.g. text) are available our method is able to leverage them to easily generate embeddings for previously unseen nodes without additional training. In other words, Graph2Gauss is inductive, which is a significant benefit over existing methods that are inherently transductive and do not naturally generalize to unseen nodes. This desirable inductive property comes from the fact that we are learning an encoder that maps the nodes’ attributes to embeddings.
+
+The main contributions of our approach are summarized as follows:
+
+a) We embed nodes as Gaussian distributions allowing us to capture uncertainty.   
+b) Our unsupervised personalized ranking formulation exploits the natural ordering of the nodes capturing the network structure at multiple scales.   
+c) We propose an inductive method that generalizes to unseen nodes and is applicable to different types of graphs: plain/attributed, directed/undirected.
+
+# 2 RELATED WORK
+
+The focus of this paper is on unsupervised learning of node embeddings for which many different approaches have been proposed. For a comprehensive recent survey see Cai et al. (2017), Hamilton et al. (2017), or Goyal & Ferrara (2017). Approaches such as DeepWalk and node2vec (Perozzi et al., 2014; Grover & Leskovec, 2016) look at plain graphs and learn an embedding based on random walks by extending or adapting the Skip-Gram (Mikolov et al., 2013) architecture. LINE (Tang et al., 2015b) uses first- and second-order proximity and trains the embedding via negative sampling. SDNE (Wang et al., 2016) similarly has a component that preserves second-order proximity and exploits first-order proximity to refine the representations. GraRep (Cao et al., 2015) is a factorization based method that considers local and global structural information.
+
+Tri-Party Deep Network Representation (TRIDNR) (Pan et al., 2016) considers node attributes, network structure and potentially node labels. CENE (Sun et al., 2016) similarly to Ganguly & Pudi (2017) treats the attributes as special kinds of nodes and learns embeddings on the augmented network. Text-Associated DeepWalk (TADW) (Yang et al., 2015) performs low-rank matrix factorization considering graph structure and text features. Heterogeneous networks are consider in (Tang et al., 2015a; Chang et al., 2015), while Huang et al. similarly to Pan et al. (2016) considers labels. GraphSAGE (Hamilton et al., 2017) is an inductive method that generates embeddings by sampling and aggregating attributes from a nodes local neighborhood and requires the edges of the new nodes.
+
+Graph convolutional networks are another family of approaches that adapt conventional CNNs to graph data (Kipf & Welling, 2016a; Defferrard et al., 2016; Henaff et al., 2015; Monti et al., 2016; Niepert et al., 2016; Pham et al., 2017). They utilize the graph Laplacian and the spectral definition of a convolution and boil down to some form of aggregation over neighbors such as averaging. They can be thought of as implicitly learning an embedding, e.g. by taking the output of the last layer before the supervised component. See Monti et al. (2016) for an overview. In contrast to this paper, most of these methods are (semi-)supervised. The graph variational autoencoder (GAE) (Kipf & Welling, 2016b) is a notable exception that learns node embeddings in an unsupervised manner.
+
+Few approaches consider the idea of learning an embedding that is a distribution. Vilnis & McCallum (2014) are the first to learn Gaussian word embeddings to capture uncertainty. Closest to our work, He et al. (2015) represent knowledge graphs and Dos Santos et al. (2016) study heterogeneous graphs for node classification. Both approaches are not applicable for the context of unsupervised learning of (attributed) graphs that we are interested in. The method in He et al. (2015) learns an embedding for each component of the triplets (head, tail, relation) in the knowledge graph. Note that we cannot naively employ this method by considering a single relation ”has an edge” and a single entity ”node”. Since their approach considers similarity between entities and relations, all nodes would be trivially similar to the single relation. Considering the semi-supervised approach proposed in Dos Santos et al. (2016), we cannot simply ”turn off” the supervised component to adapt their method for unsupervised learning, since given the defined loss we would trivially map all nodes to the same Gaussian. Additionally, both of these approaches do not consider node attributes.
+
+In this section we introduce our method Graph2Gauss (G2G) and detail how both the attributes and the network structure influence the learning of node representations. The embedding is carried out in two steps: (i) the node attributes are passed through a non-linear transformation via a deep neural network (encoder) and yield the parameters associated with the node’s embedding distribution; (ii) we formulate an unsupervised loss function that incorporates the natural ranking of the nodes as given by the network structure w.r.t. a dissimilarity measure on the embedding distributions.
+
+Problem definition. Let $G = \left( \mathbf { A } , \mathbf { X } \right)$ be a directed attributed graph, where $\mathbf { A } \in \mathbb { R } ^ { N \times N }$ is an adjacency matrix representing the edges between $N$ nodes and $\breve { \mathbf { X } } \in \mathbb { R } ^ { N \times D }$ collects the attribute information for each node where $\mathbf { x } _ { i }$ is a $D$ dimensional attribute vector of the $i ^ { t h }$ node.1 $V$ denotes the set of all nodes. We aim to find a lower-dimensional Gaussian distribution embedding $\mathbf { h } _ { i } \ =$ $\textstyle { \mathcal { N } } ( \mu _ { i } , \Sigma _ { i } )$ , $\mu _ { i } \in \mathbb { R } ^ { L } , \Sigma _ { i } \in \mathbb { R } ^ { L \times L }$ with $L \ll N , D$ , such that nodes similar w.r.t. attributes and network structure are also similar in the embedding space given a dissimilarity measure $\Delta ( \mathbf { h } _ { i } , \mathbf { h } _ { j } )$ . In Fig.5(a) for example we show nodes that are embedded as two dimensional Gaussians.
+
+# 3.1 NETWORK STRUCTURE REPRESENTATION VIA PERSONALIZED RANKING
+
+To capture the structural information of the network in the embedding space, we propose a personalized ranking approach. That is, locally per node $i$ we impose a ranking of all remaining nodes w.r.t. their distance to node $i$ in the embedding space. More precisely, in this paper we exploit the $k$ -hop neighborhoods of each node. Given some anchor node $i$ , we define ${ \cal N } _ { i k } = \{ \bar { j } \in V | i \neq j , \operatorname* { m i n } ( s p ( i , j ) , K ) = k \}$ to be the set of nodes who are exactly $k$ hops away from node $i$ , where $V$ is the set of all nodes, $K$ is a hyper-parameter denoting the maximum distance we are wiling to consider, and $s p ( i , j )$ returns either the length of the shortest path starting at node $i$ and ending in node $j$ or $\infty$ if node $j$ is not reachable.
+
+Intuitively, we want all nodes belonging to the 1-hop neighborhood of $i$ to be closer to $i$ w.r.t. their embedding, compared to the all nodes in its 2-hop neighborhood, which in turn are closer than the nodes in its 3-hop neighborhood and so on up to $K$ . Thus, the ranking that we want to ensure from the perspective of node $i$ is
+
+$$
+\Delta ( \mathbf { h } _ { i } , \mathbf { h } _ { k _ { 1 } } ) < \Delta ( \mathbf { h } _ { i } , \mathbf { h } _ { k _ { 2 } } ) < \cdots < \Delta ( \mathbf { h } _ { i } , \mathbf { h } _ { k _ { K } } ) \quad \forall k _ { 1 } \in N _ { i 1 } , \forall k _ { 2 } \in N _ { i 2 } , \dots , \forall k _ { K } \in N _ { i K }
+$$
+
+or equivalently, we aim to satisfy the following pairwise constraints
+
+$$
+\Delta ( { \bf h } _ { i } , { \bf h } _ { j } ) < \Delta ( { \bf h } _ { i } , { \bf h } _ { j ^ { \prime } } ) , \forall i \in V , \forall j \in N _ { i k } , \forall j ^ { \prime } \in N _ { i k ^ { \prime } } , \forall k < k ^ { \prime }
+$$
+
+Going beyond mere first-order and second-order proximity this enables us to capture the network structure at multiple scales incorporating local and global structure.
+
+Dissimilarity measure. To solve the above ranking task we have to define a suitable dissimilarity measure between the latent representation of two nodes. Since our latent representations are distributions, similarly to Dos Santos et al. (2016) and He et al. (2015) we employ the asymmetric KL divergence. This gives the additional benefit of handling directed graphs in a sound way. More specifically, given the latent Gaussian distribution representation of two nodes $\mathbf { h } _ { i } , \mathbf { h } _ { j }$ we define
+
+$$
+\Delta ( { \bf h } _ { i } , { \bf h } _ { j } ) = D _ { K L } ( N _ { j } | | \mathcal { N } _ { i } ) = \frac { 1 } { 2 } \bigg [ t r ( \Sigma _ { i } ^ { - 1 } \Sigma _ { j } ) + ( \mu _ { i } - \mu _ { j } ) ^ { T } \Sigma _ { i } ^ { - 1 } ( \mu _ { i } - \mu _ { j } ) - L - l o g \frac { d e t ( \Sigma _ { j } ) } { d e t ( \Sigma _ { i } ) } \bigg ]
+$$
+
+Here we use the notation $\mu _ { i } , \Sigma _ { i }$ to denote the outputs of some functions $\mu _ { \boldsymbol { \theta } } ( \mathbf { x } _ { i } )$ and $\Sigma _ { \theta } ( \mathbf { x } _ { i } )$ applied to the attributes $\mathbf { x } _ { i }$ of node $i$ and $t r ( . )$ denotes the trace of a matrix. The asymmetric KL divergence also applies to the case of an undirected graph by simply processing both directions of the edge. We could alternatively use a symmetric dissimilarity measure such as the Jensen-Shannon divergence or the expected likelihood (probability product kernel).
+
+# 3.2 DEEP ENCODER
+
+The functions $\mu _ { \boldsymbol { \theta } } ( \mathbf { x } _ { i } )$ and $\Sigma _ { \theta } ( \mathbf { x } _ { i } )$ are deep feed-forward non-linear neural networks parametrized by $\theta$ . It is important to note that these parameters are shared across instances and thus enjoy statistical strength benefits. Additionally, we design $\mu _ { \boldsymbol { \theta } } ( \mathbf { x } _ { i } )$ and $\Sigma _ { \theta } ( \mathbf { x } _ { i } )$ such that they share parameters as well. More specifically, a deep encoder $f _ { \theta } ( \mathbf { x } _ { i } )$ processes the node’s attributes and outputs an intermediate hidden representation, which is then in turn used to output $\mu _ { i }$ and $\Sigma _ { i }$ in the final layer of the architecture. We focus on diagonal covariance matrices.2 The mapping from the nodes’ attributes to their embedding via the deep encoder is precisely what enables the inductiveness of Graph2Gauss.
+
+# 3.3 LEARNING VIA ENERGY-BASED LOSS
+
+Since it is intractable to find a solution that satisfies all of the pairwise constraints defined in Sec. 3.1 we turn to an energy based learning approach. The idea is to define an objective function that penalizes ranking errors given the energy of the pairs. More specifically, denoting the KL divergence between two nodes as the respective energy, $E _ { i j } = D _ { K L } ( \mathcal { N } _ { j } | | \mathcal { N } _ { i } )$ , we define the following loss to be optimized
+
+$$
+\mathcal { L } = \sum _ { i } \sum _ { k < l } \sum _ { j _ { k } \in N _ { i k } } \sum _ { j _ { l } \in N _ { i l } } \left( E _ { i j _ { k } } { ^ { 2 } } + \exp ^ { - E _ { i j _ { l } } } \right) = \sum _ { ( i , j _ { k } , j _ { l } ) \in \mathcal { D } _ { t } } \left( E _ { i j _ { k } } { ^ { 2 } } + \exp ^ { - E _ { i j _ { l } } } \right)
+$$
+
+where $\mathcal { D } _ { t } = \{ ( i , j _ { k } , j _ { l } ) ~ | ~ s p ( i , j _ { k } ) < s p ( i , j _ { l } ) \}$ is the set of all valid triplets. The $E _ { i j _ { k } }$ terms are positive examples whose energy should be lower compared to the energy of the negative examples $E _ { i j _ { l } }$ . Here, we employed the so called square-exponential loss (LeCun et al., 2006) which unlike other typically used losses (e.g. hinge loss) does not have a fixed margin and pushes the energy of the negative terms to infinity with exponentially decreasing force. In our setting, for a given anchor node $i$ , the energy $E _ { i j }$ should be lowest for nodes $j$ in his 1-hop neighborhood, followed by a higher energy for nodes in his 2-hop neighborhood and so on.
+
+Finally, we can optimize the parameters $\theta$ of the deep encoder such that the loss $\mathcal { L }$ is minimized and the pairwise rankings are satisfied. Note again that the parameters are shared across all instances, meaning that we share statistical strength and can learn them more easily in comparison to treating the distribution parameters (e.g. $\mu _ { i } , \Sigma _ { i } )$ independently as free variables. The parameters are optimized using Adam (Kingma & Ba, 2014) with a fixed learning rate of 0.001.
+
+Sampling strategy. For large graphs, the complete loss is intractable to compute, confirming the need for a stochastic variant. The naive approach would be to sample triplets from $\mathcal { D } _ { t }$ uniformly, i.e. replace $\sum _ { \left( i , j _ { k } , j _ { l } \right) \in { \mathcal { D } } _ { t } }$ with $\mathbb { E } _ { ( i , j _ { k } , j _ { l } ) \sim \mathcal { D } _ { t } }$ in Eq. 1. However, with the naive sampling we are less likely to sample triplets that involve low-degree nodes since high degree nodes occur in many more pairwise constraints. This in turn means that we update the embedding of low-degree nodes less often which is not desirable. Therefore, we propose an alternative node-anchored sampling strategy. Intuitively, for every node $i$ , we randomly sample one other node from each of its neighborhoods (1-hop, 2-hop, etc.) and then optimize over all the corresponding pairwise constraints $( E _ { i 1 } < E _ { i 2 } , \ldots , E _ { i 1 } < E _ { i K } , E _ { i 2 } < E _ { i 3 } , \ldots E _ { i 2 } <$ $\kappa , E _ { i 2 } < E _ { i 3 } , \ldots E _ { i 2 } < E _ { i K } , \ldots , E _ { i K - 1 } < E _ { i K } )$ .
+
+Naively applying the node-anchored sampling strategy and optimizing Eq. 1, however, would lead to biased estimates of the gradient. Theorem 1 shows how to adapt the loss such that it is equal in expectation to the original loss under our new sampling strategy. As a consequence, we have unbiased estimates of the gradient using stochastic optimization of the reformulated loss.
+
+Theorem 1 For all $i$ , let $( j _ { 1 } , \dots , j _ { K } )$ be independent uniform random samples from the sets $( N _ { i 1 } , \dots , N _ { i K } )$ and $| N _ { i * } |$ the cardinality of each set. Then $\mathcal { L }$ is equal in expectation to
+
+$$
+\mathcal { L } _ { s } = \sum _ { i } \mathbb { E } _ { ( j _ { 1 } , \dots , j _ { K } ) \sim ( N _ { i 1 } , \dots , N _ { i K } ) } \left[ \sum _ { k < l } \vert N _ { i k } \vert \cdot \vert N _ { i l } \vert \cdot \left( E _ { i j _ { k } } ^ { \phantom { - } } + \exp ^ { - E _ { i j _ { l } } } \right) \right] = \mathcal { L }
+$$
+
+We provide the proof in the appendix. For cases where the number of nodes $N$ is particularly large we can further subsample mini-batches, by selecting anchor nodes $i$ at random. Furthermore, in our experimental study, we analyze the effect of the sampling strategy on convergence, as well as the quality of the stochastic variant w.r.t. the obtained solution and the reached local optima.
+
+# 3.4 DISCUSSION
+
+Inductive learning. While during learning we need both the network structure (to evaluate the ranking loss) and the attributes, once the learning concludes, the embedding for a node can be obtained solely based on its attributes. This enables our method to easily handle the issue of obtaining representations for new nodes that were not part of the network during training. To do so we simply pass the attributes of the new node through our learned deep encoder. Most approaches cannot handle this issue at all, with a notable exception being SDNE and GraphSAGE (Wang et al., 2016; Hamilton et al., 2017). However, both approaches require the edges of the new node to get the node’s representation, and cannot handle nodes that have no existing connections. In contrast, our method can handle even such nodes, since after the model is learned we rely only on the attribute information.
+
+Plain graph embedding. Even though attributed graphs are often found in the real-world, sometimes it is desirable to analyze plain graphs. As already discussed, our method easily handles plain graphs, when the attributes are not available, by using one-hot encoding of the nodes instead. As we later show in the experiments we are able to learn useful representations in this scenario, even outperforming some attributed approaches. Naturally, in this case we lose the inductive ability to handle unseen nodes. We compare the one-hot encoding version, termed G2G oh, with our full method G2G that utilizes the attributes, as well as all remaining competitors.
+
+Encoder architecture. Depending on the type of the node attributes (e.g. images, text) we could in principle use CNNs/RNNs to process them. We could also easily incorporate any of the proposed graph convolutional layers inheriting their benefits. However, we observe that in practice using simple feed-forward architecture with rectifier units is sufficient, while being much faster and easier to train. Better yet, we observed that Graph2Gauss is not sensitive to the choice of hyperparameters such as number and size of hidden layers. We provide more detailed information and sensible defaults in the appendix.
+
+Complexity. The time complexity for computing the original loss is $O ( N ^ { 3 } )$ where $N$ is the number of nodes. Using our node-anchored sampling strategy, the complexity of the stochastic version is $O ( K ^ { 2 } N )$ where $K$ is the maximum distance considered. Since a small value of $K \leq 2$ consistently showed good performance, $K ^ { 2 }$ becomes negligible and thus the complexity is $O ( N )$ , meaning linear in the number of nodes. This coupled with the small number of epochs $T$ needed for convergence $T \leq 2 0 0 0$ for all shown experiments, see e.g. Fig. 3(b)) and an efficient GPU implementation also made our method faster than most competitors in terms of wall-clock time.
+
+# 4 EMBEDDING EVALUATION
+
+We compare Graph2Gauss with and without considering attributes (G2G, G2G oh) to several competitors namely: TRIDNR and TADW (Pan et al., 2016; Yang et al., 2015) as representatives that consider attributed graphs, GAE (Kipf & Welling, 2016b) as the unsupervised graph convolutional representative, and node2vec (Grover & Leskovec, 2016) as a representative of the random walk based plain graph embeddings. Additionally, we include a strong Logistic Regression baseline that considers only the attributes. As with all other methods we train TRIDNR in a unsupervised manner, however, since it can only process raw text as attributes (rather than e.g. bag-of-words) it is not always applicable. Furthermore, since TADW, and GAE only support undirected graphs we must symmetrize the graph before using them – giving them a substantial advantage, especially in the link prediction task. Moreover, in all experiments if the competing techniques use an $L$ dimensional embedding, G2G’s embedding is actually only half of this dimensionality so that the overall number of ’parameters’ per node (mean vector $^ +$ variance terms of the diagonal $\Sigma _ { i }$ ) matches $L$ .
+
+Dataset description. We use several attributed graph datasets. Cora (McCallum et al., 2000) is a well-known citation network labeled based on the paper topic. While most approaches report on a small subset of this dataset we additionally extract from the original data the entire network and name these two datasets CORA $( N = 1 9 7 9 3 , E = 6 5 3 1 1 , D = 8 7 1 0 , K = 7 0 )$ and CORA-ML $( N = 2 9 9 5 , E = 8 4 1 6 , D = 2 8 7 9 , K = 7 )$ respectively. CITESEER $( N = 4 2 3 0 , E = 5 3 5 8 , D =$ $2 7 0 1 , K = 6 )$ (Giles et al., 1998), DBLP (Pan et al., 2016) $( N = 1 7 7 1 6 , E = 1 0 5 7 3 4 , D =$ 1639, $K = 4$ ) and PUBMBED $( N = 1 8 2 3 0 , E = 7 9 6 1 2 , D = 5 0 0 , K = 3 )$ (Sen et al., 2008) are other commonly used citation datasets. We provide all datasets, the source code of G2G, and further supplementary material (https://www.kdd.in.tum. $\mathrm { d e } / \mathrm { g } 2 \mathrm { g } )$ ).
+
+# 4.1 LINK PREDICTION
+
+Setup. Link prediction is a commonly used task to demonstrate the meaningfulness of the embeddings. To evaluate the performance we hide a set of edges/non-edges from the original graph and train on the resulting graph. Similarly to Kipf & Welling (2016b) and Wang et al. (2016) we create a validation/test set that contains $5 \% / 1 0 \%$ randomly selected edges respectively and equal number of randomly selected non-edges.We used the validation set for hyper-parameter tuning and early stopping and the test set only to report the performance. As by convention we report the area under the ROC curve (AUC) and the average precision (AP) scores for each method. To rank the candidate edges we use the negative energy $- E _ { i j }$ for Graph2Gauss, and the exact same approach as in the respective original methods (e.g. dot product of the embeddings).
+
+Performance on real-world datasets. Table 1 shows the performance on the link prediction task for different datasets and embedding size $L = 1 2 8$ . As we can see our method significantly outperforms the competitors across all datasets which is a strong sign that the learned embeddings are useful. Furthermore, even the constrained version of our method G2G oh that does not consider attributes at all outperforms the competitors on some datasets. While GAE achieves comparable performance on some of the datasets their approach doesn’t scale to large graphs. In fact, for graphs beyond $1 5 K$ nodes we had to revert to slow training on the CPU since the data did not fit on the GPU memory (12GB). The simple Logistic Regression baseline showed surprisingly strong performance, even outperforming some of the more complicated methods.
+
+Table 1: Link prediction performance for real-world datasets with $L = 1 2 8$   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Cora-ML</td><td colspan="2">Cora</td><td colspan="2">Citeseer</td><td colspan="2">DBLP</td><td colspan="2">Pubmed</td><td colspan="2">Cora-ML Easy</td></tr><tr><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td></tr><tr><td>Logistic Regression</td><td>90.01</td><td>89.75</td><td>86.58</td><td>86.51</td><td>81.70</td><td>79.10</td><td>82.04</td><td>81.91</td><td>90.50</td><td>90.99</td><td>90.28</td><td>90.99</td></tr><tr><td>node2vec(Grover &amp; Leskovec,2016)</td><td>76.80</td><td>75.26</td><td>79.95</td><td>78.98</td><td>83.04</td><td>83.74</td><td>95.42</td><td>95.33</td><td>95.42</td><td>95.33</td><td>93.47</td><td>93.53</td></tr><tr><td>TADW(Yang et al., 2015)</td><td>81.26</td><td>81.34</td><td>76.56</td><td>78.06</td><td>70.14</td><td>72.93</td><td>65.67</td><td>59.85</td><td>62.72</td><td>68.02</td><td>83.53</td><td>82.47</td></tr><tr><td>TRIDNR(Pan et al., 2016)</td><td>84.51</td><td>85.69</td><td>81.61</td><td>81.08</td><td>87.23</td><td>88.87</td><td>92.01</td><td>91.62</td><td>NTA</td><td>NTA</td><td>85.59</td><td>86.16</td></tr><tr><td>GAE(Kipf &amp; Welling,2016b)</td><td>96.65</td><td>96.67</td><td>97.91</td><td>98.07</td><td>92.31</td><td>93.88</td><td>95.78</td><td>96.67</td><td>96.07</td><td>96.12</td><td>95.97</td><td>95.17</td></tr><tr><td>G2G_oh</td><td>96.95</td><td>97.54</td><td>98.41</td><td>98.63</td><td>95.89</td><td>95.78</td><td>98.29</td><td>98.46</td><td>96.75</td><td>96.47</td><td>96.98</td><td>96.42</td></tr><tr><td>G2G</td><td>98.01</td><td>98.03</td><td>98.81</td><td>98.78</td><td>96.09</td><td>96.16</td><td>98.65</td><td>98.78</td><td>97.42</td><td>97.85</td><td>98.03</td><td>98.12</td></tr></table>
+
+We also include the performance on the so called ”Cora-ML Easy” dataset, obtained from the CoraML dataset by making it undirected and selecting the nodes in the largest connected component. We see that while node2vec struggles on the original real-world data, it significantly improves in this ”easy” setting. On the contrary, Graph2Gauss handles both settings effortlessly. This demonstrates that Graph2Gauss can be readily applied in realistic scenarios on potentially messy real-world data.
+
+Sensitivity analysis. In Figs.1(a) and 1(b) we show the performance w.r.t. the dimensionality of the embedding, averaged over 10 trials. G2G is able to learn useful embeddings with strong performance even for relatively small embedding sizes. Even for the case $L = 2$ , where we embed the points as one dimensional Gaussian distributions $( L = 1 + 1$ for the mean and the sigma of the Gaussian), G2G still outperforms all of the competitors irrespective of their much higher embedding sizes.
+
+![](images/8a932d69c197c1a3e0f8320a4a38a85a80651549fda7644ac92413355fa92960.jpg)  
+Figure 1: Link prediction performance for different embedding sizes and percentages of training edges on Cora-ML. G2G outperforms the competitors even for small sizes and percentage of edges.
+
+Finally, we evaluate the performance w.r.t. the percentage of training edges varying from $1 5 \%$ to $8 5 \%$ , averaged over 10 trials. We can see in Figs.1(c) and 1(d) Graph2Gauss strongly outperforms the competitors, especially for small number of training edges. The dashed line indicates the percentage above which we can guarantee to have every node appear at least once in the training set.3 The performance below that line is then indicative of the performance in the inductive setting. Since, the structure only methods are unable to compute meaningful embeddings for unseen nodes we cannot report their performance below the dashed line.
+
+# 4.2 NODE CLASSIFICATION
+
+Setup. Node classification is another task commonly used to evaluate the strength of the learned embeddings – after they have been trained in an unsupervised manner. We evaluate the node classification performance for three datasets (Cora-ML, Citeseer and DBLP) that have ground-truth classes. First, we train the embeddings on the entire training data in an unsupervised manner (excluding the class labels). Then, following Perozzi et al. (2014) we use varying percentage of randomly selected nodes and their learned embeddings along with their labels as training data for a logistic regression, while evaluating the performance on the rest of the nodes. We also optimize the regularization strength for each method/dataset via cross-validation. We show results averaged over 10 trials.
+
+![](images/9dfe41f5f0f942d2393078c92264f4f3c1bf45a3aee7e8643ba84bccb167a5d8.jpg)  
+Figure 2: Classification performance comparison - both G2G and G2G oh perform strongly.
+
+Performance on real-world datasets. Figs. 2 compares the methods w.r.t. the classification performance for different percentage of labeled nodes. We can see that our method clearly outperforms the competitors. Again, the constrained version of our method that does not consider attributes is able to outperform some of the competing approaches. Additionally, we can conclude that in general our method shows stable performance regardless of the percentage of labeled nodes. This is a highly desirable property since it shows that should we need to perform classification it is sufficient to train only on a small percentage of labeled nodes.
+
+# 4.3 SAMPLING STRATEGY
+
+Figure 3(a) shows the validation set ROC score for the link prediction task w.r.t. the number of triplets $( i , j _ { k } , j _ { l } )$ seen. We can see that both sampling strategies are able to reach the same performance as the full loss in significantly fewer $( < 4 . 2 \%$ ) number of pairs seen (note the log scale). It also shows that the naive random sampling converges slower than the node-anchored sampling strategy. Figures 3(b) gives us some insight as to why – our node-anchored sampling strategy achieves significantly lower loss. Finally, Fig. 3(c) shows that our node-anchored sampling strategy has lower variance of the gradient updates, which is another contributor to faster convergence.
+
+![](images/64ef383bd0bca1f97a6bdaad086b8de2a61848e9e23a02c3696b4d53b541a631.jpg)  
+Figure 3: Our sampling strategy converges significantly faster than the full loss, while maintaining good performance. It also achieves better loss and has lower variance compared to naive sampling.
+
+Learning an embedding that is a distribution rather than a point-vector allows us to capture uncertainty about the representation. We perform several experiments to evaluate the benefit of modeling uncertainty. Figure 4(a) shows that the learned uncertainty is correlated with neighborhood diversity, where for a node $i$ we define diversity as the number of distinct classes among the nodes in its $p$ -hop neighborhood $\textstyle ( \bigcup _ { 1 \leq k \leq p } N _ { i k } )$ . Since the uncertainty for a node $i$ is an $L$ -dimensional vector (diagonal covariance) we show the average across the dimensions. In line with our intuition, nodes with less diverse neighborhood have significantly lower variance compare to more diverse nodes whose immediate neighbors belong to many different classes, thus making their embedding more uncertain. The figure shows the result on the Cora dataset for $p = 3$ hop neighborhood. Similar results hold for the other datasets. This result is particularly impressive given the fact that we learn our embedding in a completely unsupervised manner, yet the uncertainty was able to capture the diversity w.r.t. the class labels of the neighbors of a node, which were never seen during training.
+
+![](images/5c593abf4a0b86b0814e86b69cc0ee34f42469e0bce9d3a752b172aed8c32348.jpg)  
+Figure 4: The benefit of modeling the uncertainty of the nodes.
+
+Figure 4(b) shows that using the learned uncertainty we are able to detect the intrinsic latent dimensionality of the graph. Each line represents the average variance (over all nodes) for a given dimension $l$ for each epoch. We can see that as the training progresses past the stopping criterion (link prediction performance on validation set) and we start to overfit, some dimensions exhibit a relatively stable average variance, while for others the variance increases with each epoch. By creating a simply rule that monitors the average change of the variance over time we were able to automatically detect these relevant latent dimensions (colored in red). This result holds for multiple datasets and is shown here for Cora-ML. Interestingly, the number of detected latent dimensions (6) is close to the number of ground-truth communities (7).
+
+The next obvious question is then how does the performance change if we remove these highly uncertain dimensions whose variance keeps increasing with training. Figure 4(c) answers exactly that. By removing progressively more and more dimensions, starting with the most uncertain first we see imperceptibly small change in performance. Only once we start removing the true latent dimension we see a noticeable degradation in performance. The dashed lines show the performance if we re-train the model, setting $L = 6$ , equal to the detected number of latent dimensions.
+
+As a last study of uncertainty, in a use case analysis, the nodes with high uncertainty reveal additional interesting patterns. For example in the Cora dataset, one of the highly uncertain nodes was the paper ”The use of word shape information for cursive script recognition” by R.J. Whitrow – surprisingly, all citations (edges) of that paper (as extracted from the dataset) were towards other papers by the same author.
+
+# 4.5 INDUCTIVE LEARNING: GENERALIZATION TO UNSEEN NODES
+
+As discussed in Sec. 3.4 G2G is able to learn embeddings even for nodes that were not part of the networks structure during training time. Thus, it not only supports transductive but also inductive learning. To evaluate how our approach generalizes to unseen nodes we perform the following experiment: (i) first we completely hide $1 0 \% / 2 5 \%$ of nodes from the network at random; (ii) we proceed to learn the node embeddings for the rest of the nodes; (iii) after learning is complete we pass the (new) unseen test nodes through our deep encoder to obtain their embedding; (iv) we evaluate by calculating the link prediction performance (AUC and AP scores) using all their edges and same number of non-edges.
+
+Table 2: Inductive link prediction performance.   
+
+<table><tr><td rowspan="2">Method (% hidden)</td><td colspan="2">Cora-ML</td><td colspan="2">Cora</td><td colspan="2">Citeseer</td><td colspan="2">DBLP</td><td colspan="2">Pubmed</td></tr><tr><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td></tr><tr><td>Log.Reg.10%</td><td>75.95</td><td>78.62</td><td>78.53</td><td>78.70</td><td>73.09</td><td>72.54</td><td>67.55</td><td>69.55</td><td>86.83</td><td>87.34</td></tr><tr><td>G2G 10%</td><td>90.93</td><td>89.37</td><td>94.18</td><td>93.40</td><td>88.58</td><td>88.31</td><td>85.06</td><td>83.75</td><td>92.22</td><td>90.45</td></tr><tr><td>G2G 25%</td><td>87.83</td><td>86.31</td><td>92.96</td><td>92.31</td><td>87.30</td><td>86.61</td><td>83.09</td><td>81.49</td><td>90.20</td><td>88.28</td></tr></table>
+
+As the results in Table 2 clearly show, since we are utilizing the rich attribute information, we are able to achieve strong performance for unseen nodes. This is true even when a quarter of the nodes are missing. This makes our method applicable in the context of large graphs where training on the entire network is not feasible. Note that SDNE (Wang et al., 2016) and GraphSAGE (Hamilton et al., 2017) cannot be applied in this scenario, since they also require the edges for the unseen nodes to produce an embedding. Graph2Gauss is the only inductive method that can obtain embeddings for a node based only on the node attributes.
+
+# 4.6 NETWORK VISUALIZATION
+
+One key application of node embedding approaches is creating meaningful visualizations of a network in 2D/3D that support tasks such as data exploration and understanding. Following Tang et al. (2015b) and Pan et al. (2016) we first learn a lower-dimensional $L = 1 2 8$ embedding for each node and then map those representations in 2D with TSNE (Maaten & Hinton, 2008). Additionally, since our method is able to learn useful representations even in low dimensions we embed the nodes as 2D Gaussians and visualize the resulting embedding. This has the added benefit of visualizing the nodes’ uncertainty as well. Fig. 5 shows the visualization for the Cora-ML dataset. We see that Graph2Gauss learns an embedding in which the different classes are clearly separated.
+
+![](images/e1a164d9a7672be7d5d2534731493901583feced1417dc1496cf52360ff653f2.jpg)  
+Figure 5: 2D visualization of the embeddings on the Cora-ML dataset. Color indicates the class label not used during training. Best viewed on screen.
+
+# 5 CONCLUSION
+
+We proposed Graph2Gauss – the first unsupervised approach that represents nodes in attributed graphs as Gaussian distributions and is therefore able to capture uncertainty. Analyzing the uncertainty reveals the latent dimensionality of a graph and gives insight into the neighborhood diversity of a node. Since we exploit the attribute information of the nodes we can effortlessly generalize to unseen nodes, enabling inductive reasoning. Graph2Gauss leverages the natural ordering of the nodes w.r.t. their neighborhoods via a personalized ranking formulation. The strength of the learned embeddings has been demonstrated on several tasks – specifically achieving high link prediction performance even in the case of low dimensional embeddings. As future work we aim to study personalized rankings beyond the ones imposed by the shortest path distance.
+
+# ACKNOWLEDGMENTS
+
+This research was supported by the German Research Foundation, Emmy Noether grant GU 1409/2- 1, and by the Technical University of Munich - Institute for Advanced Study, funded by the German Excellence Initiative and the European Union Seventh Framework Programme under grant agreement no 291763, co-funded by the European Union.
+
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+Prithviraj Sen, Galileo Namata, Mustafa Bilgic, Lise Getoor, Brian Galligher, and Tina Eliassi-Rad. Collective classification in network data. AI magazine, 29(3):93, 2008.
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+Xiaofei Sun, Jiang Guo, Xiao Ding, and Ting Liu. A general framework for content-enhanced network representation learning. arXiv preprint arXiv:1610.02906, 2016.
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+Jian Tang, Meng Qu, and Qiaozhu Mei. Pte: Predictive text embedding through large-scale heterogeneous text networks. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1165–1174. ACM, 2015a.
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+Jian Tang, Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, and Qiaozhu Mei. Line: Large-scale information network embedding. In Proceedings of the 24th International Conference on World Wide Web, pp. 1067–1077. ACM, 2015b.
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+Luke Vilnis and Andrew McCallum. Word representations via gaussian embedding. arXiv preprint arXiv:1412.6623, 2014.
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+Daixin Wang, Peng Cui, and Wenwu Zhu. Structural deep network embedding. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1225–1234. ACM, 2016.
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+Cheng Yang, Zhiyuan Liu, Deli Zhao, Maosong Sun, and Edward Y Chang. Network representation learning with rich text information. In IJCAI, pp. 2111–2117, 2015.
+
+# APPENDIX
+
+A PROOF OF THEOREM 1
+
+To prove Theorem 1 we start with the loss $\mathcal { L } _ { s }$ (Eq. 2), and show that by applying the expectation operator we will obtain the original loss $\mathcal { L }$ (Eq. 1). From there it trivially follows that taking the gradient with respect to $\mathcal { L } _ { s }$ for a set of samples gives us an unbiased estimate of the gradient of $\mathcal { L }$ .
+
+First we notice that both $\mathcal { L }$ and $\mathcal { L } _ { s }$ are summing over $i$ , thus it is sufficient to show that the losses are equal in expectation for a single node $i$ . Denoting with $\mathcal { L } _ { s } ^ { ( i ) }$ the loss for a single node $i$ and with $E _ { i , k , l } = E _ { i j _ { k } } { } ^ { 2 } + \exp ^ { - E _ { i j _ { l } } }$ for notational convenience we have:
+
+$$
+\begin{array} { r l } & { \| \nabla _ { x } ^ { \bot } \varphi \| _ { L ^ { 2 } ( \mathcal H _ { 1 } ) } ^ { 2 } = \frac { 1 } { \lambda _ { x } ^ { 2 } } \sum _ { t _ { i } ^ { \prime } \in \mathcal H _ { 1 } } ^ { 1 } \lambda _ { t _ { i } ^ { \prime } } \sum _ { \lambda _ { x } ^ { \prime } } ^ { \lambda _ { \prime } } \lambda _ { t _ { i } ^ { \prime } } \Big | \nabla _ { x } ^ { \bot } \varphi \Big | _ { L ^ { 2 } ( \mathcal H _ { 1 } ) } ^ { 2 } } \\ & { \quad - \frac { 1 } { \lambda _ { x } ^ { 2 } } \sum _ { t _ { i } ^ { \prime } \in \mathcal H _ { 1 } } \lambda _ { t _ { i } ^ { \prime } } \sum _ { \lambda _ { x } ^ { \prime } } \lambda _ { t _ { i } ^ { \prime } } \Big | \nabla _ { x } \varphi \Big | _ { L ^ { 2 } ( \mathcal H _ { 1 } ) } ^ { 2 } \cdot \nabla _ { x } \varphi \Big | _ { L ^ { 2 } ( \mathcal H _ { 1 } ) } ^ { 2 } } \\ & { \quad + \frac { 1 } { \lambda _ { x } ^ { 2 } } \sum _ { t _ { i } ^ { \prime } \in \mathcal H _ { 1 } } \lambda _ { t _ { i } ^ { \prime } } \sum _ { \lambda _ { x } ^ { \prime } } \lambda _ { t _ { i } ^ { \prime } } \Big | \nabla _ { x } \varphi \Big | _ { L ^ { 2 } ( \mathcal H _ { 1 } ) } ^ { 2 } \cdot \nabla _ { x } \varphi \Big | _ { L ^ { 2 } ( \mathcal H _ { 1 } ) } ^ { 2 } \cdot \nabla _ { x } \varphi \Big | _ { L ^ { 2 } ( \mathcal H _ { 1 } ) } ^ { 2 } } \\ & { \quad - \frac { 1 } { \lambda _ { x } ^ { 2 } } \sum _ { t _ { i } ^ { \prime } \in \mathcal H _ { 1 } } \lambda _ { t _ { i } ^ { \prime } } \Big | \nabla _ { x } \varphi \Big | _ { L ^ { 2 } ( \mathcal H _ { 1 } ) } ^ { 2 } \cdot \nabla _ { x } \varphi \Big | _ { L ^ { 2 } ( \mathcal H _ { 1 } ) } ^ { 2 } } \\ &  \quad - \frac { 1 } { \lambda _ { x } ^ { 2 } } \sum _ { t _ { i } ^ { \prime } \in \mathcal H _ { 1 } } \lambda _ { t _ { i } ^ { \prime } } \Big | \nabla _ { x } \varphi \Big | _  L ^  \end{array}
+$$
+
+In step (1) we have expanded the sum over $k \ < \ l$ in independent terms. In step (2) we have marginalized the expectation over the variables that do not appear in the expression, e.g. for the term $\mathbb { E } _ { ( j _ { 1 } , . . . , j _ { K } ) \sim ( N _ { i 1 } , . . . , N _ { i K } ) } | N _ { i 1 } | \cdot | N _ { i 2 } | \cdot E _ { i 1 2 }$ we can marginalize over $j _ { p }$ where $p \neq 1$ and $p \neq 2$ since the term doesn’t depend on them. In step (3) we have expanded the expectation term. In step (4) we have substituted $p ( j _ { p } )$ with $\frac { 1 } { | N _ { i j p } | }$ since we are sampling uniformly at random.
+
+Since $\mathcal { L } _ { s } ^ { ( i ) }$ is equal to $\mathcal { L } ^ { ( i ) }$ in expectation it follows that $\nabla { \mathcal { L } } _ { s }$ based on a set of samples is an unbiased estimate of $\nabla \mathcal { L }$ .
+
+# B IMPLEMENTATION DETAILS
+
+Architecture and hyperparameters. We observed that Graph2Gauss is not sensitive to the choice of hyperparameters such as number and size of hidden layers. Better yet, as shown in Sec. 4.4, Graphs2Gauss is also not sensitive to the size of the embedding $L$ . Thus, for a new graph, one can simply pick a relatively large embedding size and if required prune it later similarly to the analysis performed in Fig. 4(c).
+
+As a sensible default we recommend an encoder with a single hidden layer of size $s _ { 1 } = 5 1 2$ . More specifically, to obtain the embeddings for a node $i$ we have
+
+$$
+\mathbf { h } _ { i } = \mathrm { r e l u } ( \mathbf { X } _ { i } \mathbf { W } + \mathbf { b } ) \qquad \mu _ { i } = \mathbf { h } _ { \mathbf { i } } \mathbf { W } _ { \mu } + \mathbf { b } _ { \mu } \qquad \sigma _ { i } = \mathrm { e l u } ( \mathbf { h } _ { i } \mathbf { W } _ { \Sigma } + \mathbf { b } _ { \Sigma } ) + 1
+$$
+
+where $\mathbf { x } _ { i }$ are node attributes, relu and elu are the rectified linear unit and exponential linear unit respectively. In practice, we found that the softplus works equally well as the elu for making sure that $\sigma _ { i }$ are positive and in turn $\Sigma _ { i }$ is positive definite. We used Xavier initialization (Glorot & Bengio, 2010) for the weight matrices $\mathbf { W } \in \mathbb { R } ^ { D \times s _ { 1 } }$ , $\mathbf { b } \in \mathbb { R } ^ { s _ { 1 } }$ , $\mathbf { W } _ { \mu } \in \mathbb { R } ^ { s _ { 1 } \times L / 2 }$ , $\mathbf { b } _ { \pmb { \mu } } \in \mathbb { R } ^ { L / 2 }$ , $\mathbf { W _ { \Sigma } } \in$ $\mathbb { R } ^ { s _ { 1 } \times L / 2 }$ , $\mathbf { b } _ { \pm } \in \mathbb { R } ^ { L / 2 }$ . As discussed in Sec. 3.4, multiple hidden layers, or other architectures such as CNNs/RNNs can also be used based on the specific problem.
+
+Unlike other approaches using Gaussian embeddings (Vilnis & McCallum, 2014; He et al., 2015; Dos Santos et al., 2016) we do not explicitly regularize the norm of the means and we do not clip the covariance matrices. Given the self-regularizing nature of the KL divergence this is unnecessary, as was confirmed in our experiments. The parameters are optimized using Adam (Kingma & Ba, 2014) with a fixed learning rate of 0.001 and no learning rate annealing/decay.
+
+Edge cover. Some of the methods such as node2vec (Grover & Leskovec, 2016) are not able to produce an embedding for nodes that have not been seen during training. Therefore, it is important to make sure that during the train-validation-test split of the edge set, every node appears at least once in the train set. Random sampling of the edges does not guarantee this, especially when allocating a low percentage of edges in the train set during the split. To guarantee that every node appears at least once in the train set we have to find an edge cover. An edge cover of a graph is a set of edges such that every node of the graph is incident to at least one edge of the set. The minimum edge cover problem is the problem of finding an edge cover of minimum size. The dashed line in Figures 1(c) and 1(d) indicates exactly the size of the minimum edge cover. This condition had to be satisfied for the competing methods, however, since Graph2Gauss is inductive, it does not require that every node is in the train set.

parse/train/r1ZdKJ-0W/r1ZdKJ-0W_content_list.json

@@ -0,0 +1,1506 @@
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+    {
+        "type": "text",
+        "text": "DEEP GAUSSIAN EMBEDDING OF GRAPHS: UNSUPERVISED INDUCTIVE LEARNING VIA RANKING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
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+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Aleksandar Bojchevski, Stephan Gunnemann ¨ Technical University of Munich, Germany {a.bojchevski,guennemann}@in.tum.de ",
+        "bbox": [
+            184,
+            170,
+            524,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            250,
+            544,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Methods that learn representations of nodes in a graph play a critical role in network analysis since they enable many downstream learning tasks. We propose Graph2Gauss – an approach that can efficiently learn versatile node embeddings on large scale (attributed) graphs that show strong performance on tasks such as link prediction and node classification. Unlike most approaches that represent nodes as point vectors in a low-dimensional continuous space, we embed each node as a Gaussian distribution, allowing us to capture uncertainty about the representation. Furthermore, we propose an unsupervised method that handles inductive learning scenarios and is applicable to different types of graphs: plain/attributed, directed/undirected. By leveraging both the network structure and the associated node attributes, we are able to generalize to unseen nodes without additional training. To learn the embeddings we adopt a personalized ranking formulation w.r.t. the node distances that exploits the natural ordering of the nodes imposed by the network structure. Experiments on real world networks demonstrate the high performance of our approach, outperforming state-of-the-art network embedding methods on several different tasks. Additionally, we demonstrate the benefits of modeling uncertainty – by analyzing it we can estimate neighborhood diversity and detect the intrinsic latent dimensionality of a graph. ",
+        "bbox": [
+            233,
+            281,
+            764,
+            530
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            558,
+            336,
+            574
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Graphs are a natural representation for a wide variety of real-life data, from social and rating networks (Facebook, Amazon), to gene interactions and citation networks (BioGRID, arXiv). Node embeddings are a powerful and increasingly popular approach to analyze such data (Cai et al., 2017). By operating in the embedding space, one can employ proved learning techniques and bypass the difficulty of incorporating the complex node interactions. Tasks such as link prediction, node classification, community detection, and visualization all greatly benefit from these latent node representations. Furthermore, for attributed graphs by leveraging both sources of information (network structure and attributes) one is able to learn more useful representations compared to approaches that only consider the graph (Yang et al., 2015; Pan et al., 2016; Ganguly & Pudi, 2017). ",
+        "bbox": [
+            174,
+            589,
+            825,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "All existing (attributed) graph embedding approaches represent each node by a single point in a low-dimensional continuous vector space. Representing the nodes simply as points, however, has a crucial limitation: we do not have information about the uncertainty of that representation. Yet uncertainty is inherent when describing a node in a complex graph by a single point only. Imagine a node for which the different sources of information are conflicting with each other, e.g. pointing to different communities or even revealing contradicting underlying patterns. Such discrepancy should be reflected in the uncertainty of its embedding. As a solution to this problem, we introduce a novel embedding approach that represents nodes as Gaussian distributions: each node becomes a full distribution rather than a single point. Thereby, we capture uncertainty about its representation. ",
+        "bbox": [
+            174,
+            722,
+            825,
+            847
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To effectively capture the non-i.i.d. nature of the data arising from the complex interactions between the nodes, we further propose a novel unsupervised personalized ranking formulation to learn the embeddings. Intuitively, from the point of view of a single node, we want nodes in its immediate neighborhood to be closest in the embedding space, while nodes multiple hops away should become increasingly more distant. This ordering between the nodes imposed by the network structure w.r.t the distances between their embeddings naturally leads to our ranking formulation. Taking into account this natural ranking from each node’s point of view, we learn more powerful embeddings since we incorporate information about the network structure beyond first and second order proximity. ",
+        "bbox": [
+            174,
+            854,
+            823,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, when node attributes (e.g. text) are available our method is able to leverage them to easily generate embeddings for previously unseen nodes without additional training. In other words, Graph2Gauss is inductive, which is a significant benefit over existing methods that are inherently transductive and do not naturally generalize to unseen nodes. This desirable inductive property comes from the fact that we are learning an encoder that maps the nodes’ attributes to embeddings. ",
+        "bbox": [
+            174,
+            152,
+            825,
+            223
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The main contributions of our approach are summarized as follows: ",
+        "bbox": [
+            174,
+            229,
+            617,
+            244
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "a) We embed nodes as Gaussian distributions allowing us to capture uncertainty.   \nb) Our unsupervised personalized ranking formulation exploits the natural ordering of the nodes capturing the network structure at multiple scales.   \nc) We propose an inductive method that generalizes to unseen nodes and is applicable to different types of graphs: plain/attributed, directed/undirected. ",
+        "bbox": [
+            210,
+            256,
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+            337
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            357,
+            341,
+            372
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The focus of this paper is on unsupervised learning of node embeddings for which many different approaches have been proposed. For a comprehensive recent survey see Cai et al. (2017), Hamilton et al. (2017), or Goyal & Ferrara (2017). Approaches such as DeepWalk and node2vec (Perozzi et al., 2014; Grover & Leskovec, 2016) look at plain graphs and learn an embedding based on random walks by extending or adapting the Skip-Gram (Mikolov et al., 2013) architecture. LINE (Tang et al., 2015b) uses first- and second-order proximity and trains the embedding via negative sampling. SDNE (Wang et al., 2016) similarly has a component that preserves second-order proximity and exploits first-order proximity to refine the representations. GraRep (Cao et al., 2015) is a factorization based method that considers local and global structural information. ",
+        "bbox": [
+            173,
+            388,
+            825,
+            513
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Tri-Party Deep Network Representation (TRIDNR) (Pan et al., 2016) considers node attributes, network structure and potentially node labels. CENE (Sun et al., 2016) similarly to Ganguly & Pudi (2017) treats the attributes as special kinds of nodes and learns embeddings on the augmented network. Text-Associated DeepWalk (TADW) (Yang et al., 2015) performs low-rank matrix factorization considering graph structure and text features. Heterogeneous networks are consider in (Tang et al., 2015a; Chang et al., 2015), while Huang et al. similarly to Pan et al. (2016) considers labels. GraphSAGE (Hamilton et al., 2017) is an inductive method that generates embeddings by sampling and aggregating attributes from a nodes local neighborhood and requires the edges of the new nodes. ",
+        "bbox": [
+            174,
+            520,
+            825,
+            632
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Graph convolutional networks are another family of approaches that adapt conventional CNNs to graph data (Kipf & Welling, 2016a; Defferrard et al., 2016; Henaff et al., 2015; Monti et al., 2016; Niepert et al., 2016; Pham et al., 2017). They utilize the graph Laplacian and the spectral definition of a convolution and boil down to some form of aggregation over neighbors such as averaging. They can be thought of as implicitly learning an embedding, e.g. by taking the output of the last layer before the supervised component. See Monti et al. (2016) for an overview. In contrast to this paper, most of these methods are (semi-)supervised. The graph variational autoencoder (GAE) (Kipf & Welling, 2016b) is a notable exception that learns node embeddings in an unsupervised manner. ",
+        "bbox": [
+            174,
+            638,
+            825,
+            750
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Few approaches consider the idea of learning an embedding that is a distribution. Vilnis & McCallum (2014) are the first to learn Gaussian word embeddings to capture uncertainty. Closest to our work, He et al. (2015) represent knowledge graphs and Dos Santos et al. (2016) study heterogeneous graphs for node classification. Both approaches are not applicable for the context of unsupervised learning of (attributed) graphs that we are interested in. The method in He et al. (2015) learns an embedding for each component of the triplets (head, tail, relation) in the knowledge graph. Note that we cannot naively employ this method by considering a single relation ”has an edge” and a single entity ”node”. Since their approach considers similarity between entities and relations, all nodes would be trivially similar to the single relation. Considering the semi-supervised approach proposed in Dos Santos et al. (2016), we cannot simply ”turn off” the supervised component to adapt their method for unsupervised learning, since given the defined loss we would trivially map all nodes to the same Gaussian. Additionally, both of these approaches do not consider node attributes. ",
+        "bbox": [
+            174,
+            757,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section we introduce our method Graph2Gauss (G2G) and detail how both the attributes and the network structure influence the learning of node representations. The embedding is carried out in two steps: (i) the node attributes are passed through a non-linear transformation via a deep neural network (encoder) and yield the parameters associated with the node’s embedding distribution; (ii) we formulate an unsupervised loss function that incorporates the natural ranking of the nodes as given by the network structure w.r.t. a dissimilarity measure on the embedding distributions. ",
+        "bbox": [
+            173,
+            132,
+            825,
+            217
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Problem definition. Let $G = \\left( \\mathbf { A } , \\mathbf { X } \\right)$ be a directed attributed graph, where $\\mathbf { A } \\in \\mathbb { R } ^ { N \\times N }$ is an adjacency matrix representing the edges between $N$ nodes and $\\breve { \\mathbf { X } } \\in \\mathbb { R } ^ { N \\times D }$ collects the attribute information for each node where $\\mathbf { x } _ { i }$ is a $D$ dimensional attribute vector of the $i ^ { t h }$ node.1 $V$ denotes the set of all nodes. We aim to find a lower-dimensional Gaussian distribution embedding $\\mathbf { h } _ { i } \\ =$ $\\textstyle { \\mathcal { N } } ( \\mu _ { i } , \\Sigma _ { i } )$ , $\\mu _ { i } \\in \\mathbb { R } ^ { L } , \\Sigma _ { i } \\in \\mathbb { R } ^ { L \\times L }$ with $L \\ll N , D$ , such that nodes similar w.r.t. attributes and network structure are also similar in the embedding space given a dissimilarity measure $\\Delta ( \\mathbf { h } _ { i } , \\mathbf { h } _ { j } )$ . In Fig.5(a) for example we show nodes that are embedded as two dimensional Gaussians. ",
+        "bbox": [
+            173,
+            223,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 NETWORK STRUCTURE REPRESENTATION VIA PERSONALIZED RANKING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            338,
+            710,
+            352
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To capture the structural information of the network in the embedding space, we propose a personalized ranking approach. That is, locally per node $i$ we impose a ranking of all remaining nodes w.r.t. their distance to node $i$ in the embedding space. More precisely, in this paper we exploit the $k$ -hop neighborhoods of each node. Given some anchor node $i$ , we define ${ \\cal N } _ { i k } = \\{ \\bar { j } \\in V | i \\neq j , \\operatorname* { m i n } ( s p ( i , j ) , K ) = k \\}$ to be the set of nodes who are exactly $k$ hops away from node $i$ , where $V$ is the set of all nodes, $K$ is a hyper-parameter denoting the maximum distance we are wiling to consider, and $s p ( i , j )$ returns either the length of the shortest path starting at node $i$ and ending in node $j$ or $\\infty$ if node $j$ is not reachable. ",
+        "bbox": [
+            173,
+            363,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, we want all nodes belonging to the 1-hop neighborhood of $i$ to be closer to $i$ w.r.t. their embedding, compared to the all nodes in its 2-hop neighborhood, which in turn are closer than the nodes in its 3-hop neighborhood and so on up to $K$ . Thus, the ranking that we want to ensure from the perspective of node $i$ is ",
+        "bbox": [
+            174,
+            481,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8cc49e1f472d59ee63b72ec7ebe00bfa2ba448ab852a882f1cd90db422ee15d4.jpg",
+        "text": "$$\n\\Delta ( \\mathbf { h } _ { i } , \\mathbf { h } _ { k _ { 1 } } ) < \\Delta ( \\mathbf { h } _ { i } , \\mathbf { h } _ { k _ { 2 } } ) < \\cdots < \\Delta ( \\mathbf { h } _ { i } , \\mathbf { h } _ { k _ { K } } ) \\quad \\forall k _ { 1 } \\in N _ { i 1 } , \\forall k _ { 2 } \\in N _ { i 2 } , \\dots , \\forall k _ { K } \\in N _ { i K }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            200,
+            541,
+            795,
+            559
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "or equivalently, we aim to satisfy the following pairwise constraints ",
+        "bbox": [
+            171,
+            561,
+            624,
+            577
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b8ce721d499e57d59ec71bb1e8db0033b149e94e49f54e694a8f044f6624b430.jpg",
+        "text": "$$\n\\Delta ( { \\bf h } _ { i } , { \\bf h } _ { j } ) < \\Delta ( { \\bf h } _ { i } , { \\bf h } _ { j ^ { \\prime } } ) , \\forall i \\in V , \\forall j \\in N _ { i k } , \\forall j ^ { \\prime } \\in N _ { i k ^ { \\prime } } , \\forall k < k ^ { \\prime }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            276,
+            579,
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+            598
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Going beyond mere first-order and second-order proximity this enables us to capture the network structure at multiple scales incorporating local and global structure. ",
+        "bbox": [
+            174,
+            601,
+            821,
+            628
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Dissimilarity measure. To solve the above ranking task we have to define a suitable dissimilarity measure between the latent representation of two nodes. Since our latent representations are distributions, similarly to Dos Santos et al. (2016) and He et al. (2015) we employ the asymmetric KL divergence. This gives the additional benefit of handling directed graphs in a sound way. More specifically, given the latent Gaussian distribution representation of two nodes $\\mathbf { h } _ { i } , \\mathbf { h } _ { j }$ we define ",
+        "bbox": [
+            174,
+            636,
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+            707
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bffe8d8c7e49641b415066884e57dd4d02230cf97ee80aaa1a5fa19e29a0b0e4.jpg",
+        "text": "$$\n\\Delta ( { \\bf h } _ { i } , { \\bf h } _ { j } ) = D _ { K L } ( N _ { j } | | \\mathcal { N } _ { i } ) = \\frac { 1 } { 2 } \\bigg [ t r ( \\Sigma _ { i } ^ { - 1 } \\Sigma _ { j } ) + ( \\mu _ { i } - \\mu _ { j } ) ^ { T } \\Sigma _ { i } ^ { - 1 } ( \\mu _ { i } - \\mu _ { j } ) - L - l o g \\frac { d e t ( \\Sigma _ { j } ) } { d e t ( \\Sigma _ { i } ) } \\bigg ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here we use the notation $\\mu _ { i } , \\Sigma _ { i }$ to denote the outputs of some functions $\\mu _ { \\boldsymbol { \\theta } } ( \\mathbf { x } _ { i } )$ and $\\Sigma _ { \\theta } ( \\mathbf { x } _ { i } )$ applied to the attributes $\\mathbf { x } _ { i }$ of node $i$ and $t r ( . )$ denotes the trace of a matrix. The asymmetric KL divergence also applies to the case of an undirected graph by simply processing both directions of the edge. We could alternatively use a symmetric dissimilarity measure such as the Jensen-Shannon divergence or the expected likelihood (probability product kernel). ",
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+    {
+        "type": "text",
+        "text": "3.2 DEEP ENCODER ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The functions $\\mu _ { \\boldsymbol { \\theta } } ( \\mathbf { x } _ { i } )$ and $\\Sigma _ { \\theta } ( \\mathbf { x } _ { i } )$ are deep feed-forward non-linear neural networks parametrized by $\\theta$ . It is important to note that these parameters are shared across instances and thus enjoy statistical strength benefits. Additionally, we design $\\mu _ { \\boldsymbol { \\theta } } ( \\mathbf { x } _ { i } )$ and $\\Sigma _ { \\theta } ( \\mathbf { x } _ { i } )$ such that they share parameters as well. More specifically, a deep encoder $f _ { \\theta } ( \\mathbf { x } _ { i } )$ processes the node’s attributes and outputs an intermediate hidden representation, which is then in turn used to output $\\mu _ { i }$ and $\\Sigma _ { i }$ in the final layer of the architecture. We focus on diagonal covariance matrices.2 The mapping from the nodes’ attributes to their embedding via the deep encoder is precisely what enables the inductiveness of Graph2Gauss. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.3 LEARNING VIA ENERGY-BASED LOSS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Since it is intractable to find a solution that satisfies all of the pairwise constraints defined in Sec. 3.1 we turn to an energy based learning approach. The idea is to define an objective function that penalizes ranking errors given the energy of the pairs. More specifically, denoting the KL divergence between two nodes as the respective energy, $E _ { i j } = D _ { K L } ( \\mathcal { N } _ { j } | | \\mathcal { N } _ { i } )$ , we define the following loss to be optimized ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/81f1b504f8e8e103ecb32d4de77db72115e47bd35012cb410a2f5013b26ad1e8.jpg",
+        "text": "$$\n\\mathcal { L } = \\sum _ { i } \\sum _ { k < l } \\sum _ { j _ { k } \\in N _ { i k } } \\sum _ { j _ { l } \\in N _ { i l } } \\left( E _ { i j _ { k } } { ^ { 2 } } + \\exp ^ { - E _ { i j _ { l } } } \\right) = \\sum _ { ( i , j _ { k } , j _ { l } ) \\in \\mathcal { D } _ { t } } \\left( E _ { i j _ { k } } { ^ { 2 } } + \\exp ^ { - E _ { i j _ { l } } } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { D } _ { t } = \\{ ( i , j _ { k } , j _ { l } ) ~ | ~ s p ( i , j _ { k } ) < s p ( i , j _ { l } ) \\}$ is the set of all valid triplets. The $E _ { i j _ { k } }$ terms are positive examples whose energy should be lower compared to the energy of the negative examples $E _ { i j _ { l } }$ . Here, we employed the so called square-exponential loss (LeCun et al., 2006) which unlike other typically used losses (e.g. hinge loss) does not have a fixed margin and pushes the energy of the negative terms to infinity with exponentially decreasing force. In our setting, for a given anchor node $i$ , the energy $E _ { i j }$ should be lowest for nodes $j$ in his 1-hop neighborhood, followed by a higher energy for nodes in his 2-hop neighborhood and so on. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Finally, we can optimize the parameters $\\theta$ of the deep encoder such that the loss $\\mathcal { L }$ is minimized and the pairwise rankings are satisfied. Note again that the parameters are shared across all instances, meaning that we share statistical strength and can learn them more easily in comparison to treating the distribution parameters (e.g. $\\mu _ { i } , \\Sigma _ { i } )$ independently as free variables. The parameters are optimized using Adam (Kingma & Ba, 2014) with a fixed learning rate of 0.001. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Sampling strategy. For large graphs, the complete loss is intractable to compute, confirming the need for a stochastic variant. The naive approach would be to sample triplets from $\\mathcal { D } _ { t }$ uniformly, i.e. replace $\\sum _ { \\left( i , j _ { k } , j _ { l } \\right) \\in { \\mathcal { D } } _ { t } }$ with $\\mathbb { E } _ { ( i , j _ { k } , j _ { l } ) \\sim \\mathcal { D } _ { t } }$ in Eq. 1. However, with the naive sampling we are less likely to sample triplets that involve low-degree nodes since high degree nodes occur in many more pairwise constraints. This in turn means that we update the embedding of low-degree nodes less often which is not desirable. Therefore, we propose an alternative node-anchored sampling strategy. Intuitively, for every node $i$ , we randomly sample one other node from each of its neighborhoods (1-hop, 2-hop, etc.) and then optimize over all the corresponding pairwise constraints $( E _ { i 1 } < E _ { i 2 } , \\ldots , E _ { i 1 } < E _ { i K } , E _ { i 2 } < E _ { i 3 } , \\ldots E _ { i 2 } <$ $\\kappa , E _ { i 2 } < E _ { i 3 } , \\ldots E _ { i 2 } < E _ { i K } , \\ldots , E _ { i K - 1 } < E _ { i K } )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Naively applying the node-anchored sampling strategy and optimizing Eq. 1, however, would lead to biased estimates of the gradient. Theorem 1 shows how to adapt the loss such that it is equal in expectation to the original loss under our new sampling strategy. As a consequence, we have unbiased estimates of the gradient using stochastic optimization of the reformulated loss. ",
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+    {
+        "type": "text",
+        "text": "Theorem 1 For all $i$ , let $( j _ { 1 } , \\dots , j _ { K } )$ be independent uniform random samples from the sets $( N _ { i 1 } , \\dots , N _ { i K } )$ and $| N _ { i * } |$ the cardinality of each set. Then $\\mathcal { L }$ is equal in expectation to ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/b17799a820a2c306a362ebc52b7f163feebe9c4a800af3a1f620da58d9358e69.jpg",
+        "text": "$$\n\\mathcal { L } _ { s } = \\sum _ { i } \\mathbb { E } _ { ( j _ { 1 } , \\dots , j _ { K } ) \\sim ( N _ { i 1 } , \\dots , N _ { i K } ) } \\left[ \\sum _ { k < l } \\vert N _ { i k } \\vert \\cdot \\vert N _ { i l } \\vert \\cdot \\left( E _ { i j _ { k } } ^ { \\phantom { - } } + \\exp ^ { - E _ { i j _ { l } } } \\right) \\right] = \\mathcal { L }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "We provide the proof in the appendix. For cases where the number of nodes $N$ is particularly large we can further subsample mini-batches, by selecting anchor nodes $i$ at random. Furthermore, in our experimental study, we analyze the effect of the sampling strategy on convergence, as well as the quality of the stochastic variant w.r.t. the obtained solution and the reached local optima. ",
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+    {
+        "type": "text",
+        "text": "3.4 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Inductive learning. While during learning we need both the network structure (to evaluate the ranking loss) and the attributes, once the learning concludes, the embedding for a node can be obtained solely based on its attributes. This enables our method to easily handle the issue of obtaining representations for new nodes that were not part of the network during training. To do so we simply pass the attributes of the new node through our learned deep encoder. Most approaches cannot handle this issue at all, with a notable exception being SDNE and GraphSAGE (Wang et al., 2016; Hamilton et al., 2017). However, both approaches require the edges of the new node to get the node’s representation, and cannot handle nodes that have no existing connections. In contrast, our method can handle even such nodes, since after the model is learned we rely only on the attribute information. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Plain graph embedding. Even though attributed graphs are often found in the real-world, sometimes it is desirable to analyze plain graphs. As already discussed, our method easily handles plain graphs, when the attributes are not available, by using one-hot encoding of the nodes instead. As we later show in the experiments we are able to learn useful representations in this scenario, even outperforming some attributed approaches. Naturally, in this case we lose the inductive ability to handle unseen nodes. We compare the one-hot encoding version, termed G2G oh, with our full method G2G that utilizes the attributes, as well as all remaining competitors. ",
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+    {
+        "type": "text",
+        "text": "Encoder architecture. Depending on the type of the node attributes (e.g. images, text) we could in principle use CNNs/RNNs to process them. We could also easily incorporate any of the proposed graph convolutional layers inheriting their benefits. However, we observe that in practice using simple feed-forward architecture with rectifier units is sufficient, while being much faster and easier to train. Better yet, we observed that Graph2Gauss is not sensitive to the choice of hyperparameters such as number and size of hidden layers. We provide more detailed information and sensible defaults in the appendix. ",
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+    {
+        "type": "text",
+        "text": "Complexity. The time complexity for computing the original loss is $O ( N ^ { 3 } )$ where $N$ is the number of nodes. Using our node-anchored sampling strategy, the complexity of the stochastic version is $O ( K ^ { 2 } N )$ where $K$ is the maximum distance considered. Since a small value of $K \\leq 2$ consistently showed good performance, $K ^ { 2 }$ becomes negligible and thus the complexity is $O ( N )$ , meaning linear in the number of nodes. This coupled with the small number of epochs $T$ needed for convergence $T \\leq 2 0 0 0$ for all shown experiments, see e.g. Fig. 3(b)) and an efficient GPU implementation also made our method faster than most competitors in terms of wall-clock time. ",
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+    {
+        "type": "text",
+        "text": "4 EMBEDDING EVALUATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We compare Graph2Gauss with and without considering attributes (G2G, G2G oh) to several competitors namely: TRIDNR and TADW (Pan et al., 2016; Yang et al., 2015) as representatives that consider attributed graphs, GAE (Kipf & Welling, 2016b) as the unsupervised graph convolutional representative, and node2vec (Grover & Leskovec, 2016) as a representative of the random walk based plain graph embeddings. Additionally, we include a strong Logistic Regression baseline that considers only the attributes. As with all other methods we train TRIDNR in a unsupervised manner, however, since it can only process raw text as attributes (rather than e.g. bag-of-words) it is not always applicable. Furthermore, since TADW, and GAE only support undirected graphs we must symmetrize the graph before using them – giving them a substantial advantage, especially in the link prediction task. Moreover, in all experiments if the competing techniques use an $L$ dimensional embedding, G2G’s embedding is actually only half of this dimensionality so that the overall number of ’parameters’ per node (mean vector $^ +$ variance terms of the diagonal $\\Sigma _ { i }$ ) matches $L$ . ",
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+    {
+        "type": "text",
+        "text": "Dataset description. We use several attributed graph datasets. Cora (McCallum et al., 2000) is a well-known citation network labeled based on the paper topic. While most approaches report on a small subset of this dataset we additionally extract from the original data the entire network and name these two datasets CORA $( N = 1 9 7 9 3 , E = 6 5 3 1 1 , D = 8 7 1 0 , K = 7 0 )$ and CORA-ML $( N = 2 9 9 5 , E = 8 4 1 6 , D = 2 8 7 9 , K = 7 )$ respectively. CITESEER $( N = 4 2 3 0 , E = 5 3 5 8 , D =$ $2 7 0 1 , K = 6 )$ (Giles et al., 1998), DBLP (Pan et al., 2016) $( N = 1 7 7 1 6 , E = 1 0 5 7 3 4 , D =$ 1639, $K = 4$ ) and PUBMBED $( N = 1 8 2 3 0 , E = 7 9 6 1 2 , D = 5 0 0 , K = 3 )$ (Sen et al., 2008) are other commonly used citation datasets. We provide all datasets, the source code of G2G, and further supplementary material (https://www.kdd.in.tum. $\\mathrm { d e } / \\mathrm { g } 2 \\mathrm { g } )$ ). ",
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+    {
+        "type": "text",
+        "text": "4.1 LINK PREDICTION",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Setup. Link prediction is a commonly used task to demonstrate the meaningfulness of the embeddings. To evaluate the performance we hide a set of edges/non-edges from the original graph and train on the resulting graph. Similarly to Kipf & Welling (2016b) and Wang et al. (2016) we create a validation/test set that contains $5 \\% / 1 0 \\%$ randomly selected edges respectively and equal number of randomly selected non-edges.We used the validation set for hyper-parameter tuning and early stopping and the test set only to report the performance. As by convention we report the area under the ROC curve (AUC) and the average precision (AP) scores for each method. To rank the candidate edges we use the negative energy $- E _ { i j }$ for Graph2Gauss, and the exact same approach as in the respective original methods (e.g. dot product of the embeddings). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Performance on real-world datasets. Table 1 shows the performance on the link prediction task for different datasets and embedding size $L = 1 2 8$ . As we can see our method significantly outperforms the competitors across all datasets which is a strong sign that the learned embeddings are useful. Furthermore, even the constrained version of our method G2G oh that does not consider attributes at all outperforms the competitors on some datasets. While GAE achieves comparable performance on some of the datasets their approach doesn’t scale to large graphs. In fact, for graphs beyond $1 5 K$ nodes we had to revert to slow training on the CPU since the data did not fit on the GPU memory (12GB). The simple Logistic Regression baseline showed surprisingly strong performance, even outperforming some of the more complicated methods. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/ec9e50512179cb2af7552da9f40523c3ee1fd7d1a8beb338445fa3463cc08df7.jpg",
+        "table_caption": [
+            "Table 1: Link prediction performance for real-world datasets with $L = 1 2 8$ "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Cora-ML</td><td colspan=\"2\">Cora</td><td colspan=\"2\">Citeseer</td><td colspan=\"2\">DBLP</td><td colspan=\"2\">Pubmed</td><td colspan=\"2\">Cora-ML Easy</td></tr><tr><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td></tr><tr><td>Logistic Regression</td><td>90.01</td><td>89.75</td><td>86.58</td><td>86.51</td><td>81.70</td><td>79.10</td><td>82.04</td><td>81.91</td><td>90.50</td><td>90.99</td><td>90.28</td><td>90.99</td></tr><tr><td>node2vec(Grover &amp; Leskovec,2016)</td><td>76.80</td><td>75.26</td><td>79.95</td><td>78.98</td><td>83.04</td><td>83.74</td><td>95.42</td><td>95.33</td><td>95.42</td><td>95.33</td><td>93.47</td><td>93.53</td></tr><tr><td>TADW(Yang et al., 2015)</td><td>81.26</td><td>81.34</td><td>76.56</td><td>78.06</td><td>70.14</td><td>72.93</td><td>65.67</td><td>59.85</td><td>62.72</td><td>68.02</td><td>83.53</td><td>82.47</td></tr><tr><td>TRIDNR(Pan et al., 2016)</td><td>84.51</td><td>85.69</td><td>81.61</td><td>81.08</td><td>87.23</td><td>88.87</td><td>92.01</td><td>91.62</td><td>NTA</td><td>NTA</td><td>85.59</td><td>86.16</td></tr><tr><td>GAE(Kipf &amp; Welling,2016b)</td><td>96.65</td><td>96.67</td><td>97.91</td><td>98.07</td><td>92.31</td><td>93.88</td><td>95.78</td><td>96.67</td><td>96.07</td><td>96.12</td><td>95.97</td><td>95.17</td></tr><tr><td>G2G_oh</td><td>96.95</td><td>97.54</td><td>98.41</td><td>98.63</td><td>95.89</td><td>95.78</td><td>98.29</td><td>98.46</td><td>96.75</td><td>96.47</td><td>96.98</td><td>96.42</td></tr><tr><td>G2G</td><td>98.01</td><td>98.03</td><td>98.81</td><td>98.78</td><td>96.09</td><td>96.16</td><td>98.65</td><td>98.78</td><td>97.42</td><td>97.85</td><td>98.03</td><td>98.12</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "We also include the performance on the so called ”Cora-ML Easy” dataset, obtained from the CoraML dataset by making it undirected and selecting the nodes in the largest connected component. We see that while node2vec struggles on the original real-world data, it significantly improves in this ”easy” setting. On the contrary, Graph2Gauss handles both settings effortlessly. This demonstrates that Graph2Gauss can be readily applied in realistic scenarios on potentially messy real-world data. ",
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+    {
+        "type": "text",
+        "text": "Sensitivity analysis. In Figs.1(a) and 1(b) we show the performance w.r.t. the dimensionality of the embedding, averaged over 10 trials. G2G is able to learn useful embeddings with strong performance even for relatively small embedding sizes. Even for the case $L = 2$ , where we embed the points as one dimensional Gaussian distributions $( L = 1 + 1$ for the mean and the sigma of the Gaussian), G2G still outperforms all of the competitors irrespective of their much higher embedding sizes. ",
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+    {
+        "type": "image",
+        "img_path": "images/8a932d69c197c1a3e0f8320a4a38a85a80651549fda7644ac92413355fa92960.jpg",
+        "image_caption": [
+            "Figure 1: Link prediction performance for different embedding sizes and percentages of training edges on Cora-ML. G2G outperforms the competitors even for small sizes and percentage of edges. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Finally, we evaluate the performance w.r.t. the percentage of training edges varying from $1 5 \\%$ to $8 5 \\%$ , averaged over 10 trials. We can see in Figs.1(c) and 1(d) Graph2Gauss strongly outperforms the competitors, especially for small number of training edges. The dashed line indicates the percentage above which we can guarantee to have every node appear at least once in the training set.3 The performance below that line is then indicative of the performance in the inductive setting. Since, the structure only methods are unable to compute meaningful embeddings for unseen nodes we cannot report their performance below the dashed line. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4.2 NODE CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Setup. Node classification is another task commonly used to evaluate the strength of the learned embeddings – after they have been trained in an unsupervised manner. We evaluate the node classification performance for three datasets (Cora-ML, Citeseer and DBLP) that have ground-truth classes. First, we train the embeddings on the entire training data in an unsupervised manner (excluding the class labels). Then, following Perozzi et al. (2014) we use varying percentage of randomly selected nodes and their learned embeddings along with their labels as training data for a logistic regression, while evaluating the performance on the rest of the nodes. We also optimize the regularization strength for each method/dataset via cross-validation. We show results averaged over 10 trials. ",
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+    {
+        "type": "image",
+        "img_path": "images/9dfe41f5f0f942d2393078c92264f4f3c1bf45a3aee7e8643ba84bccb167a5d8.jpg",
+        "image_caption": [
+            "Figure 2: Classification performance comparison - both G2G and G2G oh perform strongly. "
+        ],
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+    {
+        "type": "text",
+        "text": "Performance on real-world datasets. Figs. 2 compares the methods w.r.t. the classification performance for different percentage of labeled nodes. We can see that our method clearly outperforms the competitors. Again, the constrained version of our method that does not consider attributes is able to outperform some of the competing approaches. Additionally, we can conclude that in general our method shows stable performance regardless of the percentage of labeled nodes. This is a highly desirable property since it shows that should we need to perform classification it is sufficient to train only on a small percentage of labeled nodes. ",
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+    {
+        "type": "text",
+        "text": "4.3 SAMPLING STRATEGY ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Figure 3(a) shows the validation set ROC score for the link prediction task w.r.t. the number of triplets $( i , j _ { k } , j _ { l } )$ seen. We can see that both sampling strategies are able to reach the same performance as the full loss in significantly fewer $( < 4 . 2 \\%$ ) number of pairs seen (note the log scale). It also shows that the naive random sampling converges slower than the node-anchored sampling strategy. Figures 3(b) gives us some insight as to why – our node-anchored sampling strategy achieves significantly lower loss. Finally, Fig. 3(c) shows that our node-anchored sampling strategy has lower variance of the gradient updates, which is another contributor to faster convergence. ",
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+        "image_caption": [
+            "Figure 3: Our sampling strategy converges significantly faster than the full loss, while maintaining good performance. It also achieves better loss and has lower variance compared to naive sampling. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Learning an embedding that is a distribution rather than a point-vector allows us to capture uncertainty about the representation. We perform several experiments to evaluate the benefit of modeling uncertainty. Figure 4(a) shows that the learned uncertainty is correlated with neighborhood diversity, where for a node $i$ we define diversity as the number of distinct classes among the nodes in its $p$ -hop neighborhood $\\textstyle ( \\bigcup _ { 1 \\leq k \\leq p } N _ { i k } )$ . Since the uncertainty for a node $i$ is an $L$ -dimensional vector (diagonal covariance) we show the average across the dimensions. In line with our intuition, nodes with less diverse neighborhood have significantly lower variance compare to more diverse nodes whose immediate neighbors belong to many different classes, thus making their embedding more uncertain. The figure shows the result on the Cora dataset for $p = 3$ hop neighborhood. Similar results hold for the other datasets. This result is particularly impressive given the fact that we learn our embedding in a completely unsupervised manner, yet the uncertainty was able to capture the diversity w.r.t. the class labels of the neighbors of a node, which were never seen during training. ",
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+        "img_path": "images/5c593abf4a0b86b0814e86b69cc0ee34f42469e0bce9d3a752b172aed8c32348.jpg",
+        "image_caption": [
+            "Figure 4: The benefit of modeling the uncertainty of the nodes. "
+        ],
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+    {
+        "type": "text",
+        "text": "Figure 4(b) shows that using the learned uncertainty we are able to detect the intrinsic latent dimensionality of the graph. Each line represents the average variance (over all nodes) for a given dimension $l$ for each epoch. We can see that as the training progresses past the stopping criterion (link prediction performance on validation set) and we start to overfit, some dimensions exhibit a relatively stable average variance, while for others the variance increases with each epoch. By creating a simply rule that monitors the average change of the variance over time we were able to automatically detect these relevant latent dimensions (colored in red). This result holds for multiple datasets and is shown here for Cora-ML. Interestingly, the number of detected latent dimensions (6) is close to the number of ground-truth communities (7). ",
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+    {
+        "type": "text",
+        "text": "The next obvious question is then how does the performance change if we remove these highly uncertain dimensions whose variance keeps increasing with training. Figure 4(c) answers exactly that. By removing progressively more and more dimensions, starting with the most uncertain first we see imperceptibly small change in performance. Only once we start removing the true latent dimension we see a noticeable degradation in performance. The dashed lines show the performance if we re-train the model, setting $L = 6$ , equal to the detected number of latent dimensions. ",
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+    {
+        "type": "text",
+        "text": "As a last study of uncertainty, in a use case analysis, the nodes with high uncertainty reveal additional interesting patterns. For example in the Cora dataset, one of the highly uncertain nodes was the paper ”The use of word shape information for cursive script recognition” by R.J. Whitrow – surprisingly, all citations (edges) of that paper (as extracted from the dataset) were towards other papers by the same author. ",
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+    {
+        "type": "text",
+        "text": "4.5 INDUCTIVE LEARNING: GENERALIZATION TO UNSEEN NODES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As discussed in Sec. 3.4 G2G is able to learn embeddings even for nodes that were not part of the networks structure during training time. Thus, it not only supports transductive but also inductive learning. To evaluate how our approach generalizes to unseen nodes we perform the following experiment: (i) first we completely hide $1 0 \\% / 2 5 \\%$ of nodes from the network at random; (ii) we proceed to learn the node embeddings for the rest of the nodes; (iii) after learning is complete we pass the (new) unseen test nodes through our deep encoder to obtain their embedding; (iv) we evaluate by calculating the link prediction performance (AUC and AP scores) using all their edges and same number of non-edges. ",
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+    {
+        "type": "table",
+        "img_path": "images/40b95400e458220b14d9a0f6036b43b08647128c288fd4c66cc325311204d8d9.jpg",
+        "table_caption": [
+            "Table 2: Inductive link prediction performance. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method (% hidden)</td><td colspan=\"2\">Cora-ML</td><td colspan=\"2\">Cora</td><td colspan=\"2\">Citeseer</td><td colspan=\"2\">DBLP</td><td colspan=\"2\">Pubmed</td></tr><tr><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td></tr><tr><td>Log.Reg.10%</td><td>75.95</td><td>78.62</td><td>78.53</td><td>78.70</td><td>73.09</td><td>72.54</td><td>67.55</td><td>69.55</td><td>86.83</td><td>87.34</td></tr><tr><td>G2G 10%</td><td>90.93</td><td>89.37</td><td>94.18</td><td>93.40</td><td>88.58</td><td>88.31</td><td>85.06</td><td>83.75</td><td>92.22</td><td>90.45</td></tr><tr><td>G2G 25%</td><td>87.83</td><td>86.31</td><td>92.96</td><td>92.31</td><td>87.30</td><td>86.61</td><td>83.09</td><td>81.49</td><td>90.20</td><td>88.28</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "As the results in Table 2 clearly show, since we are utilizing the rich attribute information, we are able to achieve strong performance for unseen nodes. This is true even when a quarter of the nodes are missing. This makes our method applicable in the context of large graphs where training on the entire network is not feasible. Note that SDNE (Wang et al., 2016) and GraphSAGE (Hamilton et al., 2017) cannot be applied in this scenario, since they also require the edges for the unseen nodes to produce an embedding. Graph2Gauss is the only inductive method that can obtain embeddings for a node based only on the node attributes. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.6 NETWORK VISUALIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "One key application of node embedding approaches is creating meaningful visualizations of a network in 2D/3D that support tasks such as data exploration and understanding. Following Tang et al. (2015b) and Pan et al. (2016) we first learn a lower-dimensional $L = 1 2 8$ embedding for each node and then map those representations in 2D with TSNE (Maaten & Hinton, 2008). Additionally, since our method is able to learn useful representations even in low dimensions we embed the nodes as 2D Gaussians and visualize the resulting embedding. This has the added benefit of visualizing the nodes’ uncertainty as well. Fig. 5 shows the visualization for the Cora-ML dataset. We see that Graph2Gauss learns an embedding in which the different classes are clearly separated. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/e1a164d9a7672be7d5d2534731493901583feced1417dc1496cf52360ff653f2.jpg",
+        "image_caption": [
+            "Figure 5: 2D visualization of the embeddings on the Cora-ML dataset. Color indicates the class label not used during training. Best viewed on screen. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We proposed Graph2Gauss – the first unsupervised approach that represents nodes in attributed graphs as Gaussian distributions and is therefore able to capture uncertainty. Analyzing the uncertainty reveals the latent dimensionality of a graph and gives insight into the neighborhood diversity of a node. Since we exploit the attribute information of the nodes we can effortlessly generalize to unseen nodes, enabling inductive reasoning. Graph2Gauss leverages the natural ordering of the nodes w.r.t. their neighborhoods via a personalized ranking formulation. The strength of the learned embeddings has been demonstrated on several tasks – specifically achieving high link prediction performance even in the case of low dimensional embeddings. As future work we aim to study personalized rankings beyond the ones imposed by the shortest path distance. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
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+        "type": "text",
+        "text": "This research was supported by the German Research Foundation, Emmy Noether grant GU 1409/2- 1, and by the Technical University of Munich - Institute for Advanced Study, funded by the German Excellence Initiative and the European Union Seventh Framework Programme under grant agreement no 291763, co-funded by the European Union. ",
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+    {
+        "type": "text",
+        "text": "Cheng Yang, Zhiyuan Liu, Deli Zhao, Maosong Sun, and Edward Y Chang. Network representation learning with rich text information. In IJCAI, pp. 2111–2117, 2015. ",
+        "bbox": [
+            173,
+            887,
+            825,
+            917
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            264,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A PROOF OF THEOREM 1 ",
+        "bbox": [
+            176,
+            138,
+            357,
+            154
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To prove Theorem 1 we start with the loss $\\mathcal { L } _ { s }$ (Eq. 2), and show that by applying the expectation operator we will obtain the original loss $\\mathcal { L }$ (Eq. 1). From there it trivially follows that taking the gradient with respect to $\\mathcal { L } _ { s }$ for a set of samples gives us an unbiased estimate of the gradient of $\\mathcal { L }$ . ",
+        "bbox": [
+            174,
+            167,
+            825,
+            212
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "First we notice that both $\\mathcal { L }$ and $\\mathcal { L } _ { s }$ are summing over $i$ , thus it is sufficient to show that the losses are equal in expectation for a single node $i$ . Denoting with $\\mathcal { L } _ { s } ^ { ( i ) }$ the loss for a single node $i$ and with $E _ { i , k , l } = E _ { i j _ { k } } { } ^ { 2 } + \\exp ^ { - E _ { i j _ { l } } }$ for notational convenience we have: ",
+        "bbox": [
+            174,
+            217,
+            825,
+            265
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c09295d6d95af7847045d4999e35f0a95cac1388e46db5c32410f9b11fc67b6e.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\| \\nabla _ { x } ^ { \\bot } \\varphi \\| _ { L ^ { 2 } ( \\mathcal H _ { 1 } ) } ^ { 2 } = \\frac { 1 } { \\lambda _ { x } ^ { 2 } } \\sum _ { t _ { i } ^ { \\prime } \\in \\mathcal H _ { 1 } } ^ { 1 } \\lambda _ { t _ { i } ^ { \\prime } } \\sum _ { \\lambda _ { x } ^ { \\prime } } ^ { \\lambda _ { \\prime } } \\lambda _ { t _ { i } ^ { \\prime } } \\Big | \\nabla _ { x } ^ { \\bot } \\varphi \\Big | _ { L ^ { 2 } ( \\mathcal H _ { 1 } ) } ^ { 2 } } \\\\ & { \\quad - \\frac { 1 } { \\lambda _ { x } ^ { 2 } } \\sum _ { t _ { i } ^ { \\prime } \\in \\mathcal H _ { 1 } } \\lambda _ { t _ { i } ^ { \\prime } } \\sum _ { \\lambda _ { x } ^ { \\prime } } \\lambda _ { t _ { i } ^ { \\prime } } \\Big | \\nabla _ { x } \\varphi \\Big | _ { L ^ { 2 } ( \\mathcal H _ { 1 } ) } ^ { 2 } \\cdot \\nabla _ { x } \\varphi \\Big | _ { L ^ { 2 } ( \\mathcal H _ { 1 } ) } ^ { 2 } } \\\\ & { \\quad + \\frac { 1 } { \\lambda _ { x } ^ { 2 } } \\sum _ { t _ { i } ^ { \\prime } \\in \\mathcal H _ { 1 } } \\lambda _ { t _ { i } ^ { \\prime } } \\sum _ { \\lambda _ { x } ^ { \\prime } } \\lambda _ { t _ { i } ^ { \\prime } } \\Big | \\nabla _ { x } \\varphi \\Big | _ { L ^ { 2 } ( \\mathcal H _ { 1 } ) } ^ { 2 } \\cdot \\nabla _ { x } \\varphi \\Big | _ { L ^ { 2 } ( \\mathcal H _ { 1 } ) } ^ { 2 } \\cdot \\nabla _ { x } \\varphi \\Big | _ { L ^ { 2 } ( \\mathcal H _ { 1 } ) } ^ { 2 } } \\\\ & { \\quad - \\frac { 1 } { \\lambda _ { x } ^ { 2 } } \\sum _ { t _ { i } ^ { \\prime } \\in \\mathcal H _ { 1 } } \\lambda _ { t _ { i } ^ { \\prime } } \\Big | \\nabla _ { x } \\varphi \\Big | _ { L ^ { 2 } ( \\mathcal H _ { 1 } ) } ^ { 2 } \\cdot \\nabla _ { x } \\varphi \\Big | _ { L ^ { 2 } ( \\mathcal H _ { 1 } ) } ^ { 2 } } \\\\ &  \\quad - \\frac { 1 } { \\lambda _ { x } ^ { 2 } } \\sum _ { t _ { i } ^ { \\prime } \\in \\mathcal H _ { 1 } } \\lambda _ { t _ { i } ^ { \\prime } } \\Big | \\nabla _ { x } \\varphi \\Big | _  L ^  \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            256,
+            291,
+            754,
+            659
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In step (1) we have expanded the sum over $k \\ < \\ l$ in independent terms. In step (2) we have marginalized the expectation over the variables that do not appear in the expression, e.g. for the term $\\mathbb { E } _ { ( j _ { 1 } , . . . , j _ { K } ) \\sim ( N _ { i 1 } , . . . , N _ { i K } ) } | N _ { i 1 } | \\cdot | N _ { i 2 } | \\cdot E _ { i 1 2 }$ we can marginalize over $j _ { p }$ where $p \\neq 1$ and $p \\neq 2$ since the term doesn’t depend on them. In step (3) we have expanded the expectation term. In step (4) we have substituted $p ( j _ { p } )$ with $\\frac { 1 } { | N _ { i j p } | }$ since we are sampling uniformly at random. ",
+        "bbox": [
+            173,
+            683,
+            825,
+            760
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Since $\\mathcal { L } _ { s } ^ { ( i ) }$ is equal to $\\mathcal { L } ^ { ( i ) }$ in expectation it follows that $\\nabla { \\mathcal { L } } _ { s }$ based on a set of samples is an unbiased estimate of $\\nabla \\mathcal { L }$ . ",
+        "bbox": [
+            174,
+            767,
+            823,
+            797
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            824,
+            390,
+            838
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Architecture and hyperparameters. We observed that Graph2Gauss is not sensitive to the choice of hyperparameters such as number and size of hidden layers. Better yet, as shown in Sec. 4.4, Graphs2Gauss is also not sensitive to the size of the embedding $L$ . Thus, for a new graph, one can simply pick a relatively large embedding size and if required prune it later similarly to the analysis performed in Fig. 4(c). ",
+        "bbox": [
+            174,
+            853,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "As a sensible default we recommend an encoder with a single hidden layer of size $s _ { 1 } = 5 1 2$ . More specifically, to obtain the embeddings for a node $i$ we have ",
+        "bbox": [
+            173,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f8a44527b2bd061fef4521219d8925761cd869356e50c9cc1fee9c745ca84864.jpg",
+        "text": "$$\n\\mathbf { h } _ { i } = \\mathrm { r e l u } ( \\mathbf { X } _ { i } \\mathbf { W } + \\mathbf { b } ) \\qquad \\mu _ { i } = \\mathbf { h } _ { \\mathbf { i } } \\mathbf { W } _ { \\mu } + \\mathbf { b } _ { \\mu } \\qquad \\sigma _ { i } = \\mathrm { e l u } ( \\mathbf { h } _ { i } \\mathbf { W } _ { \\Sigma } + \\mathbf { b } _ { \\Sigma } ) + 1\n$$",
+        "text_format": "latex",
+        "bbox": [
+            238,
+            138,
+            759,
+            156
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { x } _ { i }$ are node attributes, relu and elu are the rectified linear unit and exponential linear unit respectively. In practice, we found that the softplus works equally well as the elu for making sure that $\\sigma _ { i }$ are positive and in turn $\\Sigma _ { i }$ is positive definite. We used Xavier initialization (Glorot & Bengio, 2010) for the weight matrices $\\mathbf { W } \\in \\mathbb { R } ^ { D \\times s _ { 1 } }$ , $\\mathbf { b } \\in \\mathbb { R } ^ { s _ { 1 } }$ , $\\mathbf { W } _ { \\mu } \\in \\mathbb { R } ^ { s _ { 1 } \\times L / 2 }$ , $\\mathbf { b } _ { \\pmb { \\mu } } \\in \\mathbb { R } ^ { L / 2 }$ , $\\mathbf { W _ { \\Sigma } } \\in$ $\\mathbb { R } ^ { s _ { 1 } \\times L / 2 }$ , $\\mathbf { b } _ { \\pm } \\in \\mathbb { R } ^ { L / 2 }$ . As discussed in Sec. 3.4, multiple hidden layers, or other architectures such as CNNs/RNNs can also be used based on the specific problem. ",
+        "bbox": [
+            173,
+            170,
+            825,
+            258
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Unlike other approaches using Gaussian embeddings (Vilnis & McCallum, 2014; He et al., 2015; Dos Santos et al., 2016) we do not explicitly regularize the norm of the means and we do not clip the covariance matrices. Given the self-regularizing nature of the KL divergence this is unnecessary, as was confirmed in our experiments. The parameters are optimized using Adam (Kingma & Ba, 2014) with a fixed learning rate of 0.001 and no learning rate annealing/decay. ",
+        "bbox": [
+            174,
+            263,
+            825,
+            334
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Edge cover. Some of the methods such as node2vec (Grover & Leskovec, 2016) are not able to produce an embedding for nodes that have not been seen during training. Therefore, it is important to make sure that during the train-validation-test split of the edge set, every node appears at least once in the train set. Random sampling of the edges does not guarantee this, especially when allocating a low percentage of edges in the train set during the split. To guarantee that every node appears at least once in the train set we have to find an edge cover. An edge cover of a graph is a set of edges such that every node of the graph is incident to at least one edge of the set. The minimum edge cover problem is the problem of finding an edge cover of minimum size. The dashed line in Figures 1(c) and 1(d) indicates exactly the size of the minimum edge cover. This condition had to be satisfied for the competing methods, however, since Graph2Gauss is inductive, it does not require that every node is in the train set. ",
+        "bbox": [
+            174,
+            340,
+            825,
+            494
+        ],
+        "page_idx": 12
+    }
+]

parse/train/rkxmPgrKwB/rkxmPgrKwB.md

@@ -0,0 +1,351 @@
+# WEIGHT-SPACE SYMMETRY IN NEURAL NETWORK LOSS LANDSCAPES REVISITED
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Neural network training depends on the structure of the underlying loss landscape, i.e. local minima, saddle points, flat plateaus, and loss barriers. In relation to the structure of the landscape, we study the permutation symmetry of neurons in each layer of a deep neural network, which gives rise not only to multiple equivalent global minima of the loss function but also to critical points in between partner minima. In a network of $d - 1$ hidden layers with $n _ { k }$ neurons in layers $k = 1 , \dots , d$ , we construct continuous paths between equivalent global minima that lead through a ‘permutation point’ where the input and output weight vectors of two neurons in the same hidden layer $k$ collide and interchange. We show that such permutation points are critical points which lie inside high-dimensional subspaces of equal loss, contributing to the global flatness of the landscape. We also find that a permutation point for the exchange of neurons $i$ and $j$ transits into a flat high-dimensional plateau that enables all $n _ { k } !$ permutations of neurons in a given layer $k$ at the same loss value. Moreover, we introduce higher-order permutation points by exploiting the hierarchical structure in the loss landscapes of neural networks, and find that the number of $K$ -th order permutation points is much larger than the (already huge) number of equivalent global minima – at least by a polynomial factor of order $K$ . In two tasks, we demonstrate numerically with our path finding method that continuous paths between partner minima exist: first, in a toy network with a single hidden layer on a function approximation task and, second, in a multilayer network on the MNIST task. Our geometric approach yields a lower bound on the number of critical points generated by weight-space symmetries and provides a simple intuitive link between previous theoretical results and numerical observations.
+
+# 1 INTRODUCTION
+
+The structure of the loss landscape plays an important role in the optimization of neural network parameters. A large number of numerical (Dauphin et al., 2014; Goodfellow et al., 2014; Li et al., 2018; Sagun et al., 2014; 2016; Ballard et al., 2017; Garipov et al., 2018; Draxler et al., 2018; Sagun et al., 2017; Baity-Jesi et al., 2018) and theoretical (Choromanska et al., 2015; Rasmussen, 2003; Freeman and Bruna, 2016; Soudry and Carmon, 2016; Nguyen and Hein, 2017) studies have explored the properties of the loss landscape. In particular, in a multilayer network of $d - 1$ hidden layers with $n$ neurons each, there are $( n ! ) ^ { { d - 1 } }$ equivalent configurations corresponding to the permutation of neuron indices in each layer of the network (Goodfellow et al., 2016; Bishop, 1995). The permutation symmetries give rise to a loss landscape where any given global minimum in the weight space must have $( n ! ) ^ { d - 1 } - 1$ completely equivalent partner minima. This property of neural network landscapes is called weight-space symmetry.
+
+Several (Saad and Solla, 1995; Amari et al., 2006; Wei et al., 2008) works explored the implications of weight-space symmetry for training dynamics in two-layer networks and found that training dynamics slow down near the singular regions caused by weight-space symmetry. Dauphin et al. (2014); Orhan and Pitkow (2017) argue that optimization paths may get close to the singular regions induced by weight-space symmetry and this, in turn, slows down training for deep neural networks. Exploiting weight-space symmetries, we give insights into and partial explanations of three observations on neural network landscapes.
+
+Observation 1. Training dynamics are slow near singular regions caused by weight-space symmetry and stochastic gradient descent might travel near these regions throughout training (Saad and Solla, 1995; Wei et al., 2008; Amari et al., 2006; Dauphin et al., 2014; Orhan and Pitkow, 2017).
+
+Observation 2. The Hessian of the loss function has numerous almost-zero eigenvalues throughout training, thus the landscape is flat in many directions (Sagun et al., 2017; Papyan, 2018; Ghorbani et al., 2019).
+
+Related to observation 1 and 2, we prove the existence of numerous connected high-dimensional plateaus extending across the landscape due to weight-space symmetries.
+
+Observation 3. The number of saddles can grow exponentially in neural network landscapes (Auer et al., 1996; Dauphin et al., 2014; Choromanska et al., 2015).
+
+Related to observation 3, we prove that there are at least polynomially many more saddles than the global minima due to weight-space symmetries in neural networks, without any further assumptions.
+
+In addition, we propose a novel low-loss path finding algorithm to find barriers between partner minima. We start from the known permutation symmetries and consider continuous low-loss paths that connect two equivalent global minima by merging the weight vectors of two neurons in a specific way. At a so-called permutation point, where the distance between the input and output weight vectors of the two neurons vanishes, the indices of the two neurons can be interchanged at no extra cost. After the change, the system returns on the ‘mirrored’ path back to the original configuration except for the permutation of one pair of indices.
+
+Surprisingly, we find that we can permute all neuron indices in the same layer at the same cost as the loss at a permutation point reached by moving along the path that merges a single pair of neurons. These constant-loss permutations are possible because each permutation point lies in a high-dimensional plateau of critical points. Our theory can be extended to higher-order saddles and provides explicit lower bounds for the number of first- and higher-order permutation points. Numerically, we confirm the existence of first-order permutation saddles.
+
+In particular, the specific contributions of our work are:
+
+• A simple low-loss path-finding algorithm linking partner global minima via a permutation point, implemented by minimization under a single scalar constraint (distance of weight vectors).   
+The theoretical characterization of permutation points, for example that these are critical points and several permutation points are connected via paths at equal loss.   
+• A lower bound for the number of first- and higher-order permutation points and their corresponding plateaus.   
+• Numerical demonstrations of the path finding method in multilayer neural networks trained on MNIST.
+
+# 1.1 RELATED WORK
+
+Structure of the landscape. For linear networks, it was shown that all the critical points – except for the global minimum – are saddles in the case of two-layer (Baldi and Hornik, 1989) or multilayer networks (Freeman and Bruna, 2016; Kawaguchi, 2016; Lu and Kawaguchi, 2017). Interestingly, deep linear networks are reported to exhibit sharp transitions at the edges of extended plateaus (Saxe et al., 2013), similar to the plateaus observed in deep nonlinear networks (Goodfellow et al., 2014). For nonlinear multilayer networks, Choromanska et al. (2015) argue that all local minima lie below a certain loss value by drawing connections to the spherical spin-glass model. Improving upon this result, Soudry and Carmon (2016); Nguyen and Hein (2017) prove that almost all local minima are global minima for multilayer networks under mild over-parametrization assumptions.
+
+Bottom of the landscape. Another line of research studies the bottom of the landscape containing global minima and low-loss barriers between them. Freeman and Bruna (2016) prove the existence of low-loss paths connecting global minima for wide two-layer networks by upper-bounding the loss along the path with a parameter that depends on the number of parameters and data smoothness. Draxler et al. (2018) use Nudged Elastic Band method introduced in Jónsson et al. (1998) to connect independent minima and numerically find that the barrier vanishes consistently for increasing width and depth in DenseNet, ConvNet and ResNet architectures trained on CIFAR datasets. In a simultaneous work, Garipov et al. (2018) confirm that there is no significant barrier by connecting independent minima with polygonal chains.
+
+Training dynamics in the landscape. For general loss functions, Lee et al. (2016) show that gradient descent with sufficiently small step-size converges to local minima if all the saddles have at least one negative eigenvalue. For overparametrized neural networks, gradient descent converges to global minima without moving far from initialization (Jacot et al., 2018; Du et al., 2018a;b), thus suggesting convex-like behavior around random initialization. For the finite size networks, how training dynamics converge to a minima and in particular how fast they converge remain an open question. For soft-committee machines, it turns out that the initial learning dynamics are slowed down by correlation of hidden neurons (Saad and Solla, 1995; Engel and Van den Broeck, 2001; Inoue et al., 2003). Amari et al. (2006); Wei et al. (2008) show that training dynamics slow down near singular regions due to weight-space symmetry. Dauphin et al. (2014) empirically argue that the large number of saddle points in the landscape makes training slow. Orhan and Pitkow (2017) numerically find that stochastic gradient descent may slow down near plateaus due to weight-space symmetry for deep (30 layers) feedforward networks trained on CIFAR100.
+
+In this paper we show that there is an impressively large number of permutation points. Each permutation point is a critical point (either a local minimum or a saddle) with a large number of flat directions, potentially linked to the empirically observed plateaus. In contrast to an earlier study by Fukumizu and Amari (2000) with a scalar output for two-layered networks where a line of critical points around the permutation point was reported, we study a deep network with $d - 1$ hidden layers and find multi-dimensional equal-loss plateaus. Moreover, we give a novel lower bound on the number of permutation points and construct sample paths between global minima using an algorithm that is different from previously used methods (Garipov et al., 2018; Draxler et al., 2018), since it exploits the symmetries at the permutation point.
+
+# 1.2 PRELIMINARIES
+
+We study multilayer neural networks $f ( \pmb { x } ; \pmb { \theta } )$ with input $\pmb { x } \in \mathbb { R } ^ { n _ { 0 } }$ , $d$ layers of $n _ { 1 } , \ldots , n _ { d }$ neurons per layer, parameters $\pmb \theta = \{ \pmb W ^ { ( k ) } \in \mathbb { R } ^ { n _ { k } \times n _ { k - 1 } }$ and $b ^ { ( k ) } \in \mathbb { R } ^ { n _ { k } } : k \in \{ 1 , . . . , d \} \} \in \Theta$ and $n _ { d }$ -dimensional output
+
+$$
+\begin{array} { r } { f ( \pmb { x } ; \pmb { \theta } ) = \pmb { W } ^ { ( d ) } g \bigg ( \cdot \cdot \cdot g \Big ( \pmb { W } ^ { ( 2 ) } g \big ( \pmb { W } ^ { ( 1 ) } \pmb { x } + \pmb { b } ^ { ( 1 ) } \big ) + \pmb { b } ^ { ( 2 ) } \Big ) \cdot \cdot \cdot \bigg ) + \pmb { b } ^ { ( d ) } , } \end{array}
+$$
+
+where $g$ is a nonlinear activation function that operates component-wise on any vector.
+
+Definition 1 & 2. (Parameter vector) We define the parameter vector $\vartheta _ { m } ^ { ( k ) }$ of neuron $m$ in layer $k$ as the incoming weights to a neuron $m$ in layer $k$ concatenated with its bias term: $\pmb { \vartheta } _ { m } ^ { ( k ) } = \bigl [ \pmb { W } _ { m , 1 } ^ { ( k ) } , \ldots , \pmb { W } _ { m , n _ { k - 1 } } ^ { ( k ) } , \pmb { b } _ { m } ^ { ( k ) } \bigr ]$ . (Output weight vector) We define the output weight vector of neuron $m$ in layer $k$ as its outgoing weights from neuron $m$ in layer $k$ to the next layer: $\big [ \pmb { W } _ { 1 , m } ^ { ( k + 1 ) } , \ldots , \pmb { W } _ { n _ { k + 1 } , m } ^ { ( k + 1 ) } \big ]$ .
+
+Since one can permute the neurons within each layer without changing the network function $f ( \pmb { x } ; \pmb { \theta } )$ , any point $\pmb \theta$ induces a ‘permutation set’. Definition 3. (Permutation set)
+
+$$
+\begin{array} { r } { P ( \theta ) = \{ \theta ^ { \prime } \in \Theta : W _ { \sigma ^ { ( k ) } ( i ) , \sigma ^ { ( k - 1 ) } ( j ) } ^ { \prime ( k ) } = W _ { i , j } ^ { ( k ) } \mathrm { ~ a n d ~ } b _ { \sigma ^ { ( k ) } ( i ) } ^ { \prime ( k ) } = b _ { i } ^ { ( k ) } , k \in \{ 1 , \dots , d \} \} , } \end{array}
+$$
+
+of points $\pmb { \theta } ^ { \prime }$ with $f ( \pmb { x } ; \pmb { \theta } ) = f ( \pmb { x } ; \pmb { \theta } ^ { \prime } )$ , where $\sigma ^ { ( k ) }$ are permutations of the neuron indices $\{ 1 , \dots , n _ { k } \}$ in (hidden) layer $k$ where $\boldsymbol { \sigma } ^ { ( 0 ) }$ and $\sigma ^ { ( d ) }$ are fixed trivial permutations, since we want to permute neither the indices of the input nor that of the output. We will use the notation $\pmb { \theta } ^ { \prime } = \sigma _ { l  m } ^ { ( k ) } ( \pmb { \theta } )$ to indicate a point $\pmb { \theta } ^ { \prime }$ that differs from $\pmb \theta$ only by swapping neurons $l$ and $m$ in layer $k$ .
+
+Note that the cardinality of a permutation set is maximal with $\begin{array} { r } { | P ( \pmb { \theta } ) | = \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ only if all parameter vectors following, we will $\vartheta _ { l } ^ { ( k ) } \neq \vartheta _ { m } ^ { ( k ) }$ are distinct for every  at global minima, all p $l \neq m$ and layer er vectors $k \in \{ 1 , \ldots , d - 1 \}$ . In t layer $k$
+
+Definition 4. (Permutation point) Consider a minimum of a multilayer network with $( n _ { 1 } , \ldots , n _ { k } -$ $1 , \ldots , n _ { d } )$ neurons per layer. We can map this minimum to a configuration in the landscape of a multilayer network with $( n _ { 1 } , \ldots , n _ { k } , \ldots , n _ { d } )$ neurons per layer by duplicating one neuron $m \in$ $\{ 1 , \ldots , n _ { k } - 1 \}$ in layer $k$ as follows: (i) substitute the parameter vector of the new neuron with a copy of the parameter vector $\vartheta _ { m } ^ { ( k ) }$ of neuron $m$ , (ii) replace the output weight vector of the new neuron and the duplicated neuron $m$ with the initial output weight of neuron $m$ rescaled by $\textstyle { \frac { 1 } { 2 } }$ , and (iii) keep all the other parameters the same. This new configuration where the parameter vectors and a permutation point, denoted by the output weight vectors of the new neuron and the duplicated neuron $\pmb { \theta } _ { l  m } ^ { ( k ) }$ , i.e. $\pmb { \theta } _ { l \Leftrightarrow m } ^ { ( k ) } = \sigma _ { l \Leftrightarrow m } ^ { ( k ) } ( \pmb { \theta } _ { l  m } ^ { ( k ) } )$ . $m$ are the same will be called
+
+For training data $D = \{ ( { \pmb x } ^ { \mu } , y ^ { \mu } ) : \mu \in \{ 1 , . . . , T \} \}$ with targets $y ^ { \mu } \in \mathcal { V }$ , we define a loss function $\begin{array} { r } { L ( \pmb { \theta } ; D ) = \frac { 1 } { T } \sum _ { \mu = 1 } ^ { T } \ell \big ( y ^ { \mu } , f ( \pmb { x } ^ { \mu } ; \pmb { \theta } ) \big ) } \end{array}$ , where $\ell : \mathcal { V } \times \mathbb { R } ^ { n _ { d } } \to \mathbb { R }$ is some single-sample loss function. To simplify notation we will usually omit the explicit mentioning of the data in the loss function, i.e. $L ( \mathbf { \boldsymbol { \theta } } ) \equiv L ( \mathbf { \boldsymbol { \theta } } ; D )$ .
+
+A
+
+![](images/17479c5aadb93dc09b8ab15e07cef31df8d5f2ff4104acc3965cc0401f37224e.jpg)  
+Figure 1: A. Configuration of two parameter vectors $\vartheta _ { l } ^ { ( k ) }$ and $\vartheta _ { m } ^ { ( k ) }$ (black) at a minimum and a potential path (green) towards a permutation point $\theta _ { l  m } ^ { ( k ) } ( \star )$ m  . The path is parametrized by the distance $d .$ . Along the path the distance $d$ (blue dashed lines) decreases continuously starting at $d _ { l , m } ^ { ( k ) } ( \theta )$ . Note that the path can lead to a permutation point far away from the initial configuration, outside the linear subspace spanned by the parameter vectors ϑ(k)l and $\vartheta _ { m } ^ { ( k ) }$ . B. We exclude hypothetical paths where the distance along the path increases.
+
+# 2 MAIN RESULTS
+
+In this section, we will first present a novel method to find a low-loss path between partner minima. Our method ensures that this path passes through a ‘permutation point’. We study the properties of permutation points. Furthermore, we will introduce higher-order permutation points and provide a lower-bound on their number.
+
+# 2.1 A NOVEL METHOD TO CONSTRUCT LOW-LOSS PATHS BETWEEN PARTNER MINIMA
+
+One natural question regarding the geometry of the bottom of the landscape is the following: is it possible to find a continous low-loss path that connects two minima? In this work, we are interested in finding the barriers between partner global minima. In particular, we want to find a continuous low-loss path $\gamma : [ 0 , 1 ]  \Theta$ connecting two partner minima by first merging two parameter vectors and output weight vectors (‘permutation point’) and then completing the path using symmetry. We will first introduce some concepts to introduce this low-loss path between partner minima formally.
+
+Definition 5. (Distance function) and returns the squared Euclidean $d _ { l , m } ^ { ( k ) } : \Theta \to \mathbb { R } ^ { + }$ is a distance function that takes a the parameter vectors of neuron confiand ration at lay $\pmb \theta$ $l$ $m$ $k$ :
+
+$$
+d _ { l , m } ^ { ( k ) } ( \pmb { \theta } ) = \| \pmb { \vartheta } _ { l } ^ { ( k ) } - \pmb { \vartheta } _ { m } ^ { ( k ) } \| ^ { 2 }
+$$
+
+Our idea is to find a low-barrier path $\begin{array} { r } { \gamma _ { * } = \arg \operatorname* { m i n } _ { \gamma : [ 0 , \frac { 1 } { 4 } ]  \Theta } \operatorname* { m a x } _ { t \in [ 0 , \frac { 1 } { 4 } ] } L ( \gamma ( t ) ) } \end{array}$ under the following constraints for the initial $\mathit { t } = 0$ ) and quarter-way $\textstyle ( t = { \frac { 1 } { 4 } } )$ ) configurations: $\gamma _ { * } ( 0 ) = \theta$ , where $\pmb { \theta }$ is the parameter configuration at the minimum, and $d _ { l , m } ^ { ( k ) } ( \gamma _ { * } ( \textstyle { \frac { 1 } { 4 } } ) ) = 0$ . Furthermore, the distance between parameter vectors $\vartheta _ { l } ^ { ( k ) }$ and $\vartheta _ { m } ^ { ( k ) }$ in layer $k$ is decreasing, i.e. $d _ { l , m } ^ { ( k ) } ( \gamma _ { * } ( t ) ) < d _ { l , m } ^ { ( k ) } ( \gamma _ { * } ( t ^ { \prime } ) )$ for all $t > t ^ { \prime } \in [ 0 , \frac { 1 } { 4 } ] ^ { 1 }$ (see Fig. 1 and pseudocode in Appendix).
+
+The quarter-way configuration guarantees that the parameter vectors are identical $\vartheta _ { l } ^ { ( k ) } = \vartheta _ { m } ^ { ( k ) }$ , but puts no constraints on the output weights of neurons $m$ and $l$ in layer $k$ . We can continously move from the quarter-way configuration $\begin{array} { r } { ( \bar { t } = \frac { 1 } { 4 } ) } \end{array}$ ) to a configuration at $\begin{array} { r } { \dot { t } = \frac { 1 } { 2 } } \end{array}$ where the outputs weights of the related neurons are equal, without making any changes to the network output or the loss $L$ as follows: we will increase all output weights $\boldsymbol { W } _ { n , l } ^ { k + 1 }$ of neuron $l$ and decrease the corresponding output weights W k+1n,m of neuron $m$ by the same amount continuously so as to keep their sum fixed until W k+1n,l $\boldsymbol { W } _ { n , l } ^ { k + 1 } = \boldsymbol { W } _ { n , m } ^ { k + 1 }$ for every neuron $n \in \{ 1 , \ldots , n _ { k + 1 } \}$ at layer $k + 1$ (see Appendix Fig. 4).
+
+Lemma 1. The configuration at $\begin{array} { r } { t = \frac { 1 } { 2 } } \end{array}$ is one of the permutation points.
+
+Once we have reached $\begin{array} { r } { t = \frac { 1 } { 2 } } \end{array}$ , we interchange the neuron indices of the ‘merged’ neurons and continue on the ‘mirror’ path that results from walking the first half of the path backwards with interchanged neuron indices, until we arrive at the partner minimum at $t = 1$ , i.e. $\begin{array} { r } { \gamma ( \frac { 1 } { 2 } + \tau ) = \sigma _ { l \Leftrightarrow m } ^ { ( k ) } ( \gamma ( \frac { 1 } { 2 } - \overline { { \tau } } ) ) } \end{array}$ for $\tau \in [ 0 , \frac { 1 } { 2 } ]$ .
+
+To find such paths algorithmically, we reparametrize $\pmb { \vartheta } _ { m } ^ { ( k ) } ( t ) = \pmb { \vartheta } _ { l } ^ { ( k ) } ( t ) + d ( t ) \pmb { e } ( t )$ where $d ( t )$ is a positive scalar and $e ( t )$ is a unit-length vector. We start with $d ( 0 ) = d _ { l , m } ^ { ( k ) } ( \pmb { \theta } )$ and initialize $e$ in direction of the difference $\vartheta _ { m } ^ { ( k ) } - \vartheta _ { l } ^ { ( k ) }$ at the global minimum i.e., the initial parameter configuration. Next, we decrease $d$ infinitesimally and perform gradient descent for fixed $d$ on the loss $L$ until convergence. Note that all parameters can change, including $\vartheta _ { l } ^ { ( k ) }$ and $e$ during gradient descent. This procedure is repeated until $d = 0$ at $\begin{array} { r } { t = \frac { 1 } { 4 } } \end{array}$ . Finally we shift the respective output weights to the same value without changing the network function (see Appendix Fig. 4).
+
+Since the path connects two partner minima, there must be at least one saddle point on the path $\gamma ( t ) , t \in [ 0 , 1 ]$ , potentially but not necessarily, at the permutation point. Moreover, there is no guarantee that the highest saddle should be located at the permutation point (see Appendix Fig. 5).
+
+# 2.2 CHARACTERIZATION OF PERMUTATION POINTS
+
+In an earlier work, Fukumizu and Amari (2000) studied a specific set of critical points induced by the hierarchical structure in the neural network landscapes in two-layer neural networks. Let $L ^ { ( H ) }$ be the loss function (‘landscape’) of a two-layer neural network with $H$ neurons in the hidden layer and a single output. They showed that any critical point in the landscape of $L ^ { ( H - 1 ) }$ induces a line of critical points in the landscape of $L ^ { ( H ) }$ . We study permutation points in the general setup for the neural networks with multiple outputs and multiple layers.
+
+Theorem 1. (Fukumizu and Amari, 2000) By duplicating one parameter vector of any critical point in $L ^ { ( H - 1 ) }$ and keeping the sum of the two output weights corresponding to the duplicated parameter vectors fixed at the value of the original output weight, one obtains a line of critical points in $L ^ { ( H ) }$ .  2
+
+Proposition 1. (i) Permutation points θ(k)l⇔m are critical points of the original loss function.
+
+(ii) Any permutation point lies inside a $n _ { k + 1 }$ -dimensional equal-loss subspace of critical points.
+
+(iii) All other permutations of neuron indices in layer $k$ can be performed by continuous equal-loss transformations starting from permutation points $\pmb { \theta } _ { l  m } ^ { ( k ) }$ of neurons $l$ and $m$ , i.e. there is a continuous path $\rho : [ 0 , 1 ] \to \Theta$ such that $\pmb { \rho } ( 0 ) = \pmb { \theta } _ { l \Leftrightarrow m } ^ { ( k ) }$ and $\rho ( 1 ) = \theta _ { i \Leftrightarrow j } ^ { ( k ) }$ and $L ( \pmb { \rho } ( t ) ) = L ( \pmb { \rho } ( 0 ) )$ for all $t \in [ 0 , 1 ]$ and all $i \neq j \in \{ 1 , \ldots , n _ { k } \}$ . (Proofs: see appendix).
+
+![](images/2ac7652c42def1467d62b64cf2c5b31a5584f8ea61061652039538915455666b.jpg)  
+Figure 2: Paths to the same or to different permutation points. Top row. Configuration of the 5 weight vectors W (1)/b(1) of the first layer at the global minimum (blue) and at a permutation point (red) reached after merging the parameter vectors of two neurons. Note that the global minimum (i.e., the starting configuration) is the same for (A), (B) and (C) but the loss at the permutation point can be the same – (A) and (B) – or different – (A) and (C) – depending on the pair of neurons chosen for merging. Numbers indicate neurons. Bottom row. Quadratic loss $L$ as a function of the distance $d$ between the neurons to be merged. The distance was decreased in 200 logarithmically spaced steps from $d = d _ { l , m } ^ { ( 1 ) } ( \theta ^ { * } )$ to $1 / 1 0 ^ { 4 }$ of the initial value. For each $d$ , full batch gradient descent on the loss $L$ was performed until convergence. Training data was generated by sampling $1 0 ^ { 3 }$ two-dimensional input points $\pmb { x } ^ { \mu }$ from a standard normal distribution and computing labels $y ^ { \mu } = f ( x ^ { \mu } ; \theta ^ { * } )$ using a teacher network (shown in blue, equivalent to the configuration at the global minima). The teacher had a single layer of five hidden neurons with rectified-linear activation function $g$ and one linear output layer.
+
+Therefore, each of the $\frac { n _ { k } ( n _ { k } - 1 ) } { 2 }$ different permutation points of layer $k$ corresponds to a plateau of $n _ { k + 1 }$ dimensions. This plateau enables the exchange of all indices in layer $k$ . Note that there can be multiple plateaus on different loss levels that correspond to different local minima of the smaller networks where one of the neurons is dropped at a permutation point. For example in Fig. 2 one can exchange all indices in the hidden layer through the configuration in A and $\mathbf { B }$ or through the configuration in C that has another loss level. Note that, amongst all these permutation points embedded in different plateaus, we could for example search for the one with the lowest cost —and this lowest-cost permutation would then also connect all global minima caused by arbitrary permutations of neurons in layer $k$ .
+
+Definition 6. (Higher-order permutation point) Consider a minimum of a multilayer network with $( n _ { 1 } , \ldots , n _ { k } - K , \ldots , n _ { d } )$ neurons per layer. We can map this minimum to a configuration in the landscape of a multilayer network with $( n _ { 1 } , \ldots , n _ { k } , \ldots , n _ { d } )$ neurons per layer by replicating some neurons $m _ { j } \in \{ 1 , \ldots , n _ { k } - K \}$ in layer $k$ to fill out the parameters of new neurons as follows: (i) substitute the parameter vectors of the new neurons with one of the parameter vectors of the initial minimum, (ii) replace the output weight vectors of the new neurons and the output weight of the corresponding parameter vector $m _ { j }$ with the initial output weight of the replicated neuron $m _ { j }$ normalized so that the mentioned output weight vectors sum up to the original output weight vectors, and (iii) keep all the other parameters the same. This new configuration where the parameter vectors and the output weight vectors of the new neurons and the replicated neuron $m _ { j }$ for several $j$ in layer $k$ are the same will be called a $K$ -th order permutation point.
+
+This natural generalization on the 1-st order permutation points to higher-orders enables generalizing Proposition 1(i) and (ii) to the $K$ -th order permutation points.
+
+Proposition 2. (i) A $K$ -th order permutation point is a critical point of the original landscape. (ii) Any $K$ -th order permutation point at layer $k$ lies in a $K n _ { k + 1 }$ -dimensional subspace of equal loss parameter configurations. (Proofs: see appendix).
+
+# 2.3 COUNTING HIGHER-ORDER PERMUTATION POINTS
+
+A configuration in the landscape of $L ^ { ( H - K ) }$ can be mapped to an equivalent configuration in $L ^ { ( H ) }$ with the procedure described in Definition 6. We can then count the number of permutation points that reduce to the same configuration in $L ^ { ( H - K ) }$ combinatorially (see Appendix Fig. 6 for the explanation of combinatorial counting). For counting, we consider the cardinality of the permutation set of a permutation point. Since some parameter vectors are replicated, we have to consider permutations of sometimes identical neurons. This enables finding a lower bound on the number of critical points that have higher loss values than the global minima in general.
+
+Proposition 3. In a neural network with $( n _ { 1 } , \ldots , n _ { d } )$ neurons per layer, let $T ( K , n _ { k } )$ denote the ratio of the number of $K ^ { \mathrm { t h } }$ -order permutations points at layer $k$ to the number of global minima for $k = 1 , \ldots , d - 1$ and $K \geq 1$ .
+
+(i) For $K = 1 , 2 , 3$ and $n _ { k } \ge 2 K$ , we find $T ( K , n _ { k } )$ to be:
+
+$$
+\begin{array} { l } { \bullet \ { T ( K = 1 , n _ { k } ) = { \binom { n _ { k } - 1 } { 1 } } { \frac { 1 } { 2 } } } } \\ { \bullet \ { T ( K = 2 , n _ { k } ) = { \binom { n _ { k } - 2 } { 1 } } { \frac { 1 } { 3 ! } } + { \binom { n _ { k } - 2 } { 2 } } { \frac { 1 } { 2 ^ { 2 } } } } } \\  \bullet \ { T ( K = 3 , n _ { k } ) = { \binom { n _ { k } - 3 } { 1 } } { \frac { 1 } { 4 ! } } + { \binom { n _ { k } - 3 } { 2 } } { \frac { 1 } { 3 ! } } + { \binom { n _ { k } - 3 } { 3 } } { \frac { 1 } { 2 ^ { 3 } } } } \end{array}
+$$
+
+(ii) For general $K \leq n _ { k } / 2$ , we find the bound $T ( K , n _ { k } ) \ge \binom { n _ { k } - K } { K } \frac { 1 } { 2 ^ { K } }$ . (Proofs: see appendix)
+
+Considering all the layers, we note that the number of permutation points of order $K$ is at least $\textstyle \sum _ { k = 1 } ^ { d - 1 } { \frac { 1 } { 2 ^ { K } } } { \binom { { \bar { n } } _ { k } - K } { K } }$ times more than the global minima for $2 K \le \operatorname* { m i n } _ { k } n _ { k }$ .
+
+Lemma 2. For finite $K \in \mathbb { Z } ^ { + }$ and $n _ { k } \to \infty$ , $T ( K , n _ { k } ) > c _ { K } n _ { k } ^ { K }$ , since ${ \frac { 1 } { 2 ^ { K } } } \left( { { ^ { n _ { k } - K } } \atop K } \right) \to c _ { K } n _ { k } ^ { K }$
+
+When one layer has large number of neurons (i.e. $n _ { k }  \infty ,$ ) then the ratio $T ( K , n _ { k } )$ grows with $n _ { k } ^ { K }$
+
+Every permutation point lies inside a high-dimensional subspace of equal loss (Proposition 2(ii), see Appendix Fig. 7 for illustration). Importantly, every permutation point lies inside a distinct but connected subspace. Therefore the count for permutation points holds for the corresponding high-dimensional equal-loss subspaces of critical points.
+
+netw many h- $( n _ { 1 } , \ldots , n _ { d } )$ neurons per layer, there are (at least)ual-loss subspaces of critical points at the $\begin{array} { r } { \sum _ { k = 1 } ^ { d - 1 } T ( K , n _ { k } ) \prod _ { k = 1 } ^ { d - 1 } n _ { k } ! } \end{array}$ $K n _ { k + 1 }$ loss of a $K$ -th order permutation point for $2 K \le \operatorname* { m i n } _ { k } n _ { k }$
+
+We could start at an arbitrary configuration in consider the landscape of $L ^ { ( H - K ) }$ and the corresponding equal-loss high-dimensional subspaces in $L ^ { ( H ) }$ , where each configuration in the subspace computes the same function as the initial configuration. This procedure again would yield the same number of high-dimensional equal-loss subspaces. Therefore due to weight-space symmetry, neural network landscapes do not only exhibit numerous high-dimensional plateaus of critical points but also numerous high-dimensional plateaus (of usually non-critical points) at various loss values.
+
+# 3 EMPIRICAL RESULTS
+
+Using a similar procedure as in the toy example (see Fig. 2), we constructed paths between global minima in a fully connected three-layer network with $n _ { 1 } = n _ { 2 } = H$ and $n _ { 3 } = 1 0$ neurons (see Fig. 3). In order to study global minima we used a student-teacher setting3: the teacher network was
+
+pre-trained on the MNIST data set using negative log-likelihood loss and its parameters $\pmb { \theta } ^ { * }$ were kept fixed thereafter. We initialized the student with the parameters $\pmb { \theta } ^ { * }$ of the teacher and decreased the logarith $d$ cally spaced steps from to of the original $m$ ue. $l$ or every $k = 2$ f , t he $d _ { m , l } ^ { ( 2 ) } ( \pmb { \theta } ^ { * } )$ $1 / 1 0 ^ { 4 }$ $d$ $L$   
+output using full batch gradient descent until convergence. With $y ^ { \mu } = f ( \pmb { x } ^ { \mu } ; \pmb { \theta } ^ { * } ) \in \mathbb { R } ^ { 1 0 }$ being the output of the last layer before the softmax operation, we chose $\begin{array} { r } { L = \frac { 1 } { T } \sum _ { \mu = 1 } ^ { T } \| y ^ { \mu } - f \big ( x ^ { \mu } ; \pmb \theta \big ) \| ^ { 2 } / \langle y _ { i } ^ { \mu 2 } \rangle } \end{array}$ as the mean squared error loss between teacher and student, where $\langle . \rangle$ denotes the mean over patterns and dimensions and $\mu = 1 , \ldots , T$ enumerates the samples of the data set. Apart from a few cases, where the trajectory towards the permutation point passed through a saddle on the way, in most cases the loss increased monotonically until the permutation point. This indicates that the permutation point is a saddle, and not a minimum. As expected from theoretical results (Freeman and Bruna, 2016) and empirically observed by (Draxler et al., 2018), the barrier height (loss at saddle) decreased with the number $H$ of hidden neurons per layer.
+
+![](images/5f4fe70e0e6504f2521fc9ea04c235976c7583ed57d0d25c1d2d6af280a729ec.jpg)  
+Figure 3: A low-loss permutation path in the loss landscape of a multi-layer network using a student-teacher setup trained on MNIST. We merged the parameter vectors of two neurons with high cosine-similarity in the second hidden layer of a three-layer student network. The corresponding teacher network with $H = 1 0$ , 15, 20 or 25 was trained on MNIST. For each hidden layer size we trained 6 teacher networks with different random seeds and display one curve per hidden layer size and seed. A. In most cases, the mean squared loss $L$ between teacher and student output increases monotonically along our constructed paths from a global minimum until the permutation point. In these cases the latter corresponds to the loss barrier along the path. Note that the barrier height (loss at saddle) decreases with $H$ . B. The MNIST classification accuracy on the training set decreases only marginally when moving to a permutation point.
+
+# 4 DISCUSSION
+
+The surprising training performance of neural networks despite their highly non-convex nature has been drawing attention to the structure of the loss landscape. In this paper, we explored how weight-space symmetry induces saddles and plateaus in the neural network loss landscape. We found that special critical points, so-called permutation points, are embedded in high-dimensional flat plateaus. We proved that all permutation points in a given layer are connected with equal-loss paths, suggesting new perspectives on loss landscape topology. We provided a novel lower bound for the number of first- and higher-order permutation points and proposed a low-loss path finding method to connect equivalent minima. The empirical validation of our path finding algorithm in a multilayer network trained on MNIST showed that permutation points could indeed be reached in practice. Additionally, we observed that the loss at the permutation point (barrier) decreased with network size and thus confirmed Freeman and Bruna (2016)’s findings for loss barriers between global minima. High-dimensional flat regions around permutation points could be one of the causes of the empirically observed slow phases in training.
+
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+
+# A SUPPLEMENTARY FIGURES
+
+A
+
+B
+
+![](images/fd1d3c5c9eeb5138c4166241c6e8fa949e9a17a7e6bd0d16dbd0ac67764c5141.jpg)
+
+Figure 4: A. The loss landscape (schematic) as a function of two parameters: the difference $W _ { m , i } ^ { ( k ) } -$ W (k) between the weights from neuron $i$ to neurons $m$ and $l$ in layer $k$ and the weight $W _ { n , m } ^ { ( k + 1 ) }$ from neuron $m$ to neuron $n$ in the next layer (see $\mathbf { B }$ for network graph). The red curve indicates the path from one of the global minima (red triangbetween the input weight vectors of neurons $m$ to a and $l$ alf-way cin layer $k$ figurationvanishes $( W _ { m , i } ^ { ( k ) } = W _ { l , i } ^ { ( k ) }$ W l,i ence, red square). Along the axis W (k)m,i $W _ { m , i } ^ { ( k ) } - W _ { l , i } ^ { ( k ) } = 0$ , we can change the output weight $W _ { n , m } ^ { ( k + 1 ) }$ at constant ntal line), as long as the sum where the two output weig $W _ { n , m } ^ { ( k + 1 ) } + W _ { n , l } ^ { ( k + 1 ) } = c$ remains constant. The pointssuming the same matching $W _ { n , m } ^ { ( k + 1 ) } = W _ { n , l } ^ { ( k + 1 ) }$   
+condition for the other output weights) defines the permutation point $\theta _ { l \Leftrightarrow m } ^ { ( k ) }$ (red $\star$ ) where we can swap the indices of neurons we then shift (dashed green $m$ and e) al $l$ in layer  output w $k$ at equal loss aights of neuron continuoin layer sly in pato zero . If for $m$ $k$ $( W _ { n , m } ^ { ( k + 1 ) } = 0$ all $n$ , green filled circle), we are free to change the weight $W _ { m , i } ^ { ( k ) }$ at constant loss (green arrows) so as to perform further permutations of neurons in layer $k$ .
+
+![](images/6d1353e63e93dbf70695b69c847321ef9f55ab7870e092d92ef506177deeb8bc.jpg)  
+Figure 5: Loss $L$ (vertical axis) on the permutation path as a function of the distance $d$ between the \*two parameter vectors to be permuted (schematic). A. In the teacher network the two parameter vectors have a distance $d _ { 0 }$ . Along the path, the distance is reduced to zero. At the permutation point $( { \star } )$ , the loss reaches a maximum which corresponds to a saddle point of the total loss function. B. On the path towards the permutation point $( { \star } )$ an intermediate saddle point (S) may occur. C. The permutation point $( { \star } )$ could be a minimum along the path.
+
+![](images/7e85260ea3602df34dffe6333d204bf75b27dad99a3de397dcaf3f9579a14e30.jpg)  
+Figure 6: Visualizing how permutation points arise in between global minima for a (hypothetical) network function with scalar inputs and one hidden layer with three neurons. Here we do not consider biases for simplicity. Blue dots: 3! many equivalent global minima. A red $\star$ : One permutation point (of first order at layer $k = 1$ ) represented by its weights in the first layer, i.e. (2.5, 2.5, 0.4). All red $\star$ : One permutation set where the weight value 2.5 is duplicated. All green $\star$ : The other permutation set where the weight value 0.4 is duplicated. Note that overall, we have 6 permutation points that give rise to the same network function as the weight configuration with two hidden neurons with (2.5, 0.4). Indeed, $T ( K = 1 , n _ { 1 } = 3 ) = { \binom { 3 - 1 } { 1 } } { \frac { 1 } { 2 } } = 1$ confirms why the number permutation points corresponding to this particular weight configuration with two hidden neurons is equal to the number of global minima. Note that the weight values are assigned randomly for visualization purposes.
+
+![](images/786f7f496ba688a5916464ef28d963c9eadb8e72a0b9dbac1a098414d955d749.jpg)  
+Figure 7: A. Zooming in permutations points of one of the permutation sets (red $\star$ in Fig. 6). B. Visualizing how permutation points (at layer $k = 1$ , see A) lie inside equal-loss lines in the weight space of the layer $k + 1 = 2$ . Only two out of three lines are shown for simplicity. We observe that the number of such equal-loss lines (hyperplanes) is equal to the number of permutation points, i.e. each permutation point lies inside one distinct line.
+
+# B PATH FINDING ALGORITHM
+
+# Algorithm 1 Finding Low-Loss Paths through Permutation Points
+
+Require: parameters $\pmb \theta$ , indices $k , l , m$ , SCHEDULE of decreasing distances   
+1: $\overline { { d _ { 0 } } } \gets \bar { d } _ { l , m } ^ { ( k ) } ( \pmb { \theta } )$ t   
+2: Re-parametrize $\pmb \theta  \tilde { \pmb \theta } = ( \{ \tilde { \pmb \vartheta } _ { i } ^ { ( j ) } \} _ { ( i , j ) \neq ( m , k ) } , \pmb e , d )$ s.t. $\vartheta _ { i } ^ { ( j ) } = \tilde { \vartheta } _ { i } ^ { ( j ) }$ , $\vartheta _ { m } ^ { ( k ) } = \tilde { \vartheta } _ { l } ^ { ( k ) } + d e / \vert \vert e \vert \vert$   
+3: d ← d0, e ← ϑ(k)m − ϑ(k)   
+4: for all $d ^ { \prime }$ in SCHEDULE $( d _ { 0 } )$ do   
+5: $d  d ^ { \prime }$   
+6: $\gamma ( d ^ { \prime } )  \tilde { \pmb { \theta } } ( d ^ { \prime } )$ as found by gradient descent on $L ( \tilde { { \pmb { \theta } } } \setminus \{ d \} )$   
+7: end for   
+8: return $\gamma$
+
+# C PROOFS
+
+# C.1 HALF-WAY CONFIGURATIONS ARE PERMUTATION POINTS
+
+Proof of Lemma 1. Since the parameter vectors and output vectors of the two neurons are identical at $\begin{array} { r } { t = \frac { 1 } { 2 } } \end{array}$ , we can merge the two neurons into a single one (by appropriately rescaling the corresponding output weights) so that the number of neurons $n _ { k }$ in layer $k$ is reduced by one. Since our path-finding method minimizes the gradient, the configuration at $\begin{array} { r } { t = \frac { 1 } { 2 } } \end{array}$ is a local minimum of the smaller network, hence a permutation point.
+
+Alternatively, we will reformalize the path-finding problem with Lagrange multipliers: minimize $L ( \pmb \theta ) + \lambda d _ { l , m } ^ { ( k ) } ( \pmb \theta )$ for increasing values of $\lambda$ , starting with $\lambda \ : = \ : 0$ and increasing (potentially) until $\lambda  \infty$ . The goal here is to decrease the distance down to zero while keeping the loss minimal. For every $\lambda$ , we will obtain a minimizer $\pmb \theta$ . This sequence of $\pmb \theta$ configurations will be an approximate discretization of the ideal path $\gamma _ { * }$ . With the Lagrangian formulation, we can easily show that the half-way configuration is a critical point: it is a minimizer of $L ( \pmb \theta ) + \lambda d _ { l , m } ^ { ( k ) } ( \pmb \theta )$ when $d _ { l , m } ^ { ( k ) } ( \pmb { \theta } ) = 0$ for a big enough $\lambda$ . Therefore, $\nabla L ( \pmb \theta ) + \lambda \nabla d _ { l , m } ^ { ( k ) } ( \pmb \theta ) = 0$ . Note that the partial derivatives of $d _ { l , m }$ with respect to the parameter vectors of neuron $l$ and $m$ in layer $k$ are $\begin{array} { r } { \frac { \partial d _ { l , m } ^ { ( k ) } } { \partial \pmb { \vartheta } _ { l } ^ { ( k ) } } = 2 ( \pmb { \vartheta } _ { l } ^ { ( k ) } - \pmb { \vartheta } _ { m } ^ { ( k ) } ) } \end{array}$ $\begin{array} { r } { \frac { \partial \ v { d } _ { l , m } ^ { ( k ) } } { \partial \pmb { \vartheta } _ { l } ^ { ( k ) } } = 2 ( \pmb { \vartheta } _ { m } ^ { ( k ) } - \pmb { \vartheta } _ { l } ^ { ( k ) } ) } \end{array}$ and both are equal to zero at the half way configuration since $\| \vartheta _ { l } ^ { ( k ) } - \vartheta _ { m } ^ { ( k ) } \| ^ { 2 } = 0$ . All other partial derivatives with respect to other parameter vectors are 0 since $d _ { l , m } ^ { ( k ) }$ does not depend on them. Therefore $\nabla d _ { l , m } ^ { ( k ) } ( \pmb { \theta } ) = 0$ . Consequently, . We note that the Lagrangian formulation is an approximation on the ideal path .
+
+# C.2 PROPERTIES OF (FIRST-ORDER) PERMUTATION POINTS
+
+Proof of Proposition 1. (i) Let us assume we merged the neurons $l$ and $m$ in layer $k$ and scaled the output weight vectors of these neurons with $\begin{array} { l } { { \frac { 1 } { 2 } } } \end{array}$ . In that case, the output vector in layer $k + 1$ remains the same. The derivative with respect to an output in the layer $k + 1$ will be the same, i.e. $\frac { \partial L } { \partial a _ { n } ^ { k } }$ for $n \in [ n _ { k + 1 } ]$ will not change under this mapping. The derivatives with respect to the output weights (say $\mathbf { \Delta } w _ { i } ^ { k }$ for $i \in [ n _ { k } ] )$ of layer $k$ is $\begin{array} { r } { \sum _ { n = 1 } ^ { n _ { k + 1 } } \frac { \partial L } { \partial a _ { n } ^ { k + 1 } } \frac { \partial a _ { n } ^ { k + 1 } } { \partial \pmb { w } _ { i } ^ { k } } } \end{array}$ These derivatives would be the same as the parameter vectors of layer $k$ is $\frac { \partial L } { \partial a _ { m } ^ { k } } \frac { \partial a _ { m } ^ { k } } { \partial \pmb { \vartheta } _ { m } ^ { k } }$ remains the same up to a scaling in the output weight value.
+
+All the other derivatives remain the same (up to a scalar constant) as the corresponding derivatives before the mapping is performed. Therefore, a critical point in the landscape of the neural network with $( n _ { 1 } , \ldots , n _ { k } - 1 , \ldots , n _ { d } )$ neurons per layer will map to another critical point in the landscape of the neural network with $( n _ { 1 } , . . . , n _ { k } , . . . , n _ { d } )$ neurons per layer under the function preserving mapping described in the definition of a permutation point. This proposition is a straightforward extension to the Theorem 1 in (Fukumizu and Amari, 2000) for multiple output and multiple-layer neural networks.
+
+(ii) Once the parameter vectors of neurons $m$ and $l$ in layer $k$ are identical, they implement the same function. Any change of an output weight W (k+1)n,m preserves the network function, and criticality (see Proposition 1-(i) above), as long as the output weight W (k+1)n,l is coadapted so as to keep the sum W (k+1) $W _ { n , m } ^ { ( k + 1 ) } + W _ { n , l } ^ { ( k + 1 ) }$ constant. The sum-constraint $W _ { n , m } ^ { ( k + 1 ) } + W _ { n , l } ^ { ( k + 1 ) } = c$ for each $n$ in layer $k + 1$ defines an $n _ { k + 1 }$ -dimensional hyperplane of critical points. In particular, at each point in the hyperplane, we have $n _ { k + 1 }$ directions of the Hessian with zero Eigenvalues if the activation function $g$ is twice-differentiable.
+
+(iii) We make the following sequence of continuous transformations that are all possible at fixed loss. First, we decrease the output weights of neuron $m$ to zero while increasing those of $l$ by the same amount, keeping the sum of weights $W _ { j , m } ^ { ( k + 1 ) } + W _ { j , l } ^ { ( k + 1 ) }$ constant for each $j$ in layer $k { \pm } 1$ . Second, we change smoothly the input parameter vector of neuron $m$ to match those of an arbitrary other neuron $i$ in the same layer $k$ . Third, we increase the output weights of neuron $m$ while decreasing those of neuron i until all output weights of neuron i are zero, keeping the sum of weights W (k+1)j,m + $W _ { j , m } ^ { ( k + 1 ) } + W _ { j , i } ^ { ( k + 1 ) }$ constant for each $j$ in layer $k + 1$ . Fourth, we reduce the input parameter vector of neuron $i$ to zero. Fifth, we increase the input parameter vector of neuron $i$ to match that of neuron $l$ at the permutation point. Finally, we equally share output weights between neurons $i$ and $l$ so that $i$ has the same weights as previously neuron $m$ at the permutation point.
+
+Effectively, this procedure enables us to exchange an arbitrary neuron $i$ with neuron $m$ , but the   
+procedure can be repeated for further permutations. The permutations constructed in the proof of   
+property (ii) start at permutation points where the parameter vectors of neuroTherefore the loss associated with all the permutations constructed in the proof is $l$ $m$ merge. the one $L ( \pmb { \theta } _ { l  m } ^ { ( k ) } )$ $i$ and   
+construct the path leading to another $\theta _ { i \Leftrightarrow j } ^ { ( k ) }$ , that has, in general, a different loss.
+
+# C.3 PROPERTIES OF HIGHER-ORDER PERMUTATION POINTS
+
+Proof of Proposition 2. (i) We take a minimum in the landscape of neural network with $( n _ { 1 } , \ldots , n _ { k } -$ $K , \ldots , n _ { d } )$ neurons per layer. Using Proposition 1-(i), we map to a critical point in the landscape of neural network with $( n _ { 1 } , \ldots , n _ { k } - K + 1 , \ldots , n _ { d } )$ neurons per layer where one parameter vector in layer $k$ is duplicated and the corresponding output weights are rescaled with $\textstyle { \frac { 1 } { 2 } }$ . Repeating the same mapping starting at the latter critical point, we map to another critical point in the landscape of neural network with $( n _ { 1 } , \ldots , n _ { k } - K + 2 , \ldots , n _ { d } )$ neurons per layer. Repeating this mapping for $K$ times, we end up a $K$ -th order permutation point (up to changes in the output weights of the replicated neurons as long as the network function is preserved) and this is a critical point in the landscape of neural network with $( n _ { 1 } , . . . , n _ { k } , . . . , n _ { d } )$ neurons per layer by induction.
+
+(ii) We will denote the parameter vectors of a neural network with $( n _ { 1 } , \ldots , l = n _ { k } - K , \ldots , n _ { d } )$ neurons per layer with {ϑ(k)1 , $\{ \vartheta _ { 1 } ^ { ( k ) } , \vartheta _ { 2 } ^ { ( k ) } , \ldots , \vartheta _ { l } ^ { ( k ) } \}$ and that of $f ( n _ { 1 } , \ldots , n _ { k } , \ldots , n _ { d } )$ neurons with {ϑ0(k)1 , $\{ \vartheta _ { 1 } ^ { \prime ( k ) } , \vartheta _ { 2 } ^ { \prime ( k ) } , \ldots , \vartheta _ { n _ { k } } ^ { \prime ( k ) } \}$ . Let’s consider an unordered partition of $n _ { k } = s _ { 1 } + s _ { 2 } + \ldots + s _ { l }$ with $s _ { m } \geq 1$ for $m = 1 , \ldots , l$ . Without loss of generality, let’s assume that the first $s _ { 1 }$ parameter vectors of layer $k$ are the same, then the next $s _ { 2 }$ and so on. Equivalently, for all $m = 1 , \ldots , l$ , we have ϑ0(k) = $\vartheta _ { j } ^ { \prime ( k ) } = \vartheta _ { m } ^ { ( k ) }$ for $j = s _ { m - 1 } + 1 , s _ { m - 1 } + 2 , \ldots , s _ { m - 1 } + s _ { m }$ where $s _ { 0 } = 0$ .
+
+The only free variables are the outgoing weights of the replicated neurons- except for the constraint that the summation of these weights is fixed. These constraints correspond to Psm−1+smj=sm−1+1 $\begin{array} { r l } { ~ } & { { } \sum _ { j = s _ { m - 1 } + 1 } ^ { s _ { m - 1 } + s _ { m } } W _ { \mathrm { i , j } } ^ { \prime ( k + 1 ) } = } \end{array}$ $W _ { \mathrm { i , m } } ^ { ( k + 1 ) }$ for all neurons $i$ at layer $k { + 1 }$ . Overall, there are $\begin{array} { r } { ( \sum _ { m = 1 } ^ { l } s _ { m } ) n _ { k + 1 } = n _ { k } n _ { k + 1 } } \end{array}$ free variables constrained by $l n _ { k + 1 }$ equations. One equation defines a $n _ { k } n _ { k + 1 } - 1$ dimensional hyperplane in the $n _ { k } n _ { k + 1 }$ space. Intersecting $l n _ { k + 1 }$ of these hyperplanes, we end up having a $( n _ { k } - l ) n _ { k + 1 } = K n _ { k + 1 }$ dimensional equal-loss hyperplane.
+
+We simulate a smaller neural network with the big network of $( n _ { 1 } , . . . , n _ { k } , . . . , n _ { d } )$ neurons across the $d$ layers. We assume that a minimum of the small neural network has $n _ { k } - K$ distinct parameter vectors at layer $k$ and $n _ { j }$ distinct parameter vectors at other layers $j \neq k$ . A $K$ -th order permutation point (at layer $k$ ) of the big network implements all the $( n _ { 1 } , \ldots , n _ { k } - K , \ldots , n _ { d } )$ parameter vectors of the small network and only these. Since the big networks has $n _ { k }$ neurons in layer $k$ and the small network only $n _ { k } - K$ , the big network must reuse some of these parameter vectors of the smaller network several times. Therefore we count the number of permutations of indices to calculate $T ( K , n _ { k } )$ . We will start with $K = 1$ .
+
+# Proof of Proposition 3.
+
+# (1) The case $K = 1$ :
+
+At a first-order permutation point in layer $k$ , we have $l = n _ { k } - 1$ distinct parameter vectors $\{ \vartheta _ { 1 } ^ { ( k ) } , \vartheta _ { 2 } ^ { ( k ) } , \ldots , \bar { \vartheta } _ { l } ^ { ( k ) } \}$ for a total of $n _ { k }$ neurons. Therefore two of the $n _ { k }$ neurons must have the same parameter vector. More formally, there is only one way to partition $n _ { k }$ into $n _ { k } - 1$ positive integers without respecting order and this unordered partition can be represented as $n _ { k } = 2 + 1 + \ldots + 1$ .
+
+The duplicated parameter vector could be the first one, $\{ \pmb { \vartheta } _ { 1 } ^ { ( k ) } \}$ , or the second one, ... or the last one, $\{ \pmb { \vartheta } _ { l } ^ { ( k ) } \}$ . Therefore there are $\binom { n _ { k } - 1 } { 1 }$ choices. For each of these choices (say, we double the third parameter vector), we have $\textstyle { \frac { n _ { k } ! } { 2 ! } }$ permutations of indices of neurons in layer $k$ . If we include the permutations that are possible at all other layers, we have $\begin{array} { r } { \binom { n _ { k } - 1 } { 1 } \frac { 1 } { 2 } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ first-order permutation $\begin{array} { r } { | P ( \pmb { \theta } ) | = \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ Since the cardinali, we arrive at a ratio $T ( K = 1 , n _ { k } ) = { \binom { n _ { k } - 1 } { 1 } } { \frac { 1 } { 2 } }$ nduced by a global minimum is.
+
+# (2) The case $K = 2$
+
+There are two ways to have $n _ { k } \mathrm { ~ - ~ } 2$ distinct vectors out of $n _ { k }$ , corresponding to two unordered partitions of $n _ { k }$ : (i) $n _ { k } = 3 + 1 + . . . + 1$ , and (ii) $n _ { k } = 2 + 2 + 1 + \ldots + 1$ .
+
+For this case, we have $\textstyle { \frac { n _ { k } ! } { 3 ! } }$ permutations given by permuting the neuron indices of layer $k$ instead of the usual $n _ { k }$ ! permutations since we should eliminate the equivalent permutations corresponding to the permutations among the replicated parameter vectors with a division by 3!. Therefore, this 2-nd order permutation point induces a permutation set with cardinality $\begin{array} { r } { | P ( \pmb { \theta } ) | = \frac { 1 } { 3 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! . } \end{array}$ .
+
+Now we should consider other 2-nd order permutation points (at layer $k$ ) giving rise to the same network function and corresponding to the same unordered partition, which we did not count in this permutation set. If we had chosen another parameter vector to replicate three times, this 2-nd order permutation point would induce another permutation set. Note that we can choose the parameter vector to replicate out of one of the permutation s $n _ { k } - 2$ in ere ${ \binom { n _ { k } - 2 } { 1 } }$ ways and theree end up having 13! Qd−1j=1 nj ! ma ny points in eachmany 2-nd order $\begin{array} { r } { \left( \begin{array} { c } { n _ { k } - 2 } \\ { 1 } \end{array} \right) \frac { 1 } { 3 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ permutation points (at layer $k$ ) corresponding to this unordered partition.
+
+For this case, we have $\textstyle { \frac { n _ { k } ! } { 2 ! ^ { 2 } } }$ permutations given by permuting the neuron indices of layer $k$ instead of the usual $n _ { k }$ ! permutations since we should eliminate the equivalent permutations corresponding to the permutations among the two pairs of duplicated parameter vectors with a division by $2 ! ^ { 2 }$ . Therefore, this 2-nd order permutation point induces a permutation set with cardinality $\begin{array} { r } { | P ( \pmb { \theta } ) | = \frac { 1 } { 2 ! ^ { 2 } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$
+
+Again, we should consider other 2-nd order permutation points (at layer $k$ ) giving rise to the same network function and corresponding to the same unordered partition. Note that we can choose two parameter vectors to duplicate out of $n _ { k } - 2$ in ${ \binom { n _ { k } - 2 } { 2 } }$ ways and there are 12!2 Qd−1j=1 nj ! many points in each one of the permutation sets. Therefore, we end up having nk−22 
 12!2 Qd−1j=1 nj ! many 2-nd order permutation points (at layer $k$ ) corresponding to this unordered partition. Overall, we have $\begin{array} { r } { \binom { n _ { k } - 2 } { 1 } \frac { 1 } { 3 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } \bar { ! } + \binom { n _ { k } - 2 } { 2 } \frac { \mathrm { i } } { 2 ! ^ { 2 } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ many 2-nd order permutation points at layer $k$ .
+
+# (3) The case $K = 3$ :
+
+There are three ways to have $n _ { k } - 3$ distinct vectors out of $n _ { k }$ , corresponding to three unordered partitions of $n _ { k }$ : (i) $n _ { k } = 4 + 1 + \ldots + 1$ , (ii) $n _ { k } = 3 + 2 + 1 + \ldots + 1$ , and (iii) $n _ { k } = 2 + 2 + 2 + 1 + \ldots + 1$ .
+
+(i) $n _ { k } = 4 + 1 + . . . + 1$ For this case, we have $\textstyle { \frac { n _ { k } ! } { 4 ! } }$ permutations given by permuting the neuron indices of layer $k$ . Therefore, this 3-rd order permutation point induces a permutation set with cardinality $\begin{array} { r } { | P ( \pmb { \theta } ) | = \frac { 1 } { 4 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ .
+
+As usual, we should consider other 3-rd order permutation points (at layer $k$ ) giving rise to the same network function and corresponding to the same unordered partition. If we had chosen another parameter vector to replicate four times, this 3-rd order permutation point would induce another permutation set. Note that we can choose the parameter vector to replicate out of nk − 3 in nk−31 
 ways and there ar e 14! Qd−1j=1 nj ! many points in each one of the permutation sets. Therefore, we end up having $\begin{array} { r } { \left( { \overset { n _ { k } - 3 } { \ 1 } } \right) { \frac { 1 } { 4 ! } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ many 3-rd order permutation points (at layer $k$ ) corresponding to this unordered partition.
+
+For this case, we have $\textstyle { \frac { n _ { k } ! } { 3 ! 2 ! } }$ permutations given by permuting the neuron indices of layer $k$ . Therefore, this 3-rd order permutation point induces a permutation set with cardinality $\begin{array} { r } { | P ( \pmb { \theta } ) | = \frac { 1 } { 3 ! 2 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$
+
+As usual, we should consider other 3-rd order permutation points (at layer $k$ ) giving rise to the same network function and corresponding to the same unordered partition. If we had chosen two other parameter vector to replicate, say ϑ0(k)l and $\vartheta _ { m } ^ { \prime ( k ) }$ , this 3-rd order permutation point would induce another permutation set. Note that we can choose two parameter vectors to replicate out of $n _ { k } - 3$ in $2 ! { \binom { n _ { k } - 3 } { 2 } }$ ways. We have an extra 2! factor here since we have different permutation sets if we replicate ϑ0(k)l twice and $\vartheta _ { m } ^ { \prime ( k ) }$ three times, or vice versa. Yet, there are $\begin{array} { r } { { \frac { 1 } { 3 ! 2 ! } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ many points in each one of the permutation sets. Therefore, we end up having $\begin{array} { r } { 2 ! \binom { n _ { k } - 3 } { 2 } \frac { 1 } { 3 ! 2 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! \ = \ \binom { n _ { k } - 3 } { 2 } \frac { 1 } { 3 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ many $3 ^ { \mathrm { r d } }$ -order permutation points (at layer $k$ ) corresponding to this unordered partition.
+
+(iii) $n _ { k } = 2 + 2 + 2 + 1 + \ldots + 1$
+
+For this case, we have $\frac { n _ { k } ! } { 2 ! ^ { 3 } }$ permutations given by permuting the neuron indices of layer $k$ . Therefore, this 3-rd order permutation point induces a permutation set with cardinality $\begin{array} { r } { | P ( \pmb { \theta } ) | = \frac { 1 } { 2 ! ^ { 3 } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$
+
+As always, we should consider the other 3-rd order permutation points (at layer $k$ ) giving rise to the same network function and corresponding to the same unordered partition. Note that we can choose three parameter vectors to duplicate out of $n _ { k } - 3$ in $\binom { n _ { k } - 3 } { 3 }$ ways and there are $\begin{array} { r } { { \frac { 1 } { 2 ^ { 3 } } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ many points in each one of the permutation sets. Therefore, we end up having $\begin{array} { r } { \binom { n _ { k } - 3 } { 3 } \frac { 1 } { 2 ^ { 3 } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ many 3-rd order permutation points (at layer $k$ ) corresponding to this unordered partition. Overall, we hav e nk−31 
 14! Qd−1j=1 nj ! +  $\begin{array} { r } { \binom { n _ { k } - 3 } { 1 } \frac 1 { 4 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! + \binom { n _ { k } - 3 } { 2 } \frac 1 { 3 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! + \binom { n _ { k } - 3 } { 3 } \frac 1 { 2 ! ^ { 3 } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ nk−33 
 12!3 Qd−1j=1 nj ! many 3-rd order permutation points at layer $k$ .
+
+# (4) A note on the general closed form formula for $T ( K , n _ { k } )$ :
+
+For a general integer $K$ there is no closed-form formula for the number of partitions, although it is easy to have a closed-form formula for the number of permutation points for one given unordered partition following the counting arguments. Thus, we believe that there is no way to find a closed-form formula for $T ( K , n _ { k } )$ as a function of $K$ . However, the lower bound for $T ( K , n _ { k } )$ is also the dominating term for large $n _ { k }$ , since every other summand would be a polynomial of at most $( K 1 )$ -th order.
+
+# (5) A lower bound for general $K$ :
+
+For general $K$ , we have $l = n _ { k } - K$ distinct parameter vectors in the small network. There are many ways to partition $n _ { k }$ into $l$ positive integers without respecting order. Since we are interested in a lower bound, we only consider the following unordered partition: $n _ { k } = 2 + . . . + 2 + 1 + . . . + 1$ , i.e. we have $K$ duplicated parameter vectors and $n _ { k } - 2 K$ parameter vectors that appear once. For this unordered partition, we have ${ \binom { n _ { k } - K } { K } }$ ways to choose the duplicated parameter vectors. For each one of these choices, we can permute the neuron indices in $\textstyle { \frac { n _ { k } ! } { 2 ^ { K } } }$ different ways. Including the permutations in other layers j 6= k, we end up with nk−KK 
 $\begin{array} { r } { \left( { \overset { n _ { k } - K } { K } } \right) { \frac { 1 } { 2 ^ { K } } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ points in the permutation set. The number is a lower bound of $T ( K , n _ { k } )$ , because other unordered partitions of $n _ { k }$ give rise to other $K ^ { \mathrm { t h } }$ -order permutation points at layer $k$ .
+
+# C.5 PROOF OF LEMMA 2
+
+We can approximate the factorial an integer $n$ using Stirling’s formula
+
+$$
+n ! \to { \sqrt { 2 \pi n } } { \biggl ( } { \frac { n } { e } } { \biggr ) } ^ { n } \ \mathrm { a s } \ n \to \infty
+$$
+
+We note that this approximation leads to accurate results even for small $n$
+
+As $n \to \infty$ , both $n - K \to \infty$ and $n - 2 K \to \infty$ for finite $K$ . Therefore, we can apply Stirling’s formula both for $n - K$ and $n - 2 K$ :
+
+$$
+\frac { 1 } { 2 ! ^ { K } } { \binom { n - K } { K } } = \frac { 1 } { 2 ! ^ { K } } \frac { ( n - K ) ! } { ( n - 2 K ) ! K ! }
+$$
+
+$$
+{ \frac { 1 } { 2 ! ^ { K } } } { \frac { ( n - K ) ! } { ( n - 2 K ) ! K ! } } \to { \frac { 1 } { 2 ! ^ { K } } } { \frac { \sqrt { 2 \pi ( n - K ) } { \Big ( } { \frac { n - K } { e } } { \Big ) } ^ { n - K } } { \sqrt { 2 \pi ( n - 2 K ) } { \Big ( } { \frac { n - 2 K } { e } } { \Big ) } ^ { n - 2 K } K ! } } \operatorname { a s } n \to \infty
+$$
+
+$$
+= \frac { 1 } { 2 ! ^ { K } } \sqrt { \frac { n - K } { n - 2 K } } \frac { ( n - K ) ^ { n - K } } { ( n - 2 K ) ^ { n - 2 K } } \frac { 1 } { e ^ { K } K ! } \mathrm { a s } n \to \infty
+$$
+
+$$
+= \frac { 1 } { 2 ! ^ { K } } \sqrt { \frac { \frac { n } { K } - 1 } { \frac { n } { K } - 2 } } \frac { ( \frac { n } { K } - 1 ) ^ { n - K } } { ( \frac { n } { K } - 2 ) ^ { n - 2 K } } K ^ { K } \frac { 1 } { e ^ { K } K ! }  \frac { 1 } { 2 ! ^ { K } } ( \frac { n } { K } ) ^ { K } \frac { K ^ { K } } { e ^ { K } K ! } \mathrm { ~ a s ~ } n  \infty
+$$
+
+$$
+= { \frac { 1 } { 2 ! ^ { K } } } { \frac { 1 } { e ^ { K } K ! } } n ^ { K } = c _ { K } n ^ { K } \operatorname { a s } n \to \infty
+$$
+
+# C.6 PROOF OF LEMMA 3
+
+We already know the number of equilavent $K$ -th order permutation points (the ones that reduce to the same configuration in the landscape of the neural network with $( n _ { 1 } , \ldots , n _ { k } - K , \ldots , n _ { d } ) ;$ ) is at least $\begin{array} { r } { \sum _ { k = 1 } ^ { d - 1 } T ( K , n _ { k } ) \prod _ { k = 1 } ^ { d - 1 } n _ { k } ! } \end{array}$ (Proposition 3-(ii)). We can easily that see that every permutation point gives rise to a distinct high-dimensional subspace of critical points by observing that their parameter vectors in layer $k$ would be in distinct positions when projected on the parameters of the layer $k$ since we never repeat the same set of parameters in layer $k$ . If two such subspaces were the same, their projection on a lower dimensional space (parameters of the layer $k$ ) would be same necessarily. Therefore, the subspaces mentioned are distinct and the number of them is equivalent to the number of related permutation po