Datasets:

zai-org
/

RPC-Bench

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+# OPTIMAL ANN-SNN CONVERSION FOR HIGHACCURACY AND ULTRA-LOW-LATENCY SPIKING NEURAL NETWORKS
+Tong $\mathbf { B } \mathbf { u } ^ { 1 }$ , Wei $\mathbf { F a n g } ^ { 1 }$ , Jianhao $\mathbf { D i n g ^ { 1 } }$ , PengLin $\mathbf { D } \mathbf { a } \mathbf { i } ^ { 2 }$ , Zhaofei $\mathbf { V } \mathbf { u } ^ { 1 }$ \*, Tiejun Huang1
+1 Peking University, 2 Southwest Jiaotong University
+\* Corresponding author: yuzf12@pku.edu.cn
+# ABSTRACT
+Spiking Neural Networks (SNNs) have gained great attraction due to their distinctive properties of low power consumption and fast inference on neuromorphic hardware. As the most effective method to get deep SNNs, ANN-SNN conversion has achieved comparable performance as ANNs on large-scale datasets. Despite this, it requires long time-steps to match the firing rates of SNNs to the activation of ANNs. As a result, the converted SNN suffers severe performance degradation problems with short time-steps, which hamper the practical application of SNNs. In this paper, we theoretically analyze ANN-SNN conversion error and derive the estimated activation function of SNNs. Then we propose the quantization clipfloor-shift activation function to replace the ReLU activation function in source ANNs, which can better approximate the activation function of SNNs. We prove that the expected conversion error between SNNs and ANNs is zero, enabling us to achieve high-accuracy and ultra-low-latency SNNs. We evaluate our method on CIFAR-10/100 and ImageNet datasets, and show that it outperforms the stateof-the-art ANN-SNN and directly trained SNNs in both accuracy and time-steps. To the best of our knowledge, this is the first time to explore high-performance ANN-SNN conversion with ultra-low latency (4 time-steps). Code is available at https://github.com/putshua/SNN conversion QCFS
+# 1 INTRODUCTION
+Spiking neural networks (SNNs) are biologically plausible neural networks based on the dynamic characteristic of biological neurons (McCulloch & Pitts, 1943; Izhikevich, 2003). As the third generation of artificial neural networks (Maass, 1997), SNNs have attracted great attention due to their distinctive properties over deep analog neural networks (ANNs) (Roy et al., 2019). Each neuron transmits discrete spikes to convey information when exceeding a threshold. For most SNNs, the spiking neurons will accumulate the current of the last layer as the output within $T$ inference time steps. The binarized activation has rendered dedicated hardware of neuromorphic computing (Pei et al., 2019; DeBole et al., 2019; Davies et al., 2018). This kind of hardware has excellent advantages in temporal resolution and energy budget. Existing work has shown the potential of tremendous energy saving with considerably fast inference (Stockl & Maass, 2021). ¨
+In addition to efficiency advantages, the learning algorithm of SNNs has been improved by leaps and bounds in recent years. The performance of SNNs trained by backpropagation through time and ANN-SNN conversion techniques has gradually been comparable to ANNs on large-scale datasets (Fang et al., 2021; Rueckauer et al., 2017). Both techniques benefit from the setting of SNN inference time. Setting longer time-steps in backpropagation can make the gradient of surrogate functions more reliable (Wu et al., 2018; Neftci et al., 2019; Zenke & Vogels, 2021). However, the price is enormous resource consumption during training. Existing platforms such as TensorFlow and PyTorch based on CUDA have limited optimization for SNN training. In contrast, ANN-SNN conversion usually depends on a longer inference time to get comparable accuracy as the original ANN (Sengupta et al., 2019) because it is based on the equivalence of ReLU activation and integrateand-fire model’s firing rate (Cao et al., 2015). Although longer inference time can further reduce the conversion error, it also hampers the practical application of SNNs on neuromorphic chips.
+The dilemma of ANN-SNN conversion is that there exists a remaining potential in the conversion theory, which is hard to be eliminated in a few time steps (Rueckauer et al., 2016). Although many methods have been proposed to improve the conversion accuracy, such as weight normalization (Diehl et al., 2015), threshold rescaling (Sengupta et al., 2019), soft-reset (Han & Roy, 2020) and threshold shift (Deng & Gu, 2020), tens to hundreds of time-steps in the baseline works are still unbearable. To obtain high-performance SNNs with ultra-low latency (e.g., 4 time-steps), we list the critical errors in ANN-SNN conversion and provide solutions for each error. Our main contributions are summarized as follows:
+• We go deeper into the errors in the ANN-SNN conversion and ascribe them to clipping error, quantization error, and unevenness error. We find that unevenness error, which is caused by the changes in the timing of arrival spikes and has been neglected in previous works, can induce more spikes or fewer spikes as expected. • We propose the quantization clip-floor-shift activation function to replace the ReLU activation function in source ANNs, which better approximates the activation function of SNNs. We prove that the expected conversion error between SNNs and ANNs is zero, indicating that we can achieve high-performance converted SNN at ultra-low time-steps. • We evaluate our method on CIFAR-10, CIFAR-100, and ImageNet datasets. Compared with both ANN-SNN conversion and backpropagation training methods, the proposed method exceeds state-of-the-art accuracy with fewer time-steps. For example, we reach top-1 accuracy $9 1 . 1 8 \%$ on CIFAR-10 with unprecedented 2 time-steps.
+# 2 PRELIMINARIES
+In this section, we first briefly review the neuron models for SNNs and ANNs. Then we introduce the basic framework for ANN-SNN conversion.
+Neuron model for ANNs. For ANNs, the computations of analog neurons can be simplified as the combination of a linear transformation and a non-linear mapping:
+$$
+\pmb { a } ^ { l } = h ( \pmb { W } ^ { l } \pmb { a } ^ { l - 1 } ) , l = 1 , 2 , . . . , M
+$$
+where the vector $\mathbf { \Delta } _ { \mathbf { \alpha } \mathbf { \beta } \mathbf { a } _ { } ^ { l } }$ denotes the output of all neurons in $l$ -th layer, $\mathbf { \Delta } W ^ { l }$ denotes the weight matrix between layer $l$ and layer $l - 1$ , and $h ( \cdot )$ is the ReLU activation function.
+Neuron model for SNNs. Similar to the previous works (Cao et al., 2015; Diehl et al., 2015; Han et al., 2020), we consider the Integrate-and-Fire (IF) model for SNNs. If the IF neurons in $l$ -th layer receive the input ${ \boldsymbol x } ^ { l - 1 } ( t )$ from last layer, the temporal potential of the IF neurons can be defined as:
+$$
+\pmb { m } ^ { l } ( t ) = \pmb { v } ^ { l } ( t - 1 ) + \pmb { W } ^ { l } \pmb { x } ^ { l - 1 } ( t ) ,
+$$
+where $m ^ { l } ( t )$ and ${ \pmb v } ^ { l } ( t )$ represent the membrane potential before and after the trigger of a spike at time-step $t$ . $W ^ { l }$ denote the weight in $l$ -th layer. As soon as any element $m _ { i } ^ { l } ( t )$ of $m ^ { l } ( t )$ exceeds the firing threshold $\theta ^ { l }$ , the neuron will elicit a spike and update the membrane potential $v _ { i } ^ { l } ( t )$ . To avoid information loss, we use the “reset-by-subtraction” mechanism (Rueckauer et al., 2017; Han et al., 2020) instead of the “reset-to-zero” mechanism, which means the membrane potential $v _ { i } ^ { l } ( t )$ is subtracted by the threshold value $\theta ^ { l }$ if the neuron fires. Based on the threshold-triggered firing mechanism and the “reset-by-subtraction” of the membrane potential after firing discussed above, we can write the uplate rule of membrane potential as:
+$$
+\begin{array} { r } { \pmb { s } ^ { l } ( t ) = H ( \pmb { m } ^ { l } ( t ) - \pmb { \theta } ^ { l } ) , } \\ { \pmb { v } ^ { l } ( t ) = \pmb { m } ^ { l } ( t ) - \pmb { s } ^ { l } ( t ) \theta ^ { l } . } \end{array}
+$$
+Here $s ^ { l } ( t )$ refers to the output spikes of all neurons in layer $l$ at time $t$ , the element of which equals 1 if there is a spike and 0 otherwise. $H ( \cdot )$ is the Heaviside step function. $\pmb { \theta } ^ { l }$ is the vector of the firing threshold $\mathbf { \dot { \theta } } ^ { l }$ . Similar to Deng $\&$ Gu (2020), we suppose that the postsynaptic neuron in $l$ -th layer receives unweighted postsynaptic potential $\theta ^ { l }$ if the presynaptic neuron in $l - 1$ -th layer fires a spike, that is:
+$$
+\pmb { x } ^ { l } ( t ) = \pmb { s } ^ { l } ( t ) \pmb { \theta } ^ { l } .
+$$
+Table 1: Summary of notations in this paper
+<table><tr><td>Symbol</td><td>Definition</td><td>Symbol</td><td>Definition</td></tr><tr><td>1</td><td>Layer index</td><td>x(t）</td><td>Unweighted PSPl</td></tr><tr><td>i</td><td>Neuron index</td><td>s(t)</td><td>Output spikes</td></tr><tr><td>W</td><td>Weight</td><td>(T)</td><td>Average unweigthed PSP before time T</td></tr><tr><td>al</td><td>ANN activation values</td><td>zl</td><td>Weighted input from l-1layer</td></tr><tr><td>t</td><td>Time-steps</td><td>h()</td><td>ReLU function</td></tr><tr><td>T</td><td>Total time-step</td><td>H()</td><td>Heaviside step function</td></tr><tr><td>0l</td><td>Threshold</td><td>L</td><td>Quantization step for ANN</td></tr><tr><td>又</td><td>Trainable threshold in ANN</td><td>Errl</td><td>Conversion Error</td></tr><tr><td>ml(t)</td><td>Potential before firing</td><td>Err -l</td><td>Estimated conversion Error</td></tr><tr><td>1(t）</td><td>Potential after firing</td><td>6</td><td>Shift of quantization clip-floor function</td></tr></table>
+1 Postsynaptic potential
+ANN-SNN conversion. The key idea of ANN-SNN conversion is to map the activation value of an analog neuron in ANN to the firing rate (or average postsynaptic potential) of a spiking neuron in SNN. Specifically, we can get the potential update equation by combining Equation 2 – Equation 4:
+$$
+\pmb { v } ^ { l } ( t ) - \pmb { v } ^ { l } ( t - 1 ) = \pmb { W } ^ { l } \pmb { x } ^ { l - 1 } ( t ) - \pmb { s } ^ { l } ( t ) \pmb { \theta } ^ { l } .
+$$
+Equation 6 describes the basic function of spiking neurons used in ANN-SNN conversion. By summing Equation 6 from time 1 to $T$ and dividing $T$ on both sides, we have:
+$$
+{ \frac { { \pmb v } ^ { l } ( T ) - { \pmb v } ^ { l } ( 0 ) } { T } } = { \frac { { \pmb W } ^ { l } \sum _ { i = 1 } ^ { T } { \pmb x } ^ { l - 1 } ( i ) } { T } } - { \frac { \sum _ { i = 1 } ^ { T } s ^ { l } ( i ) \theta ^ { l } } { T } } .
+$$
+If we use from 0 to $\begin{array} { r } { \phi ^ { l - 1 } ( T ) = \frac { \sum _ { i = 1 } ^ { T } { \pmb x } ^ { l - 1 } ( i ) } { T } } \end{array}$ to denote the average postsynaptic potential during the period 5 into Equation 7, then we get: $T$
+$$
+\phi ^ { l } ( T ) = W ^ { l } \phi ^ { l - 1 } ( T ) - \frac { { \pmb v } ^ { l } ( T ) - { \pmb v } ^ { l } ( 0 ) } { T } .
+$$
+Equation 8 describes the relationship of the average postsynaptic potential of neurons in adjacent layers. Note that $\phi ^ { l } ( T ) \geqslant 0$ . If we set the initial potential $\mathbf { \boldsymbol { v } } ^ { l } ( \mathbf { \bar { 0 } } )$ to zero and neglect the remaining term vl(T )T when the simulation time-steps T is long enough, the converted SNN has nearly the same activation function as source ANN (Equation 1). However, high $T$ would cause long inference latency that hampers the practical application of SNNs. Therefore, this paper aims to implement high-performance ANN-SNN conversion with extremely low latency.
+# 3 CONVERSION ERROR ANALYSIS
+In this section, we will analyze the conversion error between the source ANN and the converted SNN in each layer in detail. In the following, we assume that both ANN and SNN receive the same input from the layer $l - 1$ , that is, ${ \pmb a } ^ { l - 1 } = \phi ^ { l - 1 } ( T )$ , and then analyze the error in layer $l$ . For simplicity, we use $z ^ { l } = W ^ { l } \phi ^ { l - 1 } ( T ) = W ^ { l } a ^ { l - 1 }$ to substitute the weighted input from layer $l - 1$ for both ANN and SNN. The absolute conversion error is exactly the outputs from converted SNN subtract the outputs from ANN:
+$$
+E r r ^ { l } = \phi ^ { l } ( T ) - a ^ { l } = z ^ { l } - \frac { { \pmb v } ^ { l } ( T ) - { \pmb v } ^ { l } ( 0 ) } { T } - h ( z ^ { l } ) ,
+$$
+where $h ( z ^ { l } ) = \mathrm { R e L U } ( z ^ { l } )$ . It can be found from Equation 9 that the conversion error is nonzero if ${ \pmb v } ^ { l } ( T ) - { \pmb v } ^ { l } ( 0 ) \neq 0$ and $z ^ { l } > 0$ . In fact, the conversion error is caused by three factors.
+Clipping error. The output ϕl(T ) of SNNs is in the range of [0, θl] as ϕl(T ) = PTi=1 xl(i)T PTi=1 sl(i)T θl (see Equation 5). However, the output al of ANNs is in a much lager range of [0, almax], where $a _ { m a x } ^ { l }$ denotes the maximum value of $\mathbf { \delta } _ { \mathbf { { a } } } l$ . As illustrated in Figure 1a, $\mathbf { \Delta } _ { \mathbf { \alpha } \mathbf { \alpha } \mathbf { \alpha } \mathbf { \alpha } \mathbf { \alpha } \mathbf { \alpha } \mathbf { \alpha } \mathbf { \beta } } \mathbf { \Delta } _  \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathrm  \langle \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathrm { \left. \frac { \partial \mathbf { \alpha } \mathbf { \beta } } { \partial \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathrm { \beta \alpha } \mathrm { \beta }  } }\right.$ can be mapped to $\phi ^ { l } ( T )$ by the following equation:
+$$
+\phi ^ { l } ( T ) = \mathrm { c l i p } \left( \frac { \theta ^ { l } } { T } \left\lfloor \frac { { \mathbf { a } } ^ { l } T } { \lambda ^ { l } } \right\rfloor , 0 , \theta ^ { l } \right) .
+$$
+![](images/ec97e07b360ac04e2fe752bfcd2992ad416384ac424d157a1b90b9e0eeac50c5.jpg)
+Figure 1: Conversion error between source ANN and converted SNN. $s _ { 1 } ^ { l - 1 }$ and $s _ { 2 } ^ { l - 1 }$ denote the output spikes of two neurons in layer $l - 1$ , and $s _ { 1 } ^ { l }$ denotes the output spikes of a neuron in layer $l$ .
+Here the clip function sets the upper bound $\theta ^ { l }$ and the lower bound 0. $\lfloor \cdot \rfloor$ denotes the floor function.
+$\lambda ^ { l }$ represents the actual maximum value of output $\mathbf { \delta } _ { \mathbf { { a } } } l$ mapped to the maximum value $\theta ^ { l }$ of $\phi ^ { l } ( T )$ .
+Considering that nearly $9 9 . 9 \%$ activations of $\mathbf { \Delta } _ { \mathbf { \alpha } \mathbf { \alpha } \mathbf { \alpha } \mathbf { \alpha } \mathbf { \alpha } \mathbf { \alpha } \mathbf { \alpha } \mathbf { \beta } } \mathbf { \Delta } _  \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathrm  \langle \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathbf { \beta } \mathrm { \left. \frac { \partial \mathbf { \alpha } \mathbf { \beta } } { \partial \mathbf { \alpha } \mathbf { \beta } \mathbf { \alpha } \mathrm { \beta \alpha } \mathrm { \beta }  } }\right.$ in ANN are in the range of $\begin{array} { r } { [ 0 , \frac { a _ { m a x } ^ { l } } { 3 } ] } \end{array}$ , Rueckauer et al. (2016) suggested to choose $\lambda ^ { l }$ according to $9 9 . 9 \%$ activations. The activations between $\lambda ^ { l }$ and $a _ { m a x } ^ { l }$ in ANN are mapped to the same value $\theta ^ { l }$ in SNN, which will cause conversion error called clipping error.
+Quantization error (flooring error). The output spikes $s ^ { l } ( t )$ are discrete events, thus $\phi ^ { l } ( T )$ are discrete with quantization resolution $\frac { \theta ^ { l } } { T }$ (see Equation 10). When mapping $\mathbf { \delta } _ { \mathbf { { a } } } l$ to $\phi ^ { l } ( T )$ , there exists unavoidable quantization error. For example, as illustrated in Figure 1a, the activations of ANN in the range of $\bigl [ \frac { \lambda ^ { l } } { T } , \frac { 2 \lambda ^ { l } } { T } \bigr )$ are mapped to the same value $\frac { \theta ^ { l } } { T }$ of SNN.
+Unevenness error. Unevenness error is caused by the unevenness of input spikes. If the timing of arrival spikes changes, the output firing rates may change, which causes conversion error. There are two situations: more spikes as expected or fewer spikes as expected. To see this, in source ANN, we suppose that two analog neurons in layer $l - 1$ are connected to an analog neuron in layer $l$ with weights 2 and $^ { - 2 }$ , and the output vector $\mathbf { a } ^ { l - 1 }$ of neurons in layer $l - 1$ is [0.6, 0.4]. Besides, in converted SNN, we suppose that the two spiking neurons in layer $l - 1$ fire 3 spikes and 2 spikes in 5 time-steps $( \mathrm { T } { = } 5 )$ ), respectively, and the threshold $\theta ^ { l - 1 } = 1$ . Thus, $\begin{array} { r } { \phi ^ { l - 1 } ( T ) = \frac { \sum _ { i = 1 } ^ { T } s ^ { l - 1 } ( i ) } { T } \theta ^ { l - 1 } = } \end{array}$ [0.6, 0.4]. Even though $\phi ^ { l - 1 } ( T ) = \pmb { a } ^ { l - 1 }$ and the weights are same for the ANN and SNN, $\phi ^ { l } ( T )$ can be different from $\mathbf { \delta } _ { \mathbf { { a } } } l$ if the timing of arrival spikes changes. According to Equation 1, the ANN output $\pmb { a } ^ { l } = \pmb { W } ^ { l } \pmb { a } ^ { l - 1 } = [ 2 , - 2 ] [ 0 . 6 , 0 . 4 ] ^ { T } = 0 . 4$ . As for SNN, supposing that the threshold $\theta ^ { l } = 1$ , there are three possible output firing rates, which are illustrated in Figure 1 (b)-(d). If the two presynaptic neurons fires at $t = 1 , 3 , 5$ and $t = 2 , 4$ (red bars) respectively with weights 2 and -2, the postsynaptic neuron will fire two spikes at $t = 1 , 3$ (red bars), and $\begin{array} { r } { \phi ^ { l } ( T ) = \frac { \sum _ { i = 1 } ^ { T } s ^ { l } \bar { ( i ) } } { T } \theta ^ { l } = 0 . 4 = a ^ { l } } \end{array}$ . However, if the presynaptic neurons fires at $t = 1 , 2 , 3$ and $t = 4 , 5$ , respectively, the postsynaptic neuron will fire four spikes at $t = 1 , 2 , 3 , 4$ , and $\phi ^ { l } ( T ) = 0 . 8 > a ^ { l }$ . If the presynaptic neurons fires at $t = 3 , 4 , 5$ and $t = 1 , 2$ , respectively, the postsynaptic neuron will fire only one spikes at $t = 5$ , and $\phi ^ { l } ( T ) = 0 . 2 < a ^ { l }$ .
+Note that the clipping error and quantization error have been proposed in Li et al. (2021). There exist interdependence between the above three kinds of errors. Specifically, the unevenness error will degenerate to the quantization error if $v ^ { l } ( T )$ is in the range of $[ 0 , \theta ^ { l } ]$ . Assuming that the potential $v ^ { l } \breve { ( T ) }$ falls into $[ 0 , \dot { \theta } ^ { l } ]$ will enable us to estimate the activation function of SNNs ignoring the effect of unevenness error. Therefore, an estimation of the output value $\phi ^ { l } ( T )$ in a converted SNN can be formulated with the combination of clip function and floor function, that is:
+$$
+\phi ^ { l } ( T ) \approx \theta ^ { l } \exp \left( \frac { 1 } { T } \left\lfloor \frac { z ^ { l } T + v ^ { l } ( 0 ) } { \theta ^ { l } } \right\rfloor , 0 , 1 \right) .
+$$
+The detailed derivation is in the Appendix. With the help of this estimation for the SNN output, the estimated conversion error $\widetilde { E r r } ^ { l }$ can be derived from Equation 9:
+$$
+\widetilde { E r r } ^ { l } = \theta ^ { l } \mathrm { c l i p } \left( \frac { 1 } { T } \left\lfloor \frac { z ^ { l } T + v ^ { l } ( 0 ) } { \theta ^ { l } } \right\rfloor , 0 , 1 \right) - h ( z ^ { l } ) \approx E r r ^ { l } .
+$$
+![](images/db665824bf0d4b31c8f99401b9613f1d2153332bb2e2e0ef3cf82f91d01e5b41.jpg)
+Figure 2: Comparison of SNN output $\phi ^ { l } ( T )$ and ANN output $\mathbf { \delta } _ { \mathbf { { a } } } l$ with same input $z ^ { l }$
+# 4 OPTIMAL ANN-SNN CONVERSION
+# 4.1 QUANTIZATION CLIP-FLOOR ACTIVATION FUNCTION
+According to the conversion error of Equation 12, it is natural to think that if the commonly used ReLU activation function $h ( z ^ { l } )$ is substituted by a clip-floor function with a given quantization steps $L$ (similar to Equation 11), the conversion error at time-steps $T = L$ will be eliminated. Thus the performance degradation problem at low latency will be solved. As shown in Equation 13, we proposed the quantization clip-floor activation function to train ANNs.
+$$
+a ^ { l } = \bar { h } ( z ^ { l } ) = \lambda ^ { l } \mathrm { c l i p } \left( \frac { 1 } { L } \left\lfloor \frac { z ^ { l } L } { \lambda ^ { l } } \right\rfloor , 0 , 1 \right) ,
+$$
+where the hyperparameter $L$ denotes quantization steps of ANNs, the trainable $\lambda ^ { l }$ decides the maximum value of $\mathbf { \delta } _ { \mathbf { { a } } } l$ in ANNs mapped to the maximum of $\phi ^ { l } ( T )$ in SNNs. Note that $z ^ { l } =$ $W ^ { l } \phi ^ { l - 1 } ( T ) = W ^ { l } a ^ { l - 1 }$ . With this new activation function, we can prove that the estimated conversion error between SNNs and ANNs is zero, and we have the following Theorem.
+Theorem 1. An ANN with activation function (13) is converted to an SNN with the same weights. If $T = L$ , $\theta ^ { l } = \lambda ^ { l }$ , and ${ \pmb v } ^ { l } ( 0 ) = { \bf 0 }$ , then:
+$$
+\widetilde { E r r } ^ { l } = \phi ^ { l } ( T ) - a ^ { l } = 0 .
+$$
+Proof. According to Equation 12, and the conditions $T = L , \theta ^ { l } = \lambda ^ { l } , \pmb { v } ^ { l } ( 0 ) = \mathbf { 0 }$ , we have $\widetilde { \pmb { E r r } } ^ { l } =$ $\begin{array} { r } { \phi ^ { l } ( T ) - a ^ { l } = \theta ^ { l } \mathrm { c l i p } \left( \frac { 1 } { T } \left\lfloor \frac { z ^ { l } T + v ^ { l } ( 0 ) } { \theta ^ { l } } \right\rfloor , 0 , 1 \right) - \lambda ^ { l } \mathrm { c l i p } \left( \frac { 1 } { L } \left\lfloor \frac { z ^ { l } L } { \lambda ^ { l } } \right\rfloor , 0 , 1 \right) = 0 . } \end{array}$ .
+Theorem 1 implies that if the time-steps $T$ of the converted SNN is the same as the quantization steps $L$ of the source ANN, the conversion error will be zero. An example is illustrated in Figure 2a, where $T = L = 4$ , $\theta ^ { l } = \lambda ^ { l }$ . The red curve presents the estimated output $\phi ^ { l } ( T )$ of the converted SNNs with respective to different input $z ^ { l }$ , while the green curve represents the out $\mathbf { \delta } _ { \mathbf { { a } } } l$ of the source ANN with respective to different input $z ^ { l }$ . As the two curve are the same, the estimated conversion error $\widetilde { E r r } ^ { l }$ is zero. Nevertheless, in practical application, we focus on the performance of SNNs at different time-steps. There is no guarantee that the conversion error is zero when $T$ is not equal to $L$ . As illustrated in Figure 2b, where $L = 4$ and $L = 8$ , we can find the conversion error is greater than zero for some $z ^ { l }$ . This error will transmit layer-by-layer and eventually degrading the accuracy of the converted SNN. One way to solve this problem is to train multiple source ANNs with different quantization steps, then convert them to SNNs with different time-steps, but it comes at a considerable cost. In the next section, we propose the quantization clip-floor activation function with a shift term to solve this problem. Such an approach can achieve high accuracy for different time-steps, without extra computation cost.
+# 4.2 QUANTIZATION CLIP-FLOOR-SHIFT ACTIVATION FUNCTION
+We propose the quantization clip-floor-shift activation function to train ANNs.
+$$
+a ^ { l } = \widehat { h } ( z ^ { l } ) = \lambda ^ { l } \mathrm { c l i p } \left( \frac { 1 } { L } \left\lfloor \frac { z ^ { l } L } { \lambda ^ { l } } + \varphi \right\rfloor , 0 , 1 \right) .
+$$
+Compared with Equation 13, there exists a hyperparameter vector $\varphi$ that controls the shift of the activation function. When $L \neq T$ , we cannot guarantee the conversion error is 0. However, we can estimate the expectation of conversion error. Similar to (Deng & Gu, 2020), we assume that $z _ { i } ^ { l }$ is uniformly distributed within intervals $[ ( t - 1 ) \lambda ^ { l } / T , ( t ) \lambda ^ { l } / T ]$ and $[ ( l - 1 ) \lambda ^ { l } / L , ( l ) \lambda ^ { l } / L ]$ for $t = 1 , 2 , . . . , T$ and $L = 1 , 2 , . . . , L$ , we have the following Theorem.
+Theorem 2. An ANN with activation function $( I 5 )$ is converted to an SNN with the same weights. $I f \theta ^ { l } = \lambda ^ { l }$ , ${ \pmb v } ^ { l } ( 0 ) = \theta ^ { l } { \pmb \varphi }$ , then for arbitrary $T$ and $L$ , the expectation of conversion error reaches 0 when the shift term $\varphi$ in source ANN is $\frac { \mathbf { 1 } } { \mathbf { 2 } }$
+$$
+\forall T , L \quad \mathbb { E } _ { z } \left( \widetilde { E r r } ^ { l } \right) \Big | _ { \varphi = \frac { 1 } { 2 } } = \mathbf { 0 } .
+$$
+The proof is in the Appendix. Theorem 2 indicates that the shift term $\frac { \mathbf { 1 } } { \mathbf { 2 } }$ is able to optimize the expectation of conversion error. By comparing Figure 2b and Figure 2c, we can find that when the shift term $\varphi = \mathbf { 0 . 5 }$ is added, the mean conversion error reaches zero, even though $L \neq T$ . These results indicate we can achieve high-performance converted SNN at ultra-low time-steps.
+$L$ is the only undetermined hyperparameter of the quantization clip-floor-shift activation. When $T = L$ , the conversion error reaches zero. So we naturally think that the parameter $L$ should be set as small as possible to get better performance at low time-steps. However, a too low quantization of the activation function will decrease the model capacity and further lead to accuracy loss when the time-steps is relatively large. Choosing the proper $L$ is a trade-off between the accuracy at low latency and the best accuracy of SNNs. We will further analyze the effects of quantization steps $L$ in the experiment section.
+# 4.3 ALGORITHM FOR TRAINING QUANTIZATION CLIP-FLOOR-SHIFT ACTIVATION FUNCTION
+Training an ANN with quantization clip-floor-shift activation instead of ReLU is also a tough problem. To direct train the ANN, we use the straight-through estimator (Bengio et al., 2013) for the derivative of the floor function, that is ${ \frac { \operatorname { d } \lfloor x \rfloor } { \operatorname { d } x } } = 1$ . The overall derivation rule is given in Equation 17.
+$$
+\frac { \partial \widehat { h } _ { i } ( z ^ { l } ) } { \partial z _ { i } ^ { l } } = \left\{ \begin{array} { l l } { 1 , \mathrm { ~ i f ~ } - \frac { \lambda ^ { l } } { 2 L } < z _ { i } ^ { l } < \lambda ^ { l } - \frac { \lambda ^ { l } } { 2 L } } \\ { 0 , \mathrm { ~ o t h e r w i s e } } \end{array} \right. , \frac { \partial \widehat { h } _ { i } ( z ^ { l } ) } { \partial \lambda ^ { l } }  &  = \left\{ \begin{array} { l l } { \frac { 1 } { 2 L } , \mathrm { ~ i f ~ } } & { - \frac { \lambda ^ { l } } { 2 L } < z _ { i } ^ { l } < \lambda ^ { l } - \frac { \lambda ^ { l } } { 2 L } } \\ { - \frac { z _ { i } ^ { l } } { ( \lambda ^ { l } ) ^ { 2 } } , \mathrm { ~ o t h e r w i s e } } \end{array} \right.
+$$
+Here $z _ { i } ^ { l }$ is the i-th element of $z ^ { l }$ . Then we can train the ANN with quantization clip-floor-shift activation using Stochastic Gradient Descent algorithm (Bottou, 2012).
+# 5 RELATED WORK
+The study of ANN-SNN conversion is first launched by Cao et al. (2015). Then Diehl et al. (2015) converted a three-layer CNN to an SNN using data-based and model-based normalization. To obtain high-performance SNNs for complex datasets and deeper networks, Rueckauer et al. (2016) and Sengupta et al. (2019) proposed more accurate scaling methods to normalize weights and scale thresholds respectively, which were later proved to be equivalent (Ding et al., 2021). Nevertheless, the converted deep SNN requires hundreds of time steps to get accurate results due to the conversion error analyzed in Sec. 3. To address the potential information loss, Rueckauer et al. (2016) and Han et al. (2020) suggested using “reset-by-subtraction” neurons rather than “reset-to-zero” neurons. Recently, many methods have been proposed to eliminate the conversion error. Rueckauer et al. (2016) recommended $9 9 . 9 \%$ percentile of activations as scale factors, and Ho & Chang (2020) added the trainable clipping layer. Besides, Han et al. (2020) rescaled the SNN thresholds to avoid the improper activation of spiking neurons. Massa et al. (2020) and Singh et al. (2021) evaluated the performance of converted SNNs on the Loihi Neuromorphic Processor. Our work share similarity with Deng & Gu (2020); Li et al. (2021), which also shed light on the conversion error. Deng & Gu (2020) minimized the layer-wise error by introducing extra bias in addition to the converted SNN biases. Li et al. (2021) further proposed calibration for weights and biases using quantized fine-tuning. They got good results with 16 and 32 time-steps without trails for more extreme time-steps. In comparison, our work aims to fit ANN into SNN with techniques eliminating the mentioned conversion error. The end-to-end training of quantization layers is implemented to get better overall performance. Our shift correction can lead to a single SNN which performs well at both ultra-low and large time-steps. Maintaining SNN performance within extremely few time-steps is difficult even for supervised learning methods like backpropagation through time (BPTT). BPTT usually requires fewer time-steps because of thorough training, yet at the cost of heavy GPU computation (Wu et al., 2018; 2019; Lee et al., 2016; Neftci et al., 2019; Lee et al., 2020; Zenke & Vogels, 2021). The timing-based backpropagation methods (Bohte et al., 2002; Tavanaei et al., 2019; Kim et al., 2020) could train SNNs over a very short temporal window, e.g. over 5-10 time-steps. However, they are usually limited to simple datasets like MNIST (Kheradpisheh & Masquelier, 2020) and CIFAR10 (Zhang & Li, 2020). Rathi et al. (2019) shortened simulation steps by initializing SNN with conversion method and then tuning SNN with STDP. In this paper, the proposed method achieves high-performance SNNs with ultra-low latency (4 time-steps).
+![](images/71f45fc75aa973a87550512c764093d0046e5d8cc9006d582b67d81062d4fda4.jpg)
+Figure 3: Compare ANNs accuracy.
+# 6 EXPERIMENTS
+In this section, we validate the effectiveness of our method and compare our method with other state-of-the-art approaches for image classification tasks on CIFAR-10 (LeCun et al., 1998), CIFAR100 (Krizhevsky et al., 2009), and ImageNet datasets (Deng et al., 2009). Similar to previous works, we utilize VGG-16 (Simonyan & Zisserman, 2014), ResNet-18 (He et al., 2016), and ResNet-20 network structures for source ANNs. We compare our method with the state-of-the-art ANN-SNN conversion methods, including Hybrid-Conversion (HC) from Rathi et al. (2019), RMP from Han et al. (2020), TSC from Han & Roy (2020), RNL from Ding et al. (2021), ReLUThresholdShift (RTS) from Deng & Gu (2020), and SNN Conversion with Advanced Pipeline (SNNC-AP) from Li et al. (2021). Comparison with different SNN training methods is also included to manifest the superiority of low latency inference, including HybridConversion-STDB (HC-STDB) from Rathi et al. (2019), STBP from Wu et al. (2018), DirectTraining (DT) from Wu et al. (2019), and TSSL from Zhang & Li (2020). The details of the proposed ANN-SNN algorithm and training configurations are provided in the Appendix.
+# 6.1 TEST ACCURACY OF ANN WITH QUANTIZATION CLIP-FLOOR-SHIFT ACTIVATION
+We first compare the performance of ANNs with quantization clip-floor activation (green curve), ANNs with quantization clip-floor-shift activation (blue curve), and original ANNs with ReLU activation (black dotted line). Figure 3(a)-(d) report the results about VGG-16 on CIFAR-10, ResNet-20 on CIFAR-10, VGG-16 on CIFAR-100 and ResNet-20 on CIFAR-100. The performance of ANNs with quantization clip-floor-shift activation is better than ANNs with quantization clip-floor activation. These two ANNs can achieve the same performance as original ANNs with ReLU activation when $L > 4$ . These results demonstrate that our quantization clip-floor-shift activation function hardly affects the performance of ANN.
+# 6.2 COMPARISON WITH THE STATE-OF-THE-ART
+Table 2 compares our method with the state-of-the-art ANN-SNN conversion methods on CIFAR10. As for low latency inference $\mathrm { ( T \leq 6 4 ) }$ ), our model outperforms all the other methods with the same time-step setting. For $\mathrm { T } = 3 2$ , the accuracy of our method is slightly better than that of ANN $( 9 5 . 5 4 \%$ vs. $9 5 . 5 2 \%$ ), whereas RMP, RTS, RNL, and SNNC-AP methods have accuracy loss of $3 3 . 3 \%$ , $1 9 . 4 8 \%$ , $7 . 4 2 \%$ , and $2 . 0 1 \%$ . Moreover, we achieve an accuracy of $9 3 . 9 6 \%$ using only 4 time-steps, which is 8 times faster than SNNC-AP that takes 32 time-steps. For ResNet-20, we achieve an accuracy of $8 3 . 7 5 \%$ with 4 time-steps. Notably, our ultra-low latency performance is comparable with other state-of-the-art supervised training methods, which is shown in Table S3 of the Appendix.
+![](images/95126425a463e43e70bdb2a808a89ade125d9a3014f499e05ecfb28a9caf2fdf.jpg)
+Figure 4: Compare quantization clip-floor activation with/without shift term
+Table 2: Comparison between the proposed method and previous works on CIFAR-10 dataset.
+<table><tr><td>Architecture</td><td>Method</td><td>ANN</td><td>T=2</td><td>T=4</td><td>T=8</td><td>T=16</td><td>T=32</td><td>T=64</td><td>T≥512</td></tr><tr><td rowspan="6">VGG-16</td><td>RMP</td><td>93.63%</td><td></td><td>-</td><td>二</td><td>-</td><td>60.30%</td><td>90.35%</td><td>93.63%</td></tr><tr><td>TSC</td><td>93.63%</td><td></td><td>二</td><td>-</td><td>-</td><td>-</td><td>92.79%</td><td>93.63%</td></tr><tr><td>RTS</td><td>95.72%</td><td></td><td>二</td><td>二</td><td>-</td><td>76.24%</td><td>90.64%</td><td>95.73%</td></tr><tr><td>RNL</td><td>92.82%</td><td>-</td><td>二</td><td>二</td><td>57.90%</td><td>85.40%</td><td>91.15%</td><td>92.95%</td></tr><tr><td>SNNC-AP</td><td>95.72%</td><td>-</td><td>-</td><td>-</td><td>-</td><td>93.71%</td><td>95.14%</td><td>95.79%</td></tr><tr><td>Ours</td><td>95.52%</td><td>91.18%</td><td>93.96%</td><td>94.95%</td><td>95.40%</td><td>95.54%</td><td>95.55%</td><td>95.59%</td></tr><tr><td rowspan="3">ResNet-20</td><td>RMP</td><td>91.47%</td><td>=</td><td>二</td><td></td><td>二</td><td>二</td><td></td><td>91.36%</td></tr><tr><td>TSC</td><td>91.47%</td><td>:</td><td>-</td><td>-</td><td>-</td><td>-</td><td>69.38%</td><td>91.42%</td></tr><tr><td>Ours</td><td>91.77%</td><td>73.20%</td><td>83.75%</td><td>89.55%</td><td>91.62%</td><td>92.24%</td><td>92.35%</td><td>92.41%</td></tr><tr><td rowspan="3">ResNet-18</td><td>RTS</td><td>95.46%</td><td>-</td><td>-</td><td>■</td><td>-</td><td>84.06%</td><td>92.48%</td><td>94.42%</td></tr><tr><td>SNNC-AP1</td><td>95.46%</td><td>-</td><td>■</td><td>-</td><td>-</td><td>94.78%</td><td>95.30%</td><td>95.45%</td></tr><tr><td>Ours</td><td>96.04%</td><td>75.44%</td><td>90.43%</td><td>94.82%</td><td>95.92%</td><td>96.08%</td><td>96.06%</td><td>96.06%</td></tr></table>
+1 RTS and SNNC-AP use altered ResNet-18, while ours use standard ResNet-18.
+We further test the performance of our method on the large-scale dataset. Table 3 reports the results on ImageNet, our method also outperforms the others both in terms of high accuracy and ultra-low latency. For ResNet-34, the accuracy of the proposed method is $4 . 8 3 \%$ higher than SNNC-AP and $6 9 . 2 8 \%$ higher than RTS when $T = 3 2$ . When the time-steps is 16, we can still achieve an accuracy of $5 9 . 3 5 \%$ . For VGG-16, the accuracy of the proposed method is $4 . 8 3 \%$ higher than SNNC-AP and $6 8 . 3 5 6 \%$ higher than RTS when $T = 3 2$ . When the time-steps is 16, we can still achieve an accuracy of $5 0 . 9 7 \%$ . These results demonstrate that our method outperforms the previous conversion methods. More experimental results on CIFAR-100 is in Table S4 of the Appendix.
+# 6.3 COMPARISON OF QUANTIZATION CLIP-FLOOR AND QUANTIZATION CLIP-FLOOR-SHIFT
+Here we further compare the performance of SNNs converted from ANNs with quantization clipfloor activation and ANN with quantization clip-floor-shift activation. In Sec. 4, we prove that the expectation of the conversion error reaches 0 with quantization clip-floor-shift activation, no matter whether $T$ and $L$ are the same or not. To verify these, we set $L$ to 4 and train ANNs with quantization clip-floor activation and quantization clip-floor-shift activation, respectively. Figure 4 shows how the accuracy of converted SNNs changes with respect to the time-steps $T$ . The accuracy of the converted SNN (green curve) from ANN with quantization clip-floor activation (green dotted line) first increases and then decreases rapidly with the increase of time-steps, because we cannot guarantee that the conversion error is zero when $T$ is not equal to $L$ . The best performance is still lower than source ANN (green dotted line). In contrast, the accuracy of the converted SNN from ANN with quantization clip-floor-shift activation (blue curve) increases with the increase of $T$ . It gets the same accuracy as source ANN (blue dotted line) when the time-steps is larger than 16.
+# 6.4 EFFECT OF QUANTIZATION STEPS L
+In our method, the quantization steps $L$ is a hyperparameter, which affects the accuracy of the converted SNN. To analyze the effect of $L$ and better determine the optimal value, we train VGG16/ResNet-20 networks with quantization clip-floor-shift activation using different quantization steps L, including 2,4,8,16 and 32, and then converted them to SNNs. The experimental results on CIFAR-10/100 dataset are shown in Table S2 and Figure 5, where the black dotted line denotes the ANN accuracy and the colored curves represent the accuracy of the converted SNN. In order to balance the trade-off between low latency and high accuracy, we evaluate the performance of converted SNN mainly in two aspects. First, we focus on the SNN accuracy at ultra-low latency (within 4 time-steps). Second, we consider the best accuracy of SNN. It is obvious to find that the SNN accuracy at ultra-low latency decreases as $L$ increases. However, a too small $L$ will decrease the model capacity and further lead to accuracy loss. When $L = 2$ , there exists a clear gap between the best accuracy of SNN and source ANN. The best accuracy of SNN approaches source ANN when $L > 4$ . In conclusion, the setting of parameter $L$ mainly depends on the aims for low latency or best accuracy. The recommend quantization step $L$ is 4 or 8, which leads to high-performance converted SNN at both small time-steps and very large time-steps.
+![](images/a0e15e069bb677f5dc4c944811f1eefdec45882c3bc1ae8c1dc7fb8a5b3c2dba.jpg)
+Figure 5: Influence of different quantization steps
+Table 3: Comparison between the proposed method and previous works on ImageNet dataset.
+<table><tr><td>Architecture</td><td>Method</td><td>ANN</td><td>T=16</td><td>T=32</td><td>T=64</td><td>T=128</td><td>T=256</td><td>T≥1024</td></tr><tr><td rowspan="5">ResNet-34</td><td>RMP</td><td>70.64%</td><td>-</td><td>二</td><td>二</td><td>二</td><td>-</td><td>65.47%</td></tr><tr><td>TSC</td><td>70.64%</td><td>-</td><td>-</td><td>:</td><td>-</td><td>61.48%</td><td>65.10%</td></tr><tr><td>RTS</td><td>75.66%</td><td>-</td><td>0.09%</td><td>0.12%</td><td>3.19%</td><td>47.11%</td><td>75.08%</td></tr><tr><td>SNNC-AP</td><td>75.66%</td><td>-</td><td>64.54%</td><td>71.12%</td><td>73.45%</td><td>74.61%</td><td>75.45%</td></tr><tr><td>Ours</td><td>74.32%</td><td>59.35%</td><td>69.37%</td><td>72.35%</td><td>73.15%</td><td>73.37%</td><td>73.39%</td></tr><tr><td rowspan="5">VGG-16</td><td>RMP</td><td>73.49%</td><td>-</td><td>-</td><td>-</td><td>-</td><td>48.32%</td><td>73.09%</td></tr><tr><td>TSC</td><td>73.49%</td><td>1</td><td>-</td><td>-</td><td>-</td><td>69.71%</td><td>73.46%</td></tr><tr><td>RTS</td><td>75.36%</td><td>1</td><td>0.114%</td><td>0.118%</td><td>0.122%</td><td>1.81%</td><td>73.88%</td></tr><tr><td>SNNC-AP</td><td>75.36%</td><td>-</td><td>63.64%</td><td>70.69%</td><td>73.32%</td><td>74.23%</td><td>75.32%</td></tr><tr><td>Ours</td><td>74.29%</td><td>50.97%</td><td>68.47%</td><td>72.85%</td><td>73.97%</td><td>74.22%</td><td>74.32%</td></tr></table>
+# 7 DISCUSSION AND CONCLUSION
+In this paper, we present ANN-SNN conversion method, enabling high-accuracy and ultra-lowlatency deep SNNs. We propose the quantization clip-floor-shift activation to replace ReLU activation, which hardly affects the performance of ANNs and is closer to SNNs activation. Furthermore, we prove that the expected conversion error is zero, no matter whether the time-steps of SNNs and the quantization steps of ANNs is the same or not. We achieve state-of-the-art accuracy with fewer time-steps on CIFAR-10, CIFAR-100, and ImageNet datasets. Our results can benefit the implementations on neuromorphic hardware and pave the way for the large-scale application of SNNs.
+Different from the work of Deng & Gu (2020), which adds the bias of the converted SNNs to shift the theoretical ANN-SNN curve to minimize the quantization error, we add the shift term in the quantization clip-floor activation function, and use this quantization clip-floor-shift function to train the source ANN. We show that the shift term can overcome the performance degradation problem when the time-steps and the quantization steps are not matched. Due to the unevenness error, there still exists a gap between ANN accuracy and SNN accuracy, even when $L = T$ . Moreover, it is hard to achieve high-performance ANN-SNN conversion when the time-steps $T = 1$ . All these problems deserve further research. One advantage of conversion-based methods is that they can reduce the overall computing cost while maintaining comparable performance as source ANN. Combining the conversion-based methods and model compression may help significantly reduce the neuron activity and thus reduce energy consumptions without suffering from accuracy loss (Kundu et al., 2021; Rathi & Roy, 2021), which is a promising direction.
+# ACKNOWLEDGEMENT
+This work was supported by the National Natural Science Foundation of China under contracts No.62176003 and No.62088102.
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+# A APPENDIX
+# A.1 NETWORK STRUCTURE AND TRAINING CONFIGURATIONS
+Before training ANNs, we first replace max-pooling with average-pooling and then replace the ReLU activation with the proposed quantization clip-floor-shift activation (Equation 15). After training, we copy all weights from the source ANN to the converted SNN, and set the threshold $\theta ^ { l }$ in each layer of the converted SNN equal to the maximum activation value $\lambda ^ { l }$ of the source ANN in the same layer. Besides, we set the initial membrane potential $\pmb { v } ^ { l } ( 0 )$ in converted SNN as $\theta ^ { l } / 2$ to match the optimal shift $\textstyle \varphi = { \frac { 1 } { 2 } }$ of quantization clip-floor-shift activation in the source ANN.
+Despite the common data normalization, we use some data pre-processing techniques. For CIFAR datasets, we resize the images into $3 2 \times 3 2$ , and for ImageNet dataset, we resize the image into $2 2 4 \times 2 2 4$ . Besides, we use random crop images, Cutout (DeVries & Taylor, 2017) and AutoAugment (Cubuk et al., 2019) for all datasets.
+We use the Stochastic Gradient Descent optimizer (Bottou, 2012) with a momentum parameter of 0.9. The initial learning rate is set to 0.1 for CIFAR-10 and ImageNet, and 0.02 for CIFAR-100. A cosine decay scheduler (Loshchilov & Hutter, 2016) is used to adjust the learning rate. We apply a $5 \times 1 0 ^ { - 4 }$ weight decay for CIFAR datasets while applying a $1 \times \mathrm { \bar { 1 0 } ^ { - 4 } }$ weight decay for ImageNet. We train all models for 300 epochs. The quantization steps $L$ is set to 4 when training all the networks on CIFAR-10, and VGG-16, ResNet-18 on CIFAR-100 dataset. When training ResNet-20 on CIFAR-100, the parameter $L$ is set to 8. When training ResNet-34 and VGG-16 on ImageNet, the parameter $L$ is set to 8, 16, respectively. We use constant input when evaluating the converted SNNs.
+# A.2 INTRODUCTION OF DATASETS
+CIFAR-10. The CIFAR-10 dataset (Krizhevsky et al., 2009) consists of $6 0 0 0 0 3 2 \times 3 2$ images in 10 classes. There are 50000 training images and 10000 test images.
+CIFAR-100. The CIFAR-100 dataset (Krizhevsky et al., 2009) consists of $6 0 0 0 0 3 2 \times 3 2$ images in 100 classes. There are 50000 training images and 10000 test images.
+ImageNet. We use the ILSVRC 2012 dataset (Russakovsky et al., 2015), which consists 1,281,167 training images and 50000 testing images.
+# A.3 DERIVATION OF EQUATION 12 AND PROOF OF THEOREM 2
+# Derivation of Equation 11
+Similar to $z ^ { l } = W ^ { l } \mathbf { a } ^ { l - 1 }$ , We define
+$$
+\pmb { u } ^ { l } ( t ) = \pmb { W } ^ { l } \pmb { x } ^ { l - 1 } ( t ) .
+$$
+We use $u _ { i } ^ { l } ( t )$ and $z _ { i } ^ { l }$ to denote the $i$ -th element in vector ${ \pmb u } ^ { l } ( t )$ and $z ^ { l }$ , respectively. To derive Equation 11, some extra assumptions on the relationship between ANN activation value and SNN postsynaptic potentials are needed, which are showed in Equation S2.
+$$
+\left\{ \begin{array} { l l } & { \mathrm { i f ~ } z _ { i } ^ { l } < 0 , \mathrm { ~ t h e n } \forall t u _ { i } ^ { l } ( t ) < 0 , } \\ & { \mathrm { i f ~ } 0 \leqslant z _ { i } ^ { l } \leqslant \theta _ { l } , \mathrm { ~ t h e n } \forall t 0 \leqslant u _ { i } ^ { l } ( t ) \leqslant \theta _ { l } , } \\ & { \mathrm { i f ~ } z _ { i } ^ { l } > \theta _ { l } , \mathrm { ~ t h e n } \forall t u _ { i } ^ { l } ( t ) > \theta _ { l } . } \end{array} \right.
+$$
+With the assumption above, we can discuss the firing behavior of the neurons in each time-step. When $z _ { i } ^ { l } < 0$ or $\overline { { z _ { i } ^ { l } } } > \theta _ { l }$ , the neuron will never fire or fire all the time-steps, which means $\phi _ { i } ^ { l } ( T ) = \bar { 0 }$ or $\phi _ { i } ^ { l } ( T ) = \theta ^ { l }$ . In this situation, we can use a clip function to denote $\phi _ { i } ^ { l } ( T )$ .
+$$
+\phi _ { i } ^ { l } ( T ) = \mathrm { c l i p } ( z _ { i } ^ { l } , 0 , \theta ^ { l } ) .
+$$
+When $0 < z _ { i } ^ { l } < \theta _ { l }$ , every input from the presynaptic neuron in SNNs falls into $[ 0 , \theta ^ { l } ]$ , then we have $\forall t$ , $v _ { i } ^ { l } ( t ) \in [ 0 , \theta ]$ . We can rewrite Equation 8 into the following equation.
+$$
+\frac { \phi _ { i } ^ { l } ( T ) T } { \theta ^ { l } } = \frac { z _ { i } ^ { l } T + v _ { i } ^ { l } ( 0 ) } { \theta ^ { l } } - \frac { v _ { i } ^ { l } ( T ) } { \theta ^ { l } } .
+$$
+Considering that $\begin{array} { r } { \frac { \phi _ { i } ^ { l } ( T ) T } { \theta ^ { l } } = \sum _ { t = 1 } ^ { T } s _ { i } ^ { l } ( t ) \in \mathbb { N } } \end{array}$ and $\begin{array} { r } { 0 < \frac { v _ { i } ^ { l } ( T ) } { \theta ^ { l } } < 1 } \end{array}$ , Equation S4 is changed to:
+$$
+\phi _ { i } ^ { l } ( T ) = \frac { \theta ^ { l } } { T } \left\lfloor \frac { z _ { i } ^ { l } T + v _ { i } ^ { l } ( 0 ) } { \theta ^ { l } } \right\rfloor .
+$$
+We combine these two situations (Equation S3 and Equation S4), and we have:
+$$
+\phi ^ { l } ( T ) = \theta ^ { l } \mathrm { c l i p } \left( \frac { 1 } { T } \left\lfloor \frac { z ^ { l } T + v ^ { l } ( 0 ) } { \theta ^ { l } } \right\rfloor , 0 , 1 \right) .
+$$
+# Proof of Theorem 2
+Before prove Theorem 2, we first introduce Lemma 1.
+Lemma 1. If random variable $x \in [ 0 , \theta ]$ is uniformly distributed in every small interval $[ m _ { t } , m _ { t + 1 } ]$ with the probability density function $p _ { t }$ $( t = 0 , 1 , . . . , T )$ , where $\begin{array} { r } { m _ { 0 } = 0 , m _ { T + 1 } = \theta , m _ { t } = \frac { ( t - \frac { 1 } { 2 } ) \theta } { T } } \end{array}$ for $t = 1 , 2 , . . . , T$ , $p _ { 0 } = p _ { T }$ , we can conclude that
+$$
+\mathbb { E } _ { x } \left( x - { \frac { \theta } { T } } \left\lfloor { \frac { T x } { \theta } } + { \frac { 1 } { 2 } } \right\rfloor \right) = 0 .
+$$
+Proof.
+$$
+\begin{array} { l } { { \mathbb { E } _ { x } \left( x - \displaystyle \frac { \theta } { T } \left\lfloor \frac { T x } { \theta } + \frac { 1 } { 2 } \right\rfloor \right) = \int _ { 0 } ^ { \theta / 2 T } p _ { 0 } \left( x - \displaystyle \frac { \theta } { T } \left\lfloor \frac { x T } { \theta } + \frac { 1 } { 2 } \right\rfloor \right) \mathrm { d } x } } \\ { { \displaystyle + \sum _ { t = 1 } ^ { T - 1 } \int _ { ( 2 t - 1 ) \theta / 2 T } ^ { ( 2 t + 1 ) \theta / 2 T } p _ { t } \left( x - \displaystyle \frac { \theta } { T } \left\lfloor \frac { x T } { \theta } + \frac { 1 } { 2 } \right\rfloor \right) \mathrm { d } x } } \\ { { \displaystyle + \int _ { ( 2 T - 1 ) \theta / 2 T } ^ { \theta } p _ { T } \left( x - \displaystyle \frac { \theta } { T } \left\lfloor \frac { x T } { \theta } + \frac { 1 } { 2 } \right\rfloor \right) \mathrm { d } x } } \\ { { \displaystyle = p _ { 0 } \int _ { 0 } ^ { \theta / 2 T } x \mathrm { d } x + \sum _ { t = 1 } ^ { T - 1 } p _ { t } \int _ { ( 2 t - 1 ) \theta / 2 T } ^ { ( 2 t + 1 ) \theta / 2 T } \left( x - \displaystyle \frac { t \theta } { T } \right) \mathrm { d } x + p _ { T } \int _ { ( 2 T - 1 ) \theta / 2 T } ^ { \theta } \left( x - \theta \right) \mathrm { d } x } } \\ { { \displaystyle = p _ { 0 } \displaystyle \frac { \theta ^ { 2 } } { 8 \pi ^ { 2 } } + 0 - p _ { T } \displaystyle \frac { \theta ^ { 2 } } { 8 \pi T ^ { 2 } } = ( p _ { 0 } - p _ { T } ) \displaystyle \frac { \theta ^ { 2 } } { 8 \pi T ^ { 2 } } = 0 . } } \end{array}
+$$
+Theorem 2. An ANN with activation function (15) is converted to an SNN with the same weights. If $\theta ^ { l } = \lambda ^ { l }$ , ${ \pmb v } ^ { l } ( 0 ) = \theta ^ { l } { \pmb \varphi } ,$ , then for arbitrary $T$ and $L$ , the expectation of conversion error reaches 0 when the shift term $\varphi$ in source ANN is $\frac { \mathbf { 1 } } { \mathbf { 2 } }$ .
+$$
+\forall T , L \quad \mathbb { E } _ { z } \left( \widetilde { E r r } ^ { l } \right) \Big | _ { \varphi = \frac { 1 } { 2 } } = \mathbf { 0 } .
+$$
+Proof.
+$$
+\mathbb { E } _ { z } \left( \widetilde { E r r } ^ { l } \right) \Big | _ { \varphi = \frac { 1 } { 2 } } = \mathbb { E } _ { z } \left( \frac { \theta ^ { l } } { T } \left\lfloor \frac { z ^ { l } T + v ^ { l } ( 0 ) } { \theta ^ { l } } \right\rfloor - \frac { \lambda ^ { l } } { L } \left\lfloor \frac { z ^ { l } L } { \lambda } + \varphi \right\rfloor \right) .
+$$
+![](images/7cf3861caece577400098d510676308fbed5f338860acbffca1d7b84dcf79a1d.jpg)
+Figure S1: More spikes than expected exists for the method of setting the maximum activation.
+As every element in vector $_ { z }$ is identical, we only need to consider one element.
+$$
+\begin{array} { r l } & { \mathbb { E } _ { z _ { i } } \left( \displaystyle \frac { \theta ^ { l } } { T } \left\lfloor \displaystyle \frac { z _ { i } ^ { l } T + v _ { i } ^ { l } ( 0 ) } { \theta ^ { l } } \right\rfloor - \displaystyle \frac { \lambda ^ { l } } { L } \left\lfloor \displaystyle \frac { z _ { i } ^ { l } L } { \lambda } + \varphi _ { i } \right\rfloor \right) } \\ & { = \mathbb { E } _ { z _ { i } } \left( \displaystyle \frac { \theta ^ { l } } { T } \left\lfloor \displaystyle \frac { z _ { i } ^ { l } T + v _ { i } ^ { l } ( 0 ) } { \theta ^ { l } } \right\rfloor - z _ { i } ^ { l } \right) + \mathbb { E } _ { z _ { i } } \left( z _ { i } ^ { l } - \displaystyle \frac { \lambda ^ { l } } { L } \left\lfloor \displaystyle \frac { z _ { i } ^ { l } L } { \lambda } + \varphi _ { i } \right\rfloor \right) . } \end{array}
+$$
+According to Lemma 1, we have
+$$
+\begin{array} { r l } & { \mathbb { E } _ { z _ { i } } \left( \frac { \theta ^ { l } } { T } \left\lfloor \frac { z _ { i } ^ { l } T + v _ { i } ^ { l } ( 0 ) } { \theta ^ { l } } \right\rfloor - z _ { i } ^ { l } \right) \Big | _ { v _ { i } ^ { l } ( 0 ) = 1 / 2 } = 0 , } \\ & { \mathbb { E } _ { z _ { i } } \left( z _ { i } ^ { l } - \frac { \lambda ^ { l } } { L } \left\lfloor \frac { z _ { i } ^ { l } L } { \lambda } + \varphi _ { i } \right\rfloor \right) \Big | _ { \varphi = 1 / 2 } = 0 . } \end{array}
+$$
+Thus the sum of both terms also equals zero.
+# A.4 COMPARISON OF THE METHODS WITH OR WITHOUT DYNAMIC THRESHOLD ON THE CIFAR-100 DATASET
+In this paper we use a training parameter $\lambda ^ { l }$ to decide the maximum value of ANN activation. The previous works suggested to set the maximum value of ANN activation after training as the threshold. If we set $\theta ^ { l } = \operatorname* { m a x } _ { s \in \{ 0 , 1 \} ^ { n } }$ $\left( \operatorname* { m a x } ( \theta ^ { l - 1 } W ^ { l } s ) \right)$ , the situation of fewer spikes as expected never happens, as we can prove that $v ^ { l } ( T ) < \theta ^ { l }$ (see Theorem 3). Despite this, there still exists the situation of more spikes as expected. An example is given in Figure S1. Here we consider the same example as in Figure 1. In source ANN, we suppose that two analog neurons in layer $l - 1$ are connected to an analog neuron in layer $l$ with weights 2 and $^ { - 2 }$ , and the output vector $\mathbf { a } ^ { l - 1 }$ of neurons in layer $l - 1$ is [0.6, 0.4]. Besides, in converted SNN, we suppose that the two spiking neurons in layer $l - 1$ fire 3 spikes and 2 spikes in 5 time-steps $( \mathrm { T } { = } 5 )$ , respectively, and the threshold $\theta ^ { l - 1 } = 1$ . Thus, $\begin{array} { r } { \phi ^ { l - 1 } ( T ) \stackrel { - } { = } \frac { \sum _ { i = 1 } ^ { T } s ^ { l - \bar { 1 } } ( i ) } { T } \theta ^ { l - 1 } = [ 0 . 6 , 0 . 4 ] } \end{array}$ ]. According to Equation 1, the ANN output $\pmb { a } ^ { l } = \pmb { W } ^ { l } \pmb { a } ^ { l - 1 } = [ 2 , - 2 ] [ 0 . \bar { 6 } , 0 . 4 ] ^ { T } = 0 . 4$ . As for SNN, we suppose that the presynaptic neurons fires at $t = 1 , 2 , 3$ and $t = 4 , 5$ , respectively. Even through we set the threshold $\mathbf { \dot { \theta } } ^ { l } = 1$ to the maximum activation 2, the postsynaptic neuron will fire three spikes at $t = 1 , 2 , 3$ , and $\phi ^ { l } ( T ) = 0 . 6 > a ^ { l }$ .
+Besides, setting $\operatorname* { m a x } _ { s \in \{ 0 , 1 \} ^ { n } }$ $\left( \operatorname* { m a x } ( \theta ^ { l - 1 } W ^ { l } s ) \right)$ as the threshold brings two other problems. First, the spiking neurons will take a long time to fire spikes because of the large value of the threshold, which makes it hard to maintain SNN performance within a few time-steps. Second, the quantization error will be large as it is proportional to the threshold. If the conversion error is not zero for one layer, it will propagate layer by layer and will be magnified by larger quantization errors. We compare our method and the method of setting the maximum activation on the CIFAR-100 dataset. The results are reported in Table S1, where DT represents the dynamic threshold in our method. The results show that our method can achieve better performance.
+Table S1: Comparison between our method and the method of setting the maximum activation.
+<table><tr><td></td><td>DT1w/o shift</td><td>T=4</td><td>T=8</td><td>T=16</td><td>T=32</td><td>T=64</td><td>T=128</td><td>T=256</td><td>T≥512</td></tr><tr><td></td><td colspan="9">VGG-16 on CIFAR-100 with L=4</td></tr><tr><td>√</td><td>√</td><td>69.62%</td><td>73.96%</td><td>76.24%</td><td>77.01%</td><td>77.10%</td><td>77.05%</td><td>77.08%</td><td>77.08%</td></tr><tr><td></td><td>&lt;×</td><td>21.57%</td><td>41.13%</td><td>58.92%</td><td>65.38%</td><td>64.19%</td><td>58.60%</td><td>52.99%</td><td>49.41%</td></tr><tr><td>×√</td><td></td><td>1.00%</td><td>0.96%</td><td>1.00%</td><td>1.10%</td><td>2.41%</td><td>13.76%</td><td>51.70%</td><td>77.10%</td></tr><tr><td>×</td><td>×</td><td>1.00%</td><td>1.00%</td><td>0.90%</td><td>1.00%</td><td>1.01%</td><td>2.01%</td><td>19.59%</td><td>70.86%</td></tr></table>
+1 Dynamic threshold.
+Theorem 3. If the threshold is set to the maximum value of ANN activation, that is $\theta ^ { l } \ =$ $\operatorname* { m a x } _ { s \in \{ 0 , 1 \} ^ { n } }$ $\left( \operatorname* { m a x } ( \theta ^ { l - 1 } W ^ { l } s ) \right)$ , and $v _ { i } ^ { l } ( 0 ) < \theta ^ { l }$ . Then at any time-step, the membrane potential of each neuron after spike $v _ { i } ^ { l } ( t )$ will be less than $\theta ^ { l }$ , where $i$ represents the index of each neuron.
+Proof. We prove it by induction. For $t = 0$ , it is easy to see $v _ { i } ^ { l } ( 0 ) < \theta ^ { l }$ . For $t > 0$ , we suppose that $v _ { i } ^ { l } ( t - 1 ) < \theta ^ { l }$ . Since we have set the threshold to the maximum possible input, and $x _ { i } ^ { l - 1 } ( t )$ represents the input from layer $l - 1$ to the $i$ -th neuron in layer $l$ , $x _ { i } ^ { l - 1 } ( t )$ will be no larger than $\theta ^ { l }$ for arbitrary $t$ . Thus we have
+$$
+\begin{array} { l } { { m _ { i } ^ { l } ( t ) = v _ { i } ^ { l } ( t - 1 ) + x _ { i } ^ { l - 1 } ( t ) < \theta ^ { l } + \theta ^ { l } = 2 \theta ^ { l } , } } \\ { { s _ { i } ^ { l } ( t ) = H ( m _ { i } ^ { l } ( t ) - \theta ^ { l } ) , } } \\ { { v _ { i } ^ { l } ( t ) = m _ { i } ^ { l } ( t ) - s _ { i } ^ { l } ( t ) \theta ^ { l } . } } \end{array}
+$$
+If $\theta ^ { l } \leqslant m _ { i } ^ { l } ( t ) < 2 \theta ^ { l }$ , then we have $v _ { i } ^ { l } ( t ) = m _ { i } ^ { l } ( t ) - \theta ^ { l } < \theta ^ { l }$ . If $m _ { i } ^ { l } ( t ) < \theta _ { l }$ , then $v _ { i } ^ { l } ( t ) = m _ { i } ^ { l } ( t ) < \theta _ { l }$ . By mathematical induction, $v _ { i } ^ { l } ( t ) < \mathsf { \bar { \theta } } ^ { l }$ holds for any $t \geqslant 0$ . □
+# A.5 EFFECT OF QUANTIZATION STEPS L
+Table S2 reports the performance of converted SNNs with different quantization steps $L$ and different time-steps $T$ . For VGG-16 and quantization steps $L = 2$ , we achieve an accuracy of $8 6 . 5 3 \%$ on CIFAR-10 dataset and an accuracy of $6 1 . 4 1 \%$ on CIFAR-100 dataset with 1 time-steps. When the quantization steps $L = 1$ , we cannot train the source ANN.
+# A.6 COMPARISON WITH STATE-OF-THE-ART SUPERVISED TRAINING METHODS ON CIFAR-10 DATASET
+Notably, our ultra-low latency performance is comparable with other state-of-the-art supervised training methods. Table S3 reports the results of hybrid training and backpropagation methods on CIFAR-10. The backpropagation methods require sufficient time-steps to convey discriminate information. Thus, the list methods need at least 5 time-steps to achieve ${ \sim } 9 1 \%$ accuracy. On the contrary, our method can achieve $9 4 . 7 3 \%$ accuracy with 4 time-steps. Besides, the hybrid training method requires 200 time-steps to obtain $9 2 . 0 2 \%$ accuracy because of further training with STDB, whereas our method achieves $9 3 . 9 6 \%$ accuracy with 4 time-steps.
+# A.7 COMPARISON ON CIFAR-100 DATASET
+Table S4 reports the results on CIFAR-100, our method also outperforms the others both in terms of high accuracy and ultra-low latency. For VGG-16, the accuracy of the proposed method is $3 . 4 6 \%$ higher than SNNC-AP and $6 9 . 3 7 \%$ higher than RTS when $T = 3 2$ . When the time-steps is only 4, we can still achieve an accuracy of $6 9 . 6 2 \%$ . These results demonstrate that our method outperforms the previous conversion methods.
+Table S2: Influence of different quantization steps.
+<table><tr><td rowspan="2">quantization steps</td><td colspan="8"></td></tr><tr><td>T=1</td><td>T=2</td><td>T=4</td><td>T=8</td><td>T=16</td><td>T=32</td><td>T=64</td><td>T=128</td></tr><tr><td colspan="9">VGG-16 on CIFAR-10</td></tr><tr><td>L=2</td><td>86.53%</td><td>91.98%</td><td>93.00%</td><td>93.95%</td><td>94.18%</td><td>94.22%</td><td>94.18%</td><td>94.14%</td></tr><tr><td>L=4</td><td>88.41%</td><td>91.18%</td><td>93.96%</td><td>94.95%</td><td>95.40%</td><td>95.54%</td><td>95.55%</td><td>95.59%</td></tr><tr><td>L=8</td><td>62.89%</td><td>83.93%</td><td>91.77%</td><td>94.45%</td><td>95.22%</td><td>95.56%</td><td>95.74%</td><td>95.79%</td></tr><tr><td>L=16</td><td>61.48%</td><td>76.76%</td><td>89.61%</td><td>93.03%</td><td>93.95%</td><td>94.24%</td><td>94.25%</td><td>94.22%</td></tr><tr><td>L=32</td><td>13.05%</td><td>73.33%</td><td>89.67%</td><td>94.13%</td><td>95.31%</td><td>95.66%</td><td>95.73%</td><td>95.77%</td></tr><tr><td colspan="9">ResNet-20 on CIFAR-10</td></tr><tr><td>L=2</td><td>77.54%</td><td>82.12%</td><td>85.77%</td><td>88.04%</td><td>88.64%</td><td>88.79%</td><td>88.85%</td><td>88.76%</td></tr><tr><td>L=4</td><td>62.43%</td><td>73.2%</td><td>83.75%</td><td>89.55%</td><td>91.62%</td><td>92.24%</td><td>92.35%</td><td>92.35%</td></tr><tr><td>L=8</td><td>46.19%</td><td>58.67%</td><td>75.70%</td><td>87.79%</td><td>92.14%</td><td>93.04%</td><td>93.34%</td><td>93.24%</td></tr><tr><td>L=16</td><td>30.96%</td><td>39.87%</td><td>57.04%</td><td>79.5%</td><td>90.87%</td><td>93.25%</td><td>93.44%</td><td>93.48%</td></tr><tr><td>L=32</td><td>22.15%</td><td>27.83%</td><td>43.56%</td><td>70.15%</td><td>88.81%</td><td>92.97%</td><td>93.48%</td><td>93.48%</td></tr><tr><td colspan="9">VGG-16 on CIFAR-100</td></tr><tr><td>L=2</td><td>61.41%</td><td>64.96%</td><td>68.0%</td><td>70.72%</td><td>71.87%</td><td>72.28%</td><td>72.35%</td><td>72.4%</td></tr><tr><td>L=4</td><td>57.5%</td><td>63.79%</td><td>69.62%</td><td>73.96%</td><td>76.24%</td><td>77.01%</td><td>77.1%</td><td>77.05%</td></tr><tr><td>L=8</td><td>44.98%</td><td>52.46%</td><td>62.09%</td><td>70.71%</td><td>74.83%</td><td>76.41%</td><td>76.73%</td><td>76.73%</td></tr><tr><td>L=16</td><td>33.12%</td><td>41.71%</td><td>53.38%</td><td>65.76%</td><td>72.80%</td><td>75.6%</td><td>76.37%</td><td>76.36%</td></tr><tr><td>L=32</td><td>15.18%</td><td>21.41%</td><td>32.21%</td><td>50.46%</td><td>67.32%</td><td>74.6%</td><td>76.18%</td><td>76.24%</td></tr><tr><td colspan="9">ResNet-20 on CIFAR-100</td></tr><tr><td>L=2</td><td>38.65%</td><td>47.35%</td><td>55.23%</td><td>59.69%</td><td>61.29%</td><td>61.5%</td><td>61.03%</td><td>60.81%</td></tr><tr><td>L=4</td><td>25.62%</td><td>36.33%</td><td>51.55%</td><td>63.14%</td><td>66.70%</td><td>67.47%</td><td>67.47%</td><td>67.41%</td></tr><tr><td>L=8</td><td>13.19%</td><td>19.96%</td><td>34.14%</td><td>55.37%</td><td>67.33%</td><td>69.82%</td><td>70.49%</td><td>70.55%</td></tr><tr><td>L=16</td><td>6.09%</td><td>9.25%</td><td>17.48%</td><td>38.22%</td><td>60.92%</td><td>68.70%</td><td>70.15%</td><td>70.20%</td></tr><tr><td>L=32</td><td>5.44%</td><td>7.41%</td><td>13.36%</td><td>31.66%</td><td>58.68%</td><td>68.12%</td><td>70.12%</td><td>70.27%</td></tr></table>
+# A.8 ENERGY CONSUMPTION ANALYSIS
+We evaluate the energy consumption of our method and the compared methods (Li et al., 2021; Deng & Gu, 2020) on CIFAR-100 datasets. Here we use the same network structure of VGG16. Following the analysis in Merolla et al. (2014), we use synaptic operation (SOP) for SNN to represent the required basic operation numbers to classify one image. We utilize 77fJ/SOP for SNN and 12.5pJ/FLOP for ANN as the power consumption baseline, which is reported from the ROLLS neuromorphic processor (Qiao et al., 2015). Note that we do not consider the memory access energy in our study because it depends on the hardware. As shown in Table S5, when the time-steps is the same, the energy consumption of our method is about two times of SNNC-AP. However, to achieve the same accuracy of $7 3 . 5 5 \%$ , our method requires less energy consumption.
+# A.9 PSEUDO-CODE FOR OVERALL CONVERSION ALGORITHM
+In this section, we summarize the entire conversion process in Algorithm 1, including training ANNs from scratch and converting ANNs to SNNs. The QCFS in the pseudo-code represents the proposed quantization clip-floor-shift function.
+Table S3: Compare with state-of-the-art supervised training methods on CIFAR-10 dataset
+<table><tr><td>Model</td><td>Method</td><td>Architecture</td><td> SNN Accuracy</td><td>Timesteps</td></tr><tr><td></td><td colspan="4">CIFAR-10</td></tr><tr><td>HC</td><td>Hybrid</td><td>VGG-16</td><td>92.02</td><td>200</td></tr><tr><td>STBP</td><td>Backprop</td><td>CIFARNet</td><td>90.53</td><td>12</td></tr><tr><td>DT</td><td>Backprop</td><td>CIFARNet</td><td>90.98</td><td>8</td></tr><tr><td>TSSL</td><td>Backprop</td><td>CIFARNet</td><td>91.41</td><td>5</td></tr><tr><td>DThIR1</td><td>ANN-SNN</td><td>cNet</td><td>77.10</td><td>256</td></tr><tr><td>Ours</td><td>ANN-SNN</td><td>VGG-16</td><td>93.96</td><td>4</td></tr><tr><td>Ours</td><td>ANN-SNN</td><td>CIFARNet2</td><td>94.73</td><td>4</td></tr></table>
+1 Implemented on Loihi neuromorphic processor 2 For CIFARNet, we use the same architecture as Wu et al. (2018).
+Table S4: Comparison between the proposed method and previous works on CIFAR-100 dataset.
+<table><tr><td>Architecture</td><td>Method</td><td>ANN</td><td>T=2</td><td>T=4</td><td>T=8</td><td>T=16</td><td>T=32</td><td>T=64</td><td>T≥512</td></tr><tr><td rowspan="5">VGG-16</td><td>RMP</td><td>71.22%</td><td>-</td><td>-</td><td>-</td><td>二</td><td>-</td><td>-</td><td>70.93%</td></tr><tr><td>TSC</td><td>71.22%</td><td>-</td><td>1</td><td>二</td><td>二</td><td>-</td><td>-</td><td>70.97%</td></tr><tr><td>RTS</td><td>77.89%</td><td>1</td><td>二</td><td>二</td><td>二</td><td>7.64%</td><td>21.84%</td><td>77.71%</td></tr><tr><td>SNNC-AP</td><td>77.89%</td><td>1</td><td>-</td><td>-</td><td>-</td><td>73.55%</td><td>76.64%</td><td>77.87%</td></tr><tr><td>Ours</td><td>76.28%</td><td>63.79%</td><td>69.62%</td><td>73.96%</td><td>76.24%</td><td>77.01%</td><td>77.10%</td><td>77.08%</td></tr><tr><td rowspan="3">ResNet-20</td><td>RMP</td><td>68.72%</td><td></td><td></td><td>-</td><td>-</td><td>27.64%</td><td>46.91%</td><td>67.82%</td></tr><tr><td>TSC</td><td>68.72%</td><td>1</td><td>-</td><td>1</td><td>:</td><td></td><td>1</td><td>68.18%</td></tr><tr><td>Ours</td><td>69.94%</td><td>19.96%</td><td>34.14%</td><td>55.37%</td><td>67.33%</td><td>69.82%</td><td>70.49%</td><td>70.50%</td></tr><tr><td rowspan="3">ResNet-18</td><td>RTS</td><td>77.16%</td><td></td><td>-</td><td>-</td><td>二</td><td>51.27%</td><td>70.12%</td><td>77.19%</td></tr><tr><td>SNNC-AP</td><td>77.16%</td><td>=</td><td>-</td><td>-</td><td>-</td><td>76.32%</td><td>77.29%</td><td>77.25%</td></tr><tr><td>Ours</td><td>78.80%</td><td>70.79%</td><td>75.67%</td><td>78.48%</td><td>79.48%</td><td>79.62%</td><td>79.54%</td><td>79.61%</td></tr></table>
+RTS and SNNC-AP use altered ResNet-18, while ours use standard ResNet-18.
+Table S5: Comparison of the energy consumption with previous works
+<table><tr><td>Method</td><td></td><td>ANN</td><td>T=2</td><td>T=4</td><td>T=8</td><td>T=16</td><td>T=32</td><td>T=64</td></tr><tr><td rowspan="3">RTS</td><td>Accuracy</td><td>77.89%</td><td>-</td><td>-</td><td></td><td>-</td><td>7.64%</td><td>21.84%</td></tr><tr><td>OP(GFLOP/GSOP)</td><td>0.628</td><td>-</td><td>-</td><td></td><td></td><td>0.508</td><td>0.681</td></tr><tr><td>Energy (mJ)</td><td>7.85</td><td>-</td><td>-</td><td></td><td></td><td>0.039</td><td>0.052</td></tr><tr><td rowspan="3">SNNC-AP</td><td>Accuracy</td><td>77.89%</td><td></td><td>■</td><td></td><td></td><td>73.55%</td><td>76.64%</td></tr><tr><td>OP (GFLOP/GSOP)</td><td>0.628</td><td>-</td><td>■</td><td></td><td></td><td>0.857</td><td>1.22</td></tr><tr><td>Energy (mJ)</td><td>7.85</td><td>-</td><td>-</td><td>-</td><td>-</td><td>0.660</td><td>0.094</td></tr><tr><td rowspan="3">Ours</td><td>Accuracy</td><td>76.28%</td><td>63.79%</td><td>69.62%</td><td>73.96%</td><td>76.24%</td><td>77.01%</td><td>77.10%</td></tr><tr><td>OP (GFLOP/GSOP)</td><td>0.628</td><td>0.094</td><td>0.185</td><td>0.364</td><td>0.724</td><td>1.444</td><td>2.884</td></tr><tr><td>Energy (mJ)</td><td>7.85</td><td>0.007</td><td>0.014</td><td>0.028</td><td>0.056</td><td>0.111</td><td>0.222</td></tr></table>
+Input: ANN model $M _ { \mathrm { A N N } } ( \pmb { x } ; \pmb { W } )$ with initial weight $W$ ; Dataset $D$ ; Quantization step $L$ ; Initial
+dynamic thresholds $\lambda$ ; Learning rate $\epsilon$ .
+Output: $M _ { \mathrm { S N N } } ( \pmb { x } ; \hat { \pmb { W } } )$
+1: for $l = 1$ to $M _ { \mathrm { A N N } }$ .layers do
+2: if is ReLU activation then
+3: Replace ReLU $( { \pmb x } )$ by $\mathrm { Q C F S } ( \pmb { x } ; L , \lambda ^ { l } )$
+4: end if
+5: if is MaxPooling layer then
+6: Replace MaxPooling layer by AvgPooling layer
+7: end if
+8: end for
+9: for $e = 1$ to epochs do
+10: for length of Dataset $D$ do
+11: Sample minibatch $( \boldsymbol { x } ^ { 0 } , \boldsymbol { y } )$ from $D$
+12: for $l = 1$ to $M _ { \mathrm { A N N } }$ .layers do
+13: $\pmb { x } ^ { l } = \mathrm { Q C F S } ( \widetilde { \pmb { W } } ^ { l } \pmb { x } ^ { l - 1 } ; L , \lambda ^ { l } )$
+14: end for
+15: $\mathrm { L o s s } = \mathrm { C r o s s E n t r o p y } ( \pmb { x } ^ { l } , \pmb { y } )$
+16: for $l = 1$ to $M _ { \mathrm { A N N } }$ .layers do
+17: 18: $\begin{array} { r l } & { W ^ { l }  W ^ { l } - \epsilon \frac { \partial L \bar { o } s s } { \partial W ^ { l } } } \\ & { \lambda ^ { l }  \lambda ^ { l } - \epsilon \frac { \partial L o s s } { \partial \lambda ^ { l } } } \\ & { - \epsilon \frac { } { \partial \lambda ^ { \complement } } } \end{array}$
+19: end for
+20: end for
+21: end for
+22: for $l = 1$ to $M _ { \mathrm { A N N } }$ .layers do
+23: $M _ { \mathrm { S N N } } . \hat { W } ^ { l } \gets M _ { \mathrm { A N N } } . W ^ { l }$
+24: $M _ { \mathrm { S N N } } . \theta ^ { l }  M _ { \mathrm { A N N } } . \lambda ^ { l }$
+25: $M _ { \mathrm { S N N } } . { \pmb v } ^ { l } ( 0 )  M _ { \mathrm { S N N } } . \theta ^ { l } / 2$
+26: end for
+27: return MSNN

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+# DIFFUSION POLICIES AS AN EXPRESSIVE POLICY CLASS FOR OFFLINE REINFORCEMENT LEARNING
+Zhendong Wang1,∗ , Jonathan J $\mathbf { H u n t } ^ { 2 , \dagger }$ , Mingyuan Zhou1,†
+1The University of Texas at Austin, 2 Twitter zhendong.wang@utexas.edu, jhunt@twitter.com mingyuan.zhou@mccombs.utexas.edu
+# ABSTRACT
+Offline reinforcement learning (RL), which aims to learn an optimal policy using a previously collected static dataset, is an important paradigm of RL. Standard RL methods often perform poorly in this regime due to the function approximation errors on out-of-distribution actions. While a variety of regularization methods have been proposed to mitigate this issue, they are often constrained by policy classes with limited expressiveness that can lead to highly suboptimal solutions. In this paper, we propose representing the policy as a diffusion model, a recent class of highly-expressive deep generative models. We introduce Diffusion Qlearning (Diffusion-QL) that utilizes a conditional diffusion model to represent the policy. In our approach, we learn an action-value function and we add a term maximizing action-values into the training loss of the conditional diffusion model, which results in a loss that seeks optimal actions that are near the behavior policy. We show the expressiveness of the diffusion model-based policy, and the coupling of the behavior cloning and policy improvement under the diffusion model both contribute to the outstanding performance of Diffusion-QL. We illustrate the superiority of our method compared to prior works in a simple 2D bandit example with a multimodal behavior policy. We then show that our method can achieve state-of-the-art performance on the majority of the D4RL benchmark tasks.
+# 1 INTRODUCTION
+Offline reinforcement learning (RL), also known as batch RL, aims at learning effective policies entirely from previously collected data without interacting with the environment (Lange et al., 2012; Fujimoto et al., 2019). Eliminating the need for online interaction with the environment makes offline RL attractive for a wide array of real-world applications, such as autonomous driving and patient treatment planning, where real-world exploration with an untrained policy is risky, expensive, or time-consuming. Instead of relying on real-world exploration, offline RL emphasizes the use of prior data, such as human demonstration, that is often available at a much lower cost than online interactions. However, relying only on previously collected data makes offline RL a challenging task. Applying standard policy improvement approaches to an offline dataset typically leads to relying on evaluating actions that have not been seen in the dataset, and therefore their values are unlikely to be estimated accurately. For this reason, naive approaches to offline RL typically learn poor policies that prefer out-of-distribution actions whose values have been overestimated, resulting in unsatisfactory performance (Fujimoto et al., 2019).
+Previous work on offline RL generally addressed this problem in one of four ways: 1) regularizing how far the policy can deviate from the behavior policy (Fujimoto et al., 2019; Fujimoto & Gu, 2021; Kumar et al., 2019; Wu et al., 2019; Nair et al., 2020; Lyu et al., 2022); 2) constraining the learned value function to assign low values to out-of-distribution actions (Kostrikov et al., 2021a; Kumar et al., 2020); 3) introducing model-based methods, which learn a model of the environment dynamics and perform pessimistic planning in the learned Markov decision process (MDP) (Kidambi et al., 2020; Yu et al., 2021); 4) treating offline RL as a problem of sequence prediction with return guidance (Chen et al., 2021; Janner et al., 2021; 2022). Our approach falls into the first category.
+Empirically, the performance of policy-regularized offline RL methods is typically slightly worse than that of other approaches, and here we show that this is largely because the policy regularization methods perform poorly due to their limited ability to accurately represent the behavior policy. This results in the regularization adversely affecting the policy improvement. For example, the policy regularization may limit the exploration space of the agent to a small region with only suboptimal actions and then the Q-learning will be induced to converge towards a suboptimal policy.
+The inaccurate policy regularization occurs for two main reasons: 1) policy classes are not expressive enough; 2) the regularization methods are improper. In most prior work, the policy is a Gaussian distribution with mean and diagonal covariance specified by the output of a neural network. However, as offline datasets are often collected by a mixture of policies, the true behavior policy may exhibit strong multi-modalities, skewness, or dependencies between different action dimensions, which cannot be well modeled by diagonal Gaussian policies (Shafiullah et al., 2022). In a particularly extreme, but not uncommon example, a Gaussian policy is used to fit bimodal training data by minimizing the Kullback–Leibler (KL) divergence from the data distribution to the policy distribution. This will result in the policy exhibiting mode-covering behavior and placing high density in the middle area of the two modes, which is actually the low-density region of the training data. In such cases, regularizing a new policy towards the behavior-cloned policy is likely to make the policy learning substantially worse. Second, the regularization, such as the KL divergence and maximum mean discrepancy (MMD) (Kumar et al., 2019), is often not well suited for offline RL. The KL divergence needs access to explicit density values and MMD needs multiple action samples at each state for optimization. These methods require an extra step by first learning a behavior cloned policy to provide density values for KL optimization or random action samples for MMD optimization. Regularizing the current policy towards the behavior cloned policy can further induce approximation errors, since the cloned behavior policy may not model the true behavior policy well, due to limitations in the expressiveness of the policy class. We conduct a simple bandit experiment in Section 4, which illustrates these issues can occur even on a simple bandit task.
+In this work, we propose a method to perform policy regularization using diffusion (or score-based) models (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Ho et al., 2020). Specifically, we use a multilayer perceptron (MLP) based denoising diffusion probabilistic model (DDPM) (Ho et al., 2020) as our policy. We construct an objective for the diffusion loss which contains two terms: 1) a behavior-cloning term that encourages the diffusion model to sample actions in the same distribution as the training set, and 2) a policy improvement term that attempts to sample high-value actions (according to a learned Q-value). Our diffusion model is a conditional model with states as the condition and actions as the outputs. Applying a diffusion model here has several appealing properties. First, diffusion models are very expressive and can well capture multi-modal distributions. Second, the diffusion model loss constitutes a strong distribution matching technique and hence it could be seen as a powerful sample-based policy regularization method without the need for extra behavior cloning. Third, diffusion models perform generation via iterative refinement, and the guidance from maximizing the Q-value function can be added at each reverse diffusion step.
+In summary, our contribution is Diffusion-QL, a new offline RL algorithm that leverages diffusion models to do precise policy regularization and successfully injects the Q-learning guidance into the reverse diffusion chain to seek optimal actions. We test Diffusion-QL on the D4RL benchmark tasks for offline RL and show this method outperforms prior methods on the majority of tasks. We also visualize the method on a simple bandit task to illustrate why it can outperform prior methods. Code is available at https://github.com/Zhendong-Wang/ Diffusion-Policies-for-Offline-RL.
+# 2 PRELIMINARIES AND RELATED WORK
+Offline RL. The environment in RL is typically defined by a Markov decision process (MDP): $M = \{ S , A , P , R , \gamma , d _ { 0 } \}$ , with state space $S$ , action space $\mathcal { A }$ , environment dynamics ${ \mathcal { P } } ( s ^ { \prime } \mid s , a ) :$ $S \times S \times \mathcal { A }  [ 0 , 1 ]$ , reward function $R : S \times \mathcal { A }  \mathbb { R }$ , discount factor $\gamma \in [ 0 , 1 )$ , and initial state distribution (Sutton & Barto, 2018). The goal is to learn policy $\pi _ { \boldsymbol { \theta } } ( \mathbf { a } \mid s )$ , parameterized by $\theta$ , that 0 maximizes the cumulative discounted reward $\begin{array} { r } { \mathbb { E } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r \overline { { \left( \pmb { s } _ { t } , \dot { \pmb { a } } _ { t } \right) } } \right] } \end{array}$ θ  . The action-value or $\mathrm { Q }$ -value of a policy $\pi$ is defined as $\begin{array} { r } { Q ^ { \pi } ( s _ { t } , a _ { t } ) = \mathbb { E } _ { a _ { t + 1 } , a _ { t + 2 } , \dots \sim \pi } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r ( s _ { t } , a _ { t } ) \right] . } \end{array}$ . In the offline setting $\operatorname { F u }$ et al., 2020), instead of the environment, a static dataset $\mathcal { D } \triangleq \{ ( \pmb { s } , \pmb { a } , r , \pmb { s } ^ { \prime } ) \}$ , collected by a behavior policy $\pi _ { b }$ , is provided. Offline RL algorithms learn a policy entirely from this static offline dataset $\mathcal { D }$ , without online interactions with the environment.
+Diffusion Model. Diffusion-based generative models (Ho et al., 2020; Sohl-Dickstein et al., 2015; Song & Ermon, 2019) assume $\begin{array} { r } { p _ { \theta } ( \pmb { x } _ { 0 } ) : = \int p _ { \theta } ( \pmb { x } _ { 0 : T } ) d \pmb { x } _ { 1 : T } } \end{array}$ , where $\pmb { x } _ { 1 } , \ldots , \pmb { x } _ { T }$ are latent variables of the same dimensionality as the data ${ \pmb x } _ { 0 } \sim p ( { \pmb x } _ { 0 } )$ . A forward diffusion chain gradually adds noise to the data ${ \pmb x } _ { 0 } \sim { \pmb q } ( { \pmb x } _ { 0 } )$ in $T$ steps with a pre-defined variance schedule $\beta _ { i }$ , expressed as
+$$
+\begin{array} { r } { q ( { \pmb x } _ { 1 : T } \mid { \pmb x } _ { 0 } ) : = \prod _ { t = 1 } ^ { T } q ( { \pmb x } _ { t } \mid { \pmb x } _ { t - 1 } ) , \quad q ( { \pmb x } _ { t } \mid { \pmb x } _ { t - 1 } ) : = \mathcal { N } ( { \pmb x } _ { t } ; \sqrt { 1 - \beta _ { t } } { \pmb x } _ { t - 1 } , \beta _ { t } I ) . } \end{array}
+$$
+A reverse diffusion chain, constructed as $\begin{array} { r } { p _ { \theta } ( { \pmb x } _ { 0 : T } ) : = \mathcal { N } ( { \pmb x } _ { T } ; { \bf 0 } , { \pmb I } ) \prod _ { t = 1 } ^ { T } p _ { \theta } ( { \pmb x } _ { t - 1 } \mid { \pmb x } _ { t } ) } \end{array}$ , is then optimized by maximizing the evidence lower bound defined as $\begin{array} { r } { \mathbb { E } _ { q } [ \ln { \frac { p _ { \theta } ( { \pmb x } _ { 0 : T } ) } { q ( { \pmb x } _ { 1 : T } \mid { \pmb x } _ { 0 } ) } } ] } \end{array}$ (Jordan et al., 1999; Blei et al., 2017). After training, sampling from the diffusion model consists of sampling ${ \pmb x } _ { T } \sim p ( { \pmb x } _ { T } )$ and running the reverse diffusion chain to go from $t = T$ to $t = 0$ . Diffusion models can be straightforwardly extended to conditional models by conditioning $p _ { \theta } ( \pmb { x } _ { t - 1 } \mid \pmb { x } _ { t } , c )$ .
+Related Work: Policy Regularization. Most prior methods for offline RL in the class of regularized policies rely on behavior cloning for policy regularization: BCQ (Fujimoto et al., 2019) constructs the policy as a learnable and maximum-value-constrained deviation from a separately learned Conditional-VAE (CVAE, Sohn et al. (2015)) behavior-cloning model; BEAR (Kumar et al., 2019) adds a weighted behavior-cloning loss via minimizing MMD into the policy improvement step; $\mathrm { T D } 3 { + } \mathrm { B C }$ (Fujimoto & Gu, 2021) applies the same trick as BEAR via maximum likelihood estimation (MLE); BRAC (Wu et al., 2019) evaluates multiple methods for behavior-cloning regularization, such as the KL divergence, MMD, and Wasserstein dual form; IQL (Kostrikov et al., 2021b) is an advantage weighted behavior-cloning method with “in-sample” learned Q-value functions. Goo & Niekum (2022) emphasize the necessity of conducting explicit behavioral cloning in offline RL, while Ajay et al. (2022) admit the power of conditional generative models for decision making.
+Related Work: Diffusion Models in RL. Pearce et al. (2023) propose to better imitate human behaviors via diffusion models which are expressive and stable. Diffuser (Janner et al., 2022) applies a diffusion model as a trajectory generator. The full trajectory of state-action pairs form a single sample for the diffusion model. A separate return model is learned to predict the cumulative rewards of each trajectory sample. The guidance of the return model is then injected into the reverse sampling stage. This approach is similar to Decision Transformer (Chen et al., 2021), which also learns a trajectory generator through GPT2 (Radford et al., 2019) with the help of the true trajectory returns. When used online, sequence models can no longer predict actions from states autoregressively (since the states are an outcome of the environment). Thus, in the evaluation stage, a whole trajectory is predicted for each state while only the first action is applied, which incurs a large computational cost. Our approach employs diffusion models for RL in a distinct manner.
+We apply the diffusion model to the action space and we form it as a conditional diffusion model with states as the condition. This approach is model-free and the diffusion model is sampling a single action at a time. Further, our Q-value function guidance is injected during training, which provides good empirical performance in our case. While both Diffuser (Janner et al., 2022) and our work apply diffusion models in Offline RL, Diffuser is from the model-based trajectory-planning perspective while our method is from the offline model-free policy-optimization perspective.
+# 3 DIFFUSION Q-LEARNING
+Below we explain how we apply a conditional diffusion model as an expressive policy for behavior cloning. Then, we introduce how we add Q-learning guidance into the learning of our diffusion model in the training stage with the behavior cloning term acting as a form of policy regularization.
+# 3.1 DIFFUSION POLICY
+Notation: Since there are two different types of timesteps in this work, one for the diffusion process and one for reinforcement learning we use superscripts $i \in \{ 1 , \ldots , N \}$ to denote diffusion timestep and subscripts $t \in \{ 1 , \ldots , T \}$ to denote trajectory timestep.
+We represent our RL policy via the reverse process of a conditional diffusion model as
+$$
+\begin{array} { r } { \pi _ { \boldsymbol { \theta } } ( \boldsymbol { a } | \boldsymbol { s } ) = p _ { \boldsymbol { \theta } } ( \boldsymbol { a } ^ { 0 : N } | \boldsymbol { s } ) = \mathcal { N } ( \boldsymbol { a } ^ { N } ; \mathbf { 0 } , I ) \prod _ { i = 1 } ^ { N } p _ { \boldsymbol { \theta } } ( \boldsymbol { a } ^ { i - 1 } | \boldsymbol { a } ^ { i } , \boldsymbol { s } ) } \end{array}
+$$
+where the end sample of the reverse chain, $\mathbf { \delta } _ { \mathbf { { a } } } ^ { 0 }$ , is the action used for RL evaluation. Generally, $p _ { \theta } ( \pmb { a } ^ { i - 1 } | \pmb { a } ^ { i } , \pmb { s } )$ could be modeled as a Gaussian distribution $\begin{array} { r } { \mathcal { N } ( { \pmb a } ^ { i - 1 } ; \pmb { \mu } _ { \boldsymbol { \theta } } ( { \pmb a } ^ { i } , { \pmb s } , i ) , \pmb { \Sigma } _ { \boldsymbol { \theta } } ( { \pmb a } ^ { i } , { \pmb s } , i ) ) } \end{array}$ . We follow Ho et al. (2020) to parameterize $p _ { \theta } ( \pmb { a } ^ { i - 1 } | \pmb { a } ^ { i } , \pmb { s } )$ as a noise prediction model with the covariance matrix fixed as $\Sigma _ { \theta } ( \bar { \pmb { a } ^ { i } } , \pmb { s } , i ) = \beta _ { i } \bar { \pmb { I } }$ and mean constructed as
+$$
+\begin{array} { r } { \mu _ { \theta } ( { \pmb a } ^ { i } , { \pmb s } , i ) = \frac { 1 } { \sqrt { \alpha _ { i } } } \big ( { \pmb a } ^ { i } - \frac { \beta _ { i } } { \sqrt { 1 - \bar { \alpha } _ { i } } } { \pmb \epsilon } _ { \theta } ( { \pmb a } ^ { i } , { \pmb s } , i ) \big ) . } \end{array}
+$$
+We first sample $\pmb { a } ^ { N } \sim \mathcal { N } ( \mathbf { 0 } , I )$ and then from the reverse diffusion chain parameterized by $\theta$ as
+$$
+\begin{array} { r } { { \pmb a } ^ { i - 1 } \parallel { \pmb a } ^ { i } = \frac { { \pmb a } ^ { i } } { \sqrt { \alpha _ { i } } } - \frac { \beta _ { i } } { \sqrt { \alpha _ { i } ( 1 - \bar { \alpha } _ { i } ) } } \epsilon _ { \theta } ( { \pmb a } ^ { i } , { \pmb s } , i ) + \sqrt { \beta _ { i } } \epsilon , \ { \epsilon } \sim \mathcal { N } ( \mathbf { 0 } , I ) , \ \mathrm { f o r } \ i = N , \dots , 1 . } \end{array}
+$$
+Following DDPM (Ho et al., 2020), when $i = 1$ , $\epsilon$ is set as $\mathbf { 0 }$ to improve the sampling quality.
+We mimic the simplified objective proposed by $\mathrm { H o }$ et al. (2020) to train our conditional $\epsilon$ -model v
+$$
+\mathcal { L } _ { d } ( \theta ) = \mathbb { E } _ { i \sim \mathcal { U } , \epsilon \sim \mathcal { N } ( \mathbf { 0 } , I ) , ( s , a ) \sim \mathcal { D } } \left[ | | \epsilon - \epsilon _ { \theta } \big ( \sqrt { \bar { \alpha } _ { i } } a + \sqrt { 1 - \bar { \alpha } _ { i } } \epsilon , s , i \big ) | | ^ { 2 } \right] ,
+$$
+where $\mathcal { U }$ is a uniform distribution over the discrete set as $\{ 1 , \ldots , N \}$ and $\mathcal { D }$ denotes the offline dataset, collected by behavior policy $\pi _ { b }$ . This diffusion model loss $\dot { \mathcal { L } } _ { d } ( \theta )$ is a behavior-cloning loss, which aims to learn the behavior policy $\pi _ { b } ( { \pmb a } | { \pmb s } )$ (i.e. it seeks to sample actions from the same distribution as the training data). Note the marginal of the reverse diffusion chain provides an implicit, expressive distribution that can capture complex distribution properties, such as skewness and multi-modality, exhibited by the offline datasets. In addition, the regularization is sampling-based that only requires taking random samples from both $\mathcal { D }$ and the current policy (i.e. this method does not require us to know the behavior policy, which may be infeasible when the dataset is collected by human demonstrations). Different from the usual two-step strategy, our strategy provides a clean and effective way of applying regularization on a flexible policy.
+$\mathcal { L } _ { d } ( \theta )$ can efficiently be optimized by sampling a single diffusion step $i$ for each data point, but the reverse sampling in Equation (1), which requires iteratively computing $\epsilon _ { \theta }$ networks $N$ times, can become a bottleneck for the running time. Thus we may want to limit $N$ to a relatively small value. To work with small $N$ , with $\beta _ { \mathrm { m i n } } = 0 . 1$ and $\beta _ { \mathrm { m a x } } = 1 0 . 0$ , we follow Xiao et al. (2021) to define
+$$
+\begin{array} { r } { \beta _ { i } = 1 - \alpha _ { i } = 1 - e ^ { - \beta _ { \mathrm { m i n } } \left( \frac { 1 } { N } \right) - 0 . 5 \left( \beta _ { \mathrm { m a x } } - \beta _ { \mathrm { m i n } } \right) \frac { 2 i - 1 } { N ^ { 2 } } } , } \end{array}
+$$
+which is a noise schedule obtained under the variance preserving SDE of Song et al. (2021).
+# 3.2 Q-LEARNING
+The policy-regularization loss $\mathcal { L } _ { d } ( \theta )$ is a behavior-cloning term, but would not result in learning a policy that can outperform the behavior policy that generated the training data. To improve the policy, we inject Q-value function guidance into the reverse diffusion chain in the training stage in order to learn to preferentially sample actions with high values. The final policy-learning objective is a linear combination of policy regularization and policy improvement:
+$$
+\pi = \underset { \pi _ { \theta } } { \arg \operatorname* { m i n } } \ : \mathcal { L } ( \theta ) = \mathcal { L } _ { d } ( \theta ) + \mathcal { L } _ { q } ( \theta ) = \mathcal { L } _ { d } ( \theta ) - \alpha \cdot \mathbb { E } _ { s \sim \mathcal { D } , a ^ { 0 } \sim \pi _ { \theta } } \left[ Q _ { \phi } ( s , a ^ { 0 } ) \right] .
+$$
+Note that $\mathbf { \delta } _ { \mathbf { { a } } ^ { 0 } }$ is reparameterized by Equation (1) and hence the gradient of the Q-value function with respect to the action is backpropagated through the whole diffusion chain.
+As the scale of the Q-value function varies in different offline datasets, to normalize it, we follow Fujimoto & Gu (2021) to set $\alpha$ as $\begin{array} { r } { \alpha = \frac { \eta } { \mathbb { E } _ { ( s , a ) \sim \mathcal { D } } \left[ | Q _ { \phi } ( s , a ) | \right] } } \end{array}$ , where $\eta$ is a hyperparameter that balances the two loss terms and the $\mathrm { Q }$ in the denominator is for normalization only and not differentiated over.
+The Q-value function itself is learned in a conventional way, minimizing the Bellman operator (Lillicrap et al., 2015; Fujimoto et al., 2019) with the double Q-learning trick (Hasselt, 2010). We built two Q-networks, $Q _ { \phi _ { 1 } }$ , $Q _ { \phi _ { 2 } }$ , and target networks $Q _ { \phi _ { 1 } ^ { \prime } }$ , $Q _ { \phi _ { 2 } ^ { \prime } }$ and $\pi _ { \theta ^ { \prime } }$ . We then optimize $\phi _ { i }$ for $i = \{ 1 , 2 \}$ by minimizing the objective,
+$$
+\begin{array} { r } { \mathbb { E } _ { ( s _ { t } , a _ { t } , s _ { t + 1 } ) \sim \mathcal { D } , a _ { t + 1 } ^ { 0 } \sim \pi _ { \theta ^ { \prime } } } \left[ \left| \left| \left( r ( s _ { t } , a _ { t } ) + \gamma \operatorname* { m i n } _ { i = 1 , 2 } Q _ { \phi _ { i } ^ { \prime } } ( s _ { t + 1 } , a _ { t + 1 } ^ { 0 } ) \right) - Q _ { \phi _ { i } } ( s _ { t } , a _ { t } ) \right| \right| ^ { 2 } \right] . } \end{array}
+$$
+We conduct extensive experiments in Sections 4 and 5 and show that $\mathcal { L } _ { d }$ and $\mathcal { L } _ { q }$ work together to achieve the best performance. We summarize our implementation in Algorithm 1.
+Algorithm 1 Diffusion Q-learning
+<table><tr><td>Initialize policy network πθ, critic networks Q𝜙1 and Q, and target networks Tθ, Q1 and Q2 for each iteration do</td></tr><tr><td>Sample transition mini-batch B = {(st, at,rt, St+1)} ~ D. # Q-value function learning</td></tr><tr><td>Sample at+1 ~ πθ(at+1|St+1) by Equation (1).</td></tr><tr><td>Update Qφ1 and QΦ2 by Equation (4). (max Q backup by Kumar et al. (202O) could be added)</td></tr><tr><td>#Policy learning Sample a𝑙 ~ πθ(at |St) by Equation (1).</td></tr><tr><td>Update policy by minimizing Equation (3).</td></tr><tr><td># Update target networks</td></tr><tr><td>0&#x27;=p0&#x27;+(1-p)0,Φ²=pΦ+(1-p)Φifori={1,2}. end for</td></tr></table>
+# 4 POLICY REGULARIZATION
+In this section, we illustrate how the previous policy regularization methods work compared to our conditional diffusion-based approach on a simple bandit task with a 2D continuous action space. Below we first provide a brief review of prior methods.
+BC-MLE. The policy $\pi _ { \boldsymbol { \theta } } ( \mathbf { a } \mid s )$ is modeled by a Gaussian distribution $\mathcal { N } ( \pmb { a } ; \pmb { \mu } _ { \theta } ( \pmb { s } _ { t } ) , \pmb { \Sigma } _ { \theta } ( \pmb { s } ) )$ , where usually $\pmb { \mu } _ { \theta }$ and $\Sigma _ { \theta }$ are parameterized by multi-layer perceptrons (MLPs) and for simplicity $\Sigma _ { \theta }$ is assumed to be a diagonal matrix. The policy is optimized by maximizing $\mathbb { E } _ { ( s , a ) \sim \mathcal { D } } [ \log \pi _ { \theta } ( { \pmb a } \mid s ) ]$ . $\mathrm { T D } 3 { + } \mathrm { B C }$ (Fujimoto & Gu, 2021) directly add the behavior-cloning (BC) loss as an additional term in policy learning, while IQL (Kostrikov et al., 2021b) proposes using “in-sample” learned advantage functions to reweigh the log term inside the expectation.
+BC-CVAE. The policy $\pi _ { \boldsymbol { \theta } } ( \mathbf { a } \mid s )$ is modeled by a CVAE model with an encoder network $q _ { \theta } ( z \mid s , \mathbf { a } )$ and a decoder network $p _ { \theta } ( \pmb { a } | \pmb { s } , \pmb { z } )$ . The two networks are optimized by maximizing the evidence lower bound $\mathbb { E } _ { ( s , a ) \sim \mathcal { D } } [ \mathbb { E } _ { z \sim q ( \cdot \mid s , a ) } [ \log p ( a \mid s , z ) ] - \mathrm { K L } ( q ( \bar { \boldsymbol { z } } \mid s , a ) | | p ( \boldsymbol { z } ) ) ]$ , where $p ( z )$ is a prior distribution that is usually set as standard Gaussian. BCQ (Fujimoto et al., 2019) trains a CVAE model as an approximation of the behavior policy and trains another deviation model to guide the actions drawn from the CVAE approximation towards the regions with high learned $\mathrm { Q }$ -values.
+BC-MMD. BEAR (Kumar et al., 2019) also mimics the behavior policy via a CVAE model and proposes to limit the current policy $\pi _ { \boldsymbol { \theta } } ( \mathbf { a } \vert \mathbf { \boldsymbol { s } } )$ to be close to the cloned behavior policy via MMD minimization. A Tanh-Gaussian policy is used, which is a Gaussian network with a Tanh activation function at the output layer.
+BCQ and BEAR can be seen as a two-step regularization: First, an approximation of the behavior policy is learned (behavior cloning), and then the policy learned by policy improvement is regularized towards the cloned behavior policy. However, such a two-step approach means that the efficacy of the second-step policy regularization heavily depends on the cloning quality, and an inaccurate regularization could misguide the subsequent policy improvement step. We illustrate this weakness in a 2D continuous action space bandit example.
+Example. We consider a simple bandit task with real-valued actions in a 2D space, $\pmb { a } \in [ - 1 , 1 ] ^ { 2 }$ . We construct an offline dataset $\mathcal { D } \ : = \ : \{ ( \boldsymbol { a } _ { j } ) \} _ { j = 1 } ^ { M }$ with $M \ : = \ : 1 0 0 0 0$ action examples, where the actions are collected from an equal mixture of four Gaussian distributions with centers $\mu \in$ $\{ ( 0 . 0 , 0 . 8 ) , ( 0 . 8 , 0 . 0 ) , ( 0 . 0 , - 0 . 8 ) , ( - 0 . 8 , 0 . 0 ) \}$ and standard deviations ${ \pmb \sigma } _ { d } = ( 0 . 0 5 , 0 . 0 5 )$ , as depicted in the first panel of Figure 1. Note that this example exhibits strong multi-modality in the behavior policy distribution, which is often the case when the dataset is collected by different policies. For example, if multiple humans are involved, some may be experts and choose actions from a different mode from amateur demonstrators.
+To evaluate the strength of prior regularization methods, we first compare them to our diffusionbased approach on a behavior-cloning task, where the goal is to just clone the behavior policy that generated the data, not improve on it. As shown in the first row of Figure 1, we observe that the diffusion model captures all the four density modes of the behavior policy. The policy of BC-MLE is limited to a single mode and hence exhibits a strong mode-covering behavior. It fits the four density modes by one Gaussian distribution with a large standard deviation, whose high-density regions are actually the low-density regions of the true behavior policy. The CVAE model is shown to exhibit mode-covering behavior even though it itself is an implicit model with enough expressiveness: We see that CVAE captures the four modes but high densities are assigned between them to cover all the modes. Note we also observe the CVAE model sometimes fails to capture all four modes with different random seeds. The Tanh-Gaussian policy optimized under MMD learns to align the densities near the boundary lines and, due to its limited policy expressiveness, fails to capture the true distribution. These observed failures on behavior cloning in this simple example illustrate the limitations of prior regularization methods in the common case of multi-modal behavior policies.
+![](images/ed09db3bbd86938ac9aa97c248e0496269d39170d15a159ad90b289acd4865ff.jpg)
+Figure 1: Offline RL experiments on a simple bandit task. The first row shows the comparison of behavior cloning between our method (BC-Diffusion, $N = 5 0$ ) and prior methods. Prior methods struggle to capture the multi-modal behavior policy (ground truth). The second row shows the comparison results when policy improvement is added (first figure shows the rewards). The policy regularization of prior methods results in a poor policy, since the behavior-cloning step fails to capture the multi-modal behavior policy, while our method (Diffusion-QL) correctly identifies the high reward behavior mode.
+![](images/bb1968b51077654b97fd926738cfa357f9929a9f4a3cdb3c7aaf5dfa991f785a.jpg)
+Figure 2: Experiment examining the effect of varying the number of diffusion steps $N$ on the simple bandit talks. The first row shows the ablation study of $N$ for BC-Diffusion. The second row shows the ablation study of $N$ for Diffusion-QL. Large $N$ results in a better fit to the data distribution, but leads to a higher computational cost. By combining the behavior cloning and Q-value losses during training, we are able to learn near-optimal policies with fewer diffusion steps.
+Next, we investigate how the policy improvement will be impacted by the corresponding policy regularization. We assign each data point a reward sampled from a Gaussian distribution, whose mean is determined by the data center and standard deviation is fixed as 0.5, as shown in the second row of Figure 1. Note here we mimic the offline RL setting, under which the underlying reward function is unknown and needs to be learned. We compare Diffusion Q-learning (QL) with prior methods, including $\mathrm { T D } 3 { + } \mathrm { B C }$ , BCQ, and BEAR-MMD. We train all methods with 1000 epochs to ensure convergence. Due to the strong policy constraint applied by each method, we observe that the policy improvement of prior methods is constrained to suboptimal or even wrong exploration regions, induced by the corresponding behavior-cloning regularization. $\mathrm { T D } 3 { + } \mathrm { B C }$ cannot converge to the optimal mode since the behavior policy places most density in the region where no offline data exists. BCQ learns to place major actions on the four diagonal corners discovered by its CVAE-based behavior cloning. The policy of BEAR-MMD learns to place actions randomly since the exploration region is constrained by inaccurate policy regularization. We observe that the prior regularizations typically push the policy to converge to sub-optimal solutions, such as BCQ, or prevent the policy from being concentrated on the optimal corner, such as $\mathrm { T D } 3 { + } \mathrm { B C }$ and BEAR-MMD. By contrast, the policy of Diffusion-QL successfully converges to the optimal bottom corner. This is because 1) diffusion policy is expressive enough to recover the behavior policy, which covers all modes for further exploration; 2) the Q-learning guidance, directly through linearly combined loss functions in Equation (3), helps diffusion policy seek optimal actions in the region. The two components are working together to produce good performance.
+Diffusions steps. We further investigated how the diffusion policy performs as the number of diffusion timesteps $N$ is varied. As expected, the first row of Figure 2 shows that as $N$ increases, the diffusion model becomes more expressive and learns more details about the underlying data distribution. When $N$ is increased to 50, the true data distribution is accurately recovered. The second row shows that with Q-learning applied, a moderately small $N$ is able to deliver good performance due to our loss coupling defined in Equation (3). However, we can see with a larger $N$ the policy regularization imposed by the diffusion model-based cloning becomes stronger. For example, when $N = 2$ , there are still a few action points sampled near regions with no training data, whereas when $N = 5 0$ , the policy is constrained in the correct data region for further exploration. The number of timesteps $N$ serves as a trade-off between policy expressiveness and computational cost for Diffusion-QL. We found $N = 5$ performs well on D4RL (Fu et al., 2020) datasets, which is also a small enough value for cost-effective training and deployment.
+# 5 EXPERIMENTS
+We evaluate our method on the popular D4RL (Fu et al., 2020) benchmark. Further, we conduct empirical studies on the number of timesteps required for our diffusion model and also perform an ablation study for analyzing the contribution of the two main components of Diffusion-QL.
+Datasets. We consider four different domains of tasks in D4RL benchmark: Gym, AntMaze, Adroit, and Kitchen. The Gym-MuJoCo locomotion tasks are the most commonly used standard tasks for evaluation and are relatively easy, since they usually include a significant fraction of near-optimal trajectories in the dataset and the reward function is quite smooth. AntMaze consists of more challenging tasks, which have sparse rewards and explicitly need the agent to stitch various sub-optimal trajectories to find a path towards the goal of the maze (Fu et al., 2020). Adroit datasets are mostly collected by human behavior and the state-action region reflected by the offline data is often very narrow, so strong policy regularization is needed to ensure that the agent stays in the expected region. The Kitchen environment requires the agent to complete 4 target subtasks in order to reach a desired state configuration, and hence long-term value optimization is important for it.
+Baselines. We consider different classes of baselines that perform well in each domain of tasks. For policy regularization-based methods, we include the classic BC, BEAR (Kumar et al., 2019), BRAC (Wu et al., 2019), BCQ (Fujimoto et al., 2019), $\mathrm { T D } 3 { + } \mathrm { B C }$ (Fujimoto & Gu, 2021), AWR (Peng et al., 2019), AWAC (Nair et al., 2020), and IQL (Kostrikov et al., 2021b), along with the Onestep RL (Brandfonbrener et al., 2021), which is based on single-step improvement. For Q-value constraint methods, we include REM (Agarwal et al., 2020) and CQL (Kumar et al., 2020). For model-based offline RL, we consider MoRel (Kidambi et al., 2020). For sequence modelling approaches, we compare with Decision Transformer (DT) (Chen et al., 2021) and Diffuser (Janner et al., 2022). We report the performance of baseline methods either using the best results reported from their own paper, Fu et al. (2020) or Kostrikov et al. (2021b).
+Experimental details. We train for 1000 epochs (2000 for Gym tasks). Each epoch consists of 1000 gradient steps with batch size 256. The training is usually quite stable as shown in Figure 3 except that we observe the training for AntMaze tasks has variations due to its sparse reward setting and lack of optimal trajectories in the offline datasets. Hence, we save multiple model checkpoints during training and use a completely offline method, as described in Appendix D, to select the best checkpoint for performance evaluation. Using $\mathcal { L } _ { d }$ loss as a lagging indicator of online performance, we perform early stopping and select the checkpoint with the second or third lowest $\mathcal { L } _ { d }$ value. The results in our main paper are all based on the offline model selection. When a small amount of online interaction can be used for model selection, we can achieve even better results (Appendix Table 4).
+Effect of hyperparameter $N$ . We conduct an empirical study on the effect of the number of timesteps $N$ of our diffusion model on real tasks. As shown in Figure 3, empirically we find as
+![](images/f343b16b9f8b5deebcda6772a764fdb500a4ec4932c81a38841b7e83ebe90d12.jpg)
+Figure 3: Ablation study of $N$ on selected Gym tasks. We consider a $N$ grid [2, 5, 10, 20] and we find that $N = 5$ is good enough for the selected tasks.
+Table 1: The performance of Diffusion-QL and SOTA baselines on D4RL Gym, AntMaze, Adroit, and Kitchen tasks. Results for Diffusion-QL correspond to the mean and standard errors of normalized scores over 50 random rollouts (5 independently trained models and 10 trajectories per model) for Gym tasks, which generally exhibit low variance in performance, and over 500 random rollouts (5 independently trained models and 100 trajectories per model) for the other tasks. Note the standard error of AntMaze is usually large since the return of trajectories is binomial (1 for success while 0 for failure). Our method outperforms all prior methods by a clear margin on all domain, even on the challenging AntMaze, for which behavior cloning methods would fail and some form of policy improvement is essential.
+<table><tr><td>Gym Tasks</td><td>BC</td><td>AWAC</td><td>Diffuser</td><td>MoRel</td><td>Onestep RL</td><td>TD3+BC</td><td>DT</td><td>CQL</td><td>IQL</td><td>Diffusion-QL</td></tr><tr><td>halfcheetah-medium-v2</td><td>42.6</td><td>43.5</td><td>44.2</td><td>42.1</td><td>48.4</td><td>48.3</td><td>42.6</td><td>44.0</td><td>47.4</td><td>51.1 ± 0.5</td></tr><tr><td>hopper-medium-v2</td><td>52.9</td><td>57.0</td><td>58.5</td><td>95.4</td><td>59.6</td><td>59.3</td><td>67.6</td><td>58.5</td><td>66.3</td><td>90.5 ±4.6</td></tr><tr><td>walker2d-medium-v2</td><td>75.3</td><td>72.4</td><td>79.7</td><td>77.8</td><td>81.8</td><td>83.7</td><td>74.0</td><td>72.5</td><td>78.3</td><td>87.0±0.9</td></tr><tr><td>halfcheetah-medium-replay-v2</td><td>36.6</td><td>40.5</td><td>42.2</td><td>40.2</td><td>38.1</td><td>44.6</td><td>36.6</td><td>45.5</td><td>44.2</td><td>47.8 ± 0.3</td></tr><tr><td>hopper-medium-replay-v2</td><td>18.1</td><td>37.2</td><td>96.8</td><td>93.6</td><td>97.5</td><td>60.9</td><td>82.7</td><td>95.0</td><td>94.7</td><td>101.3 ± 0.6</td></tr><tr><td>walker2d-medium-replay-v2</td><td>26.0</td><td>27.0</td><td>61.2</td><td>49.8</td><td>49.5</td><td>81.8</td><td>66.6</td><td>77.2</td><td>73.9</td><td>95.5 ± 1.5</td></tr><tr><td>halfcheetah-medium-expert-v2</td><td>55.2</td><td>42.8</td><td>79.8</td><td>53.3</td><td>93.4</td><td>90.7</td><td>86.8</td><td>91.6</td><td>86.7</td><td>96.8 ± 0.3</td></tr><tr><td>hopper-medium-expert-v2</td><td>52.5</td><td>55.8</td><td>107.2</td><td>108.7</td><td>103.3</td><td>98.0</td><td>107.6</td><td>105.4</td><td>91.5</td><td>111.1 ± 1.3</td></tr><tr><td>walker2d-medium-expert-v2</td><td>107.5</td><td>74.5</td><td>108.4</td><td>95.6</td><td>113.0</td><td>110.1</td><td>108.1</td><td>108.8</td><td>109.6</td><td>110.1 ± 0.3</td></tr><tr><td>Average</td><td>51.9</td><td>50.1</td><td>75.3</td><td>72.9</td><td>76.1</td><td>75.3</td><td>74.7</td><td>77.6</td><td>77.0</td><td>88.0</td></tr><tr><td>AntMaze Tasks</td><td>BC</td><td>AWAC</td><td>BCQ</td><td>BEAR</td><td>Onestep RL</td><td>TD3+BC</td><td>DT</td><td>CQL</td><td>IQL</td><td>Diffusion-QL</td></tr><tr><td>antmaze-umaze-v0</td><td>54.6</td><td>56.7</td><td>78.9</td><td>73.0</td><td>64.3</td><td>78.6</td><td>59.2</td><td>74.0</td><td>87.5</td><td>93.4±3.4</td></tr><tr><td>antmaze-umaze-diverse-v0</td><td>45.6</td><td>49.3</td><td>55.0</td><td>61.0</td><td>60.7</td><td>71.4</td><td>53.0</td><td>84.0</td><td>62.2</td><td>66.2± 8.6</td></tr><tr><td>antmaze-medium-play-v0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.3</td><td>10.6</td><td>0.0</td><td>61.2</td><td>71.2</td><td>76.6 ± 10.8</td></tr><tr><td>antmaze-medium-diverse-v0</td><td>0.0</td><td>0.7</td><td>0.0</td><td>8.0</td><td>0.0</td><td>3.0</td><td>0.0</td><td>53.7</td><td>70.0</td><td>78.6 ± 10.3</td></tr><tr><td>antmaze-large-play-v0</td><td>0.0</td><td>0.0</td><td>6.7</td><td>0.0</td><td>0.0</td><td>0.2</td><td>0.0</td><td>15.8</td><td>39.6</td><td>46.4 ± 8.3</td></tr><tr><td>antmaze-large-diverse-v0</td><td>0.0</td><td>1.0</td><td>2.2</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>14.9</td><td>47.5</td><td>56.6± 7.6</td></tr><tr><td>Average</td><td>16.7</td><td>18.0</td><td>23.8</td><td>23.7</td><td>20.9</td><td>27.3</td><td>18.7</td><td>50.6</td><td>63.0</td><td>69.6</td></tr><tr><td>Adroit Tasks</td><td>BC</td><td>SAC</td><td>BCQ</td><td>BEAR</td><td>BRAC-p</td><td>BRAC-v</td><td>REM</td><td>CQL</td><td>IQL</td><td>Diffusion-QL</td></tr><tr><td>pen-human-v1</td><td>25.8</td><td>4.3</td><td>68.9</td><td>-1.0</td><td>8.1</td><td>0.6</td><td>5.4</td><td>35.2</td><td>71.5</td><td>72.8± 9.6</td></tr><tr><td>pen-cloned-v1</td><td>38.3</td><td>-0.8</td><td>44.0</td><td>26.5</td><td>1.6</td><td>-2.5</td><td>-1.0</td><td>27.2</td><td>37.3</td><td>57.3 ± 11.9</td></tr><tr><td>Average</td><td>32.1</td><td>1.8</td><td>56.5</td><td>12.8</td><td>4.9</td><td>-1.0</td><td>2.2</td><td>31.2</td><td>54.4</td><td>65.1</td></tr><tr><td>Kitchen Tasks</td><td>BC</td><td>SAC</td><td>BCQ</td><td>BEAR</td><td>BRAC-p</td><td>BRAC-v</td><td>AWR</td><td>CQL</td><td>IQL</td><td>Diffusion-QL</td></tr><tr><td>kitchen-complete-v0</td><td>33.8</td><td>15.0</td><td>8.1</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>43.8</td><td>62.5</td><td>84.0 ± 7.4</td></tr><tr><td>kitchen-partial-vo</td><td>33.8</td><td>0.0</td><td>18.9</td><td>13.1</td><td>0.0</td><td>0.0</td><td>15.4</td><td>49.8</td><td>46.3</td><td>60.5 ± 6.9</td></tr><tr><td>kitchen-mixed-v0</td><td>47.5</td><td>2.5</td><td>8.1</td><td>47.2</td><td>0.0</td><td>0.0</td><td>10.6</td><td>51.0</td><td>51.0</td><td>62.6 ± 5.1</td></tr><tr><td>Average</td><td>38.4</td><td>5.8</td><td>11.7</td><td>20.1</td><td>0.0</td><td>0.0</td><td>8.7</td><td>48.2</td><td>53.3</td><td>69.0</td></tr></table>
+the $N$ increases, the model converges faster and the performance becomes more stable. In the following D4RL tasks, we set a moderate value, $N = 5$ , to balance the performance and computational cost. With $N = 5$ , the training time of our method is similar to that of CQL (Kumar et al., 2020). The other hyperparameters, such as the learning rate and $\eta$ , are provided in Appendix E.
+# 5.1 COMPARISON TO OTHER METHODS
+We compare our Diffusion-QL with the baselines on four domains of tasks and report the results in Table 1. We give the analysis based on each specific domain.
+Results for Gym Domain. We can see while most baselines already work well on the Gym tasks, Diffusion-QL can often further improve their performance by a clear margin, especially in ‘medium’ and ‘medium-replay’ tasks. Note the ‘medium’ datasets include the trajectories collected by an online SAC (Haarnoja et al., 2018) agent trained to approximately 1/3 the performance of the expert. Hence, the Tanh-Gaussian policy at that time is usually exploratory and not concentrated, which makes the collected data distribution hard to learn. As shown in Section 4, the diffusion model has the expressivity to mimic the behavior policy even in complicated cases and then the policy improvement term will guide the policy to converge to the optimal actions in the subset of explored action space. These two components are key to the good empirical performance of Diffusion-QL.
+Table 2: Ablation study. We conduct an ablation study to compare our diffusion model with CVAE models, and our Q-learning method with BCQ policy improvement.
+<table><tr><td>Gym Tasks</td><td>BC-CVAE</td><td>BC-Diffusion</td><td>BCQ-CVAE</td><td>BCQ-Diffusion</td><td>CVAE-QL</td><td>Diffusion-QL</td></tr><tr><td>halfcheetah-medium-expert-v2</td><td>67.3± 7.4</td><td>76.6± 7.0</td><td>96.1 ± 0.5</td><td>94.6±1.0</td><td>70.3± 5.5</td><td>97.2 ± 0.4</td></tr><tr><td>hopper-medium-expert-v2</td><td>69.9 ± 8.6</td><td>78.0±8.9</td><td>108.5 ±0.6</td><td>109.3 ± 0.9</td><td>109.2 ± 4.6</td><td>112.3 ± 0.8</td></tr><tr><td>walker2d-medium-expert-v2</td><td>102.5 ± 4.4</td><td>103.1 ± 4.4</td><td>110.7 ± 0.2</td><td>114.0 ± 0.5</td><td>50.5 ±38.2</td><td>111.2 ± 0.9</td></tr><tr><td>Average</td><td>79.9</td><td>85.9</td><td>105.1</td><td>106.0</td><td>76.6</td><td>106.9</td></tr></table>
+Results for AntMaze Domain. The sparse reward and large amount of sub-optimal trajectories make the AntMaze tasks especially difficult. Strong and stable Q-learning is required to achieve good performance. For example, BC-based methods could easily fail on ‘medium’ and ‘large’ tasks. We show that the proposed Q-learning guidance added during the training of a conditional diffusion model is stable and effective. Empirically, with a proper $\eta$ , we find Diffusion-QL outperforms the prior methods by a clear margin, especially in harder tasks, such as ‘large-diverse’.
+Results for Adroit and Kitchen Domain. We find the Adroit domain needs strong policy regularization to overcome the extrapolation error (Fujimoto et al., 2019) in offline RL, due to the narrowness of the human demonstrations. With a smaller $\eta$ , Diffusion-QL easily beats the other baselines by its reverse diffusion-based policy, which has high expressiveness and better policy regularization. Moreover, long-term value optimization is required for the Kitchen tasks, and we find Diffusion-QL also performs very well in this domain.
+# 5.2 ABLATION STUDY
+In this section, we analyze why Diffusion-QL outperforms the other policy-constraint based methods quantitatively on D4RL tasks. We conduct an ablation study on the two main components of Diffusion-QL: use of a diffusion model as an expressive policy and Q-learning guidance. For the policy part, we compare our diffusion model with the popular CVAE model for behavior cloning. For the Q-learning component, we compare with the BCQ approach on policy improvement.
+As shown in Table 2, BC-Diffusion model outperforms BC-CVAE, validating that our diffusionbased policy is more expressive and better at capturing the data distributions (as we would expect from the results in figure 1). Under the same BCQ framework, which explicitly limits how far the action samples from polices could deviate from the cloned action samples, BCQ-Diffusion still works better than BCQ-CVAE. The hard and physical value constraints on the deviation by BCQ actually limits the policy improvement, as we can see that, Diffusion-QL further boosts the performance. Note Diffusion-QL applies the diffusion model learning itself as a soft policy regularization and guides the policy optimization via additive Q-learning. The weak expressiveness and poor cloning behavior of CVAE makes it fail when coupling with a free Q-learning guidance, as shown by the results of CVAE-QL. In a nutshell, the ablation study shows that the two components of Diffusion-QL are working together to produce good performance.
+# 6 CONCLUSION
+In this work, we present Diffusion-QL, which is a conditional diffusion-based offline RL algorithm. First, its policy is built by the reverse chain of a conditional diffusion model, which allows for a highly expressive policy class and whose learning itself acts as a strong policy regularization method. Second, Q-learning guidance through a jointly learned Q-value function is injected in the learning of the diffusion policy, which guides the denoising sampling towards the optimal region in its exploration area. The two key components contribute to its state-of-the-art performance across all tasks in the D4RL benchmark.
+# ACKNOWLEDGEMENTS
+Z. Wang and M. Zhou acknowledge the support of NSF-IIS 2212418 and the Texas Advanced Computing Center (TACC) for providing HPC resources that have contributed to the research results reported within this paper.
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+# Appendix
+# A MORE TOY EXPERIMENTS
+Here we describe an additional toy experiment on a bandit task. Actions are again in a real-valued 2D space, $\begin{array} { r l r } { { \bf a } } & { { } \in } & { [ - 1 , 1 ] ^ { 2 } } \end{array}$ . The offline data $\begin{array} { r c l } { \mathcal { D } } & { = } & { \{ ( { \bf a } _ { i } ) \} _ { i = 1 } ^ { 1 0 0 0 0 } } \end{array}$ are collected by sampling actions equally from four Gaussian distributions with centers $\mu \in$ $\{ ( - 0 . 8 , \bar { 0 } . 8 ) , ( \bar { 0 . 8 } , \bar { 0 . 8 } ) , ( 0 . 8 , - 0 . \bar { 8 } ) , ( - 0 . 8 , - 0 . 8 ) \}$ and standard deviations ${ \pmb \sigma } _ { d } = ( 0 . 0 5 , 0 . 0 5 )$ , as depicted in the first panel of Figure 4. We conduct the same experiments as the ones in our main paper (see figure 1) and show the performance in Figure 4. The only difference in this experiment is that the samples are now in the corners of the ation space.
+For behavior-cloning experiments, we observe that only our diffusion model could recover the original data distribution while the prior regularization methods fail in some way. For example, CVAE could only capture the two diagonal modes and place density between them, while MMD tends to align density around the boundaries because of its Tanh-Gaussian policy. For Q-learning experiments, we observe that the prior regularization methods typically push the policy to converge to sub-optimal solutions in BCQ and BEAR while preventing the policy of $\mathrm { T D } 3 { + } \mathrm { B C }$ from being concentrated on the right corner. However, the policy of Diffusion-QL successfully converges to the optimal bottom corner. The ablation study experiments are consistent with our conclusion in the main paper.
+![](images/181d726c98a3785888a1597f23d5e1c8bef7ef5621f2032a76f6096ceb356228.jpg)
+Figure 4: Bandit experiment with a strong multi-modal behavior policy. The first row shows the comparison of behavior-cloning results between our method and prior methods. The second row shows the comparison results with Q-learning involved. The third row shows the ablation study of $N$ for BC-Diffusion. The fourth row shows the learned reward function and the ablation study of $N$ for Diffusion-QL.
+# B IMPLEMENTATION DETAILS
+Diffusion policy. We build our policy as an MLP-based conditional diffusion model. Following the parameterization of Ho et al. (2020), the model itself is a residual model, $\epsilon _ { \theta } ( a ^ { i } , s , i )$ , where the $i$ is the last timestep and $\pmb { s }$ is the state condition. We model $\epsilon _ { \theta }$ as a 3-layer MLPs with Mish activations and we use 256 hidden units for all networks. The input of $\epsilon _ { \theta }$ is the concatenation of the last step action vector, the current state vector, and the sinusoidal positional embedding of timestep $i$ . The output of $\epsilon _ { \theta }$ is the predicted residual at diffusion timestep $i$ .
+Q networks. We build two Q networks with the same MLP setting as our diffusion policy, which has 3-layer MLPs with Mish activations and 256 hidden units for all networks.
+We use the Adam (Kingma & Ba, 2014) optimizer for the training of both Diffusion policy and Q networks.
+# C EXPERIMENTAL DETAILS
+We train our algorithm with 2000 epochs for Gym tasks and 1000 epochs for the other tasks, where each epoch consists of 1000 gradient steps. For the Gym locomotion tasks, we average mean returns over 6 independently trained models and 10 trajectories per mode. For the other tasks, we average over 6 independently trained models and 100 evaluation trajectories.
+We investigate two model selection methods, online and offline model selection. For online model selection, following the convention of supervised learning, the best-performed model is saved during training and used for final evaluation. For offline case, we select the model based on our lagging indicator $\mathcal { L } _ { d }$ and more details are in Appendix D.
+We use the original task rewards from MuJoCo Gym and Kitchen tasks. We standardize Adroit task rewards for training stability. We modify the rewards according to the suggestion of CQL (Kumar et al., 2020) for the AntMaze datasets.
+# D OFFLINE MODEL SELECTION
+For reducing the training cost and picking the best model during training without any interaction with the real environment, we provide a way to properly conduct early stopping for DiffusionQL. Empirically, we found that $\mathcal { L } _ { d }$ loss is a lagging indicator of online performance. Note $\mathcal { L } _ { d }$ is the behavior cloning loss of our diffusion model-based policy and measures how close the current policy is to the behavior policy. For offline RL, we want $\mathcal { L } _ { d }$ to be small to get rid of the OOD issue but we also don’t expect $\mathcal { L } _ { d }$ to be the smallest since our goal is policy learning while not behavior cloning. Hence, we monitor the $\mathcal { L } _ { d }$ and save multiple model checkpoints during training. We stop the training whenever $\mathcal { L } _ { d }$ increases in our evaluation stage for early stopping. When the training stopped, we picked the checkpoint according to our metric, the second or third lowest $\mathcal { L } _ { d }$ value. Note our model selection part is totally offline and only based on $\mathcal { L } _ { d }$ without any access to the environment. The selection of the $\mathcal { L } _ { d }$ is not very sensitive. Always using the 2nd smallest checkpoints doesn’t impact the performance much. For example, with 2nd checkpoints selected, for the Gym domain, the average score is 87.6, and for the AntMaze domain, the average score is 69.1.
+# E HYPERPARAMETERS
+For Diffusion- $\mathrm { Q L }$ , we consider three hyperparameters in total: learning rate, Q-learning weight $\eta$ , and whether to use max Q backup from CQL (Kumar et al., 2020). For the learning rate of Adam, we consider values in the grid $\{ 1 \tilde { \times } 1 0 ^ { - 3 } , \bar { 3 \times } 1 0 ^ { - 4 } , 3 \times 1 0 ^ { - 5 } \}$ for the policy, while we use a fixed learning rate, $3 \times 1 0 ^ { - 4 }$ , for Q-networks. For $\eta$ , we consider values according to the characteristics of different domains, as we mentioned in the description of datasets that the Adroit and Kitchen tasks require more policy regularization and the AntMaze tasks require more Q-learning. For max Q backup, we only apply it on the AntMaze tasks. Based on these considerations, we provide our hyperparameter setting in Table 3.
+Table 3: Hyperparameter settings of all selected tasks.
+<table><tr><td>Tasks</td><td>learning rate</td><td>n</td><td>max Qbackup</td></tr><tr><td>halfcheetah-medium-v2</td><td>3×10-4</td><td>1.0</td><td>False</td></tr><tr><td>hopper-medium-v2</td><td>3×10-4</td><td>1.0</td><td>False</td></tr><tr><td>walker2d-medium-v2</td><td>3×10-4</td><td>1.0</td><td>False</td></tr><tr><td>halfcheetah-medium-replay-v2</td><td>3×10-4</td><td>1.0</td><td>False</td></tr><tr><td>hopper-medium-replay-v2</td><td>3×10-4</td><td>1.0</td><td>False</td></tr><tr><td>walker2d-medium-replay-v2</td><td>3×10-4</td><td>1.0</td><td>False</td></tr><tr><td>halfcheetah-medium-expert-v2</td><td>3×10-4</td><td>1.0</td><td>False</td></tr><tr><td>hopper-medium-expert-v2</td><td>3×10-4</td><td>1.0</td><td>False</td></tr><tr><td>walker2d-medium-expert-v2</td><td>3×10-4</td><td>1.0</td><td>False</td></tr><tr><td>antmaze-umaze-vo</td><td>3×10-4</td><td>0.5</td><td>False</td></tr><tr><td>antmaze-umaze-diverse-v0</td><td>3×10-4</td><td>2.0</td><td>True</td></tr><tr><td>antmaze-medium-play-v0</td><td>1×10-3</td><td>2.0</td><td>True</td></tr><tr><td>antmaze-medium-diverse-vO</td><td>3×10-4</td><td>3.0</td><td>True</td></tr><tr><td>antmaze-large-play-vo</td><td>3×10-4</td><td>4.5</td><td>True</td></tr><tr><td>antmaze-large-diverse-vO</td><td>3×10-4</td><td>3.5</td><td>True</td></tr><tr><td>pen-human-v1</td><td>3×10-5</td><td>0.15</td><td>False</td></tr><tr><td>pen-cloned-v1</td><td>3×10-5</td><td>0.1</td><td>False</td></tr><tr><td>kitchen-complete-v0</td><td>3×10-4</td><td>0.005</td><td>False</td></tr><tr><td>kitchen-partial-vo</td><td>3×10-4</td><td>0.005</td><td>False</td></tr><tr><td>kitchen-mixed-v0</td><td>3×10-4</td><td>0.005</td><td>False</td></tr></table>
+# F OPTIMAL RESULTS
+If a small amount of online experience is provided during the evaluation stage for model selection, we can pick the best models during training via online evaluations (similar to early stopping in supervised learning). This regime provides a further boost in the performance of Diffusion-QL as shown in Table 4.
+# G LIMITATIONS AND FUTURE WORK
+Diffusion policies are highly expressive and hence they can capture multi-modal distributions well. We have shown this results in learning better policies in offlineRL. However, at the inference time, the reverse sampling defined in Equation (1) requires iteratively computing $\epsilon _ { \theta }$ networks $N$ times, and this can become a bottleneck for the running time. In our case, we couple the learning of diffusion policies with Q-learning, and achieve good performance with small $N = 5$ . Diffusion policies with $N = 5$ could be four to five times slower in action inference compared to previous one-step feedforward policies, and hence the inference cost could prevent the approach from deployment in some real-world scenarios, where fast response is necessary.
+This motivates potential future works. There have been many recent works focusing on improving the sampling speed of diffusion models (Lu et al., 2022; Wang et al., 2022; Xiao et al., 2021; Salimans & Ho, 2022; Song et al., 2020; Zheng et al., 2022), which could be applied to improve the sampling efficiency of Diffusion-QL. For example, the diffusion policy may be able to be distilled into a simpler feedforward policy after training.
+# H GAUSSIAN MIXTURE POLICY
+We test a Gaussian Mixture policy as a classic policy class that can capture multi-modal distributions as an additional baseline. We modify $\mathrm { T D } 3 { + } \mathrm { B C }$ (Fujimoto & Gu, 2021) by replacing the original deterministic actor with a mixture density network (Bishop, 1994), where each mixture component is a Gaussian. Since a Gaussian mixture policy is applied, we replaced minimizing the L2 loss (from $\mathrm { T D } 3 { + } \mathrm { B C } )$ between predicted actions and real actions, with maximizing the likelihood estimate of Gaussian mixtures on real state-action pairs. We keep all the other parts the same as $\mathrm { T D } 3 { + } \mathrm { B C }$ .
+Table 4: Performance comparison with online model selection and offline model selection.
+<table><tr><td>Gym Tasks</td><td>Diffusion-QL (Offline)</td><td>Diffusion-QL (Online)</td></tr><tr><td>halfcheetah-medium-v2</td><td>51.1 ± 0.5</td><td>51.5 ± 0.3</td></tr><tr><td>hopper-medium-v2</td><td>90.5 ± 4.6</td><td>96.6 ± 3.4</td></tr><tr><td>walker2d-medium-v2</td><td>87.0 ± 0.9</td><td>87.3 ± 0.5</td></tr><tr><td>halfcheetah-medium-replay-v2</td><td>47.8 ± 0.3</td><td>48.3 ± 0.2</td></tr><tr><td>hopper-medium-replay-v2</td><td>101.3 ± 0.6</td><td>102.0 ± 0.4</td></tr><tr><td>walker2d-medium-replay-v2</td><td>95.5 ± 1.5</td><td>98.0 ± 0.5</td></tr><tr><td>halfcheetah-medium-expert-v2</td><td>96.8 ± 0.3</td><td>97.2 ± 0.4</td></tr><tr><td>hopper-medium-expert-v2</td><td>111.1 ± 1.3</td><td>112.3 ± 0.8</td></tr><tr><td>walker2d-medium-expert-v2</td><td>110.1 ± 0.3</td><td>111.2 ± 0.9</td></tr><tr><td>Average</td><td>88.0</td><td>89.3</td></tr><tr><td>AntMaze Tasks</td><td>Diffusion-QL (Offline)</td><td>Diffusion-QL (Online)</td></tr><tr><td>antmaze-umaze-vo</td><td>93.4 ± 3.4</td><td>96.0 ± 3.3</td></tr><tr><td>antmaze-umaze-diverse-v0</td><td>66.2 ± 8.6</td><td>84.0 ± 10.1</td></tr><tr><td>antmaze-medium-play-v0</td><td>76.6 ± 10.8</td><td>79.8 ± 8.7</td></tr><tr><td>antmaze-medium-diverse-v0</td><td>78.6 ± 10.3</td><td>82.0 ± 9.5</td></tr><tr><td>antmaze-large-play-v0</td><td>46.4 ± 8.3</td><td>49.0 ± 9.4</td></tr><tr><td>antmaze-large-diverse-v0</td><td>56.6 ± 7.6</td><td>61.7 ± 8.2</td></tr><tr><td>Average</td><td>69.6</td><td>75.4</td></tr><tr><td>Adroit Tasks</td><td>Diffusion-QL (Offline)</td><td>Diffusion-QL (Online)</td></tr><tr><td>pen-human-v1</td><td>72.8 ± 9.6</td><td>75.7± 9.0</td></tr><tr><td>pen-cloned-v1</td><td>57.3 ± 11.9</td><td>60.8 ± 11.8</td></tr><tr><td>Average</td><td>65.1</td><td>68.3</td></tr><tr><td>Kitchen Tasks</td><td>Diffusion-QL (Offline)</td><td>Diffusion-QL (Online)</td></tr><tr><td>kitchen-complete-v0</td><td>84.0 ± 7.4</td><td>84.5 ± 6.1</td></tr><tr><td>kitchen-partial-v0</td><td>60.5 ± 6.9</td><td>63.7 ± 5.2</td></tr><tr><td>kitchen-mixed-v0</td><td>62.6 ± 5.1</td><td>66.6 ± 3.3</td></tr><tr><td>Average</td><td>69.0</td><td>71.6</td></tr></table>
+Table 5: Performance comparison for Gaussian Mixture ablation study.
+<table><tr><td>Gym Tasks</td><td>TD3+BC</td><td>TD3+BC-GM</td><td>Diffusion-QL</td></tr><tr><td>halfcheetah-medium-expert-v2</td><td>90.4</td><td>49.0</td><td>96.8</td></tr><tr><td>hopper-medium-expert-v2</td><td>98.0</td><td>40.0</td><td>111.1</td></tr><tr><td>walker2d-medium-expert-v2</td><td>110.1</td><td>66.5</td><td>110.1</td></tr></table>
+We evaluated the model (we call it $\mathrm { T D } 3 { + } \mathrm { B C } { \cdot } \mathrm { G M } \rangle$ ) on both our bandit toy experiment and selected D4RL tasks. As shown in Figure 5, with a properly selected number of mixtures, the Gaussian mixture could capture the multi-modal distribution in our behavior cloning experiment. However, in the Q-learning experiment, it fails to converge to the optimal target location, and always places some density on the suboptimal modes, resulting in a suboptimal policy with multi-modes. For D4RL experiments, we set the number of mixtures to be 3, and seen in Table 5, we observed that $\mathrm { T D } 3 { + } \mathrm { B C } { - } \mathrm { G M }$ does not perform well on the three D4RL tasks, which is consistent with our previous observation that $\mathrm { T D } 3 { + } \mathrm { B C } { - } \mathrm { G M }$ is prone to have suboptimal actions.
+One reason for the poor performance could be that Gaussian mixture models are challenging to fit well, particularly in higher dimensional spaces. We are also forced to choose the number of mixture components, which for the D4RL experiments we don’t know a priori how many components there are. Moreover, each mixture component is often parameterized with a diagonal Gaussian that has limited ability in capturing the dependence between different dimensions.
+![](images/bb9d33f44ed325e7df174e40e32ef904e23fcb4b55194a931d7cb5f3c2def080.jpg)
+Figure 5: Gaussian Mixture ablation study. The first row shows the behavior cloning experiment and the second row shows the Q-learning experiment, which are the same set of experiments described in Section 4. Here, $K$ is the number of Gaussian mixtures.

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+# MONODISTILL: LEARNING SPATIAL FEATURES FOR MONOCULAR 3D OBJECT DETECTION
+Zhiyu Chong∗1, Xinzhu $\mathbf { M _ { a } } { ^ { * 2 } }$ , Hong Zhang1, Yuxin Yue1, Haojie $\mathbf { L i } ^ { 1 }$ , Zhihui Wang1, and Wanli Ouyang2
+1Dalian University of Technology 2The University of Sydney {czydlut, jingshui, 22017036}@mail.dlut.edu.cn {xinzhu.ma, wanli.ouyang}@sydney.edu.au {hjli, zhwang}@dlut.edu.cn
+# ABSTRACT
+3D object detection is a fundamental and challenging task for 3D scene understanding, and the monocular-based methods can serve as an economical alternative to the stereo-based or LiDAR-based methods. However, accurately detecting objects in the 3D space from a single image is extremely difficult due to the lack of spatial cues. To mitigate this issue, we propose a simple and effective scheme to introduce the spatial information from LiDAR signals to the monocular 3D detectors, without introducing any extra cost in the inference phase. In particular, we first project the LiDAR signals into the image plane and align them with the RGB images. After that, we use the resulting data to train a 3D detector (LiDAR Net) with the same architecture as the baseline model. Finally, this LiDAR Net can serve as the teacher to transfer the learned knowledge to the baseline model. Experimental results show that the proposed method can significantly boost the performance of the baseline model and ranks the $1 ^ { s t }$ place among all monocularbased methods on the KITTI benchmark. Besides, extensive ablation studies are conducted, which further prove the effectiveness of each part of our designs and illustrate what the baseline model has learned from the LiDAR Net. Our code will be released at https://github.com/monster-ghost/MonoDistill.
+# 1 INTRODUCTION
+3D object detection is an indispensable component for 3D scene perception, which has wide applications in the real world, such as autonomous driving and robotic navigation. Although the algorithms with stereo (Li et al., 2019b; Wang et al., 2019; Chen et al., 2020a) or LiDAR sensors (Qi et al., 2018; Shi et al., 2019; 2020) show promising performances, the heavy dependence on the expensive equipment restricts the application of these algorithms. Accordingly, the methods based on the cheaper and more easy-to-deploy monocular cameras (Xu & Chen, 2018; Ma et al., 2019; 2021; Brazil & Liu, 2019; Ding et al., 2020) show great potentials and have attracted lots of attention.
+As shown in Figure 1 (a), some prior works (Brazil & Liu, 2019; Simonelli et al., 2019; Chen et al., 2020b) directly estimate the 3D bounding boxes from monocular images. However, because of the lack of depth cues, it is extremely hard to accurately detect the objects in the 3D space, and the localization error is the major issue of these methods (Ma et al., 2021). To mitigate this problem, an intuitive idea is to estimate the depth maps from RGB images, and then use them to augment the input data (Figure 1 (b)) (Xu & Chen, 2018; Ding et al., 2020) or directly use them as the input data (Figure 1 (c)) (Wang et al., 2019; Ma et al., 2019). Although these two strategies have made significant improvement in performance, the drawbacks of them can not be ignored: (1) These methods generally use an off-the-shelf depth estimator to generate the depth maps, which introduce lots of computational cost (e.g. the most commonly used depth estimator (Fu et al., 2018) need about $4 0 0 \mathrm { m s }$ to process a standard KITTI image). (2) The depth estimator and detector are trained separately, which may lead to a sub-optimal optimization. Recently, Reading et al. (2021) propose an end-to-end framework (Figure 1 (d)) for monocular 3D detection, which can also leverage depth estimator to provide depth cues. Specifically, they introduce a sub-network to estimate the depth distribution and use it to enrich the RGB features. Although this model can be trained end-to-end and achieves better performance, it still suffer from the low inference speed (630ms per image), mainly caused by the depth estimator and the complicated network architecture. Note that the welldesigned monocular detectors, like Ma et al. (2021); Zhang et al. (2021b); Lu et al. (2021), only take about $4 0 \mathrm { m s }$ per image.
+![](images/3fa330cfff050e5dc2ed89024e93f34bd6cce756c45abec7b51ede2082a6377c.jpg)
+Figure 1: Comparison on the high-level paradigms of monocular 3D detectors.
+In this paper, we aim to introduce the depth cues to the monocular 3D detectors without introducing any extra cost in the inference phase. Inspired by the knowledge distillation (Hinton et al.), which can transfer the learned knowledge from a well-trained CNN to another one without any changes in the model design, we propose that the spatial cues may also be transferred by this way from the LiDAR-based models. However, the main problem for this proposal is the difference of the feature representations used in these two kinds of methods (2D images features vs. 3D voxel features). To bridge this gap, we propose to project the LiDAR signals into the image plane and use the 2D CNN, instead of the commonly used 3D CNN or point-wise CNN, to train a ‘image-version’ LiDAR-based model. After this alignment, the knowledge distillation can be friendly applied to enrich the features of our monocular detector.
+Based on the above-mentioned motivation and strategy, we propose the distillation based monocular 3D detector (MonoDistill): We first train a teacher net using the projected LiDAR maps (used as the ground truths of the depth estimator in previous works), and then train our monocular 3D detector under the guidance of the teacher net. We argue that, compared with previous works, the proposed method has the following two advantages: First, our method directly learn the spatial cues from the teacher net, instead of the estimated depth maps. This design performs better by avoiding the information loss in the proxy task. Second, our method does not change the network architecture of the baseline model, and thus no extra computational cost is introduced.
+Experimental results on the most commonly used KITTI benchmark, where we rank the $1 ^ { s t }$ place among all monocular based models by applying the proposed method on a simple baseline, demonstrate the effectiveness of our approach. Besides, we also conduct extensive ablation studies to present each design of our method in detail. More importantly, these experiments clearly illustrate the improvements are achieved by the introduction of spatial cues, instead of other unaccountable factors in CNN.
+# 2 RELATED WORKS
+3D detection from only monocular images. 3D detection from only monocular image data is challenging due to the lack of reliable depth information. To alleviate this problem, lots of scholars propose their solutions in different ways, including but not limited to network design (Roddick et al., 2019; Brazil & Liu, 2019; Zhou et al., 2019; Liu et al., 2020; Luo et al., 2021), loss formulation (Simonelli et al., 2019; Ma et al., 2021), 3D prior (Brazil & Liu, 2019), geometric constraint (Mousavian et al., 2017; Qin et al., 2019; Li et al., 2019a; Chen et al., 2020b), or perspective modeling (Zhang et al., 2021a; Lu et al., 2021; Shi et al., 2021).
+Depth augmented monocular 3D detection. To provide the depth information to the 3D detectors, several works choose to estimate the depth maps from RGB images. According to the usage of the estimated depth maps, these methods can be briefly divided into three categories. The first class of these methods use the estimated depth maps to augment the RGB images (Figure 1 (b)). In particular, Xu & Chen (2018) propose three fusion strategies of the RGB images and depth maps, while Ding et al. (2020) and Wang et al. (2021a) focus on how to enrich the RGB features with depth maps in the latent feature space. Besides, Wang et al. (2019); Ma et al. (2019) propose another pipeline (Figure 1 (c)): They back-project the depth maps into the 3D space, and then train a LiDAR-based model and use the resulting data (pseudo-LiDAR signal) to predict the 3D boxes. This framework shows promising performance, and lots of works (Weng & Kitani, 2019; Cai et al., 2020; Wang et al., 2020a; Chu et al., 2021) are built on this solid foundation. Recently, Reading et al. (2021) propose another way to leverage the depth cues for monocular 3D detection (Figure 1 (d)). Particularly, they first estimate the depth distribution using a sub-network, and then use it to lift the 2D features into 3D features, which is used to generate the final results. Compared with the previous two families, this model can be trained in the end-to-end manner, avoiding the sub-optimal optimization. However, a common disadvantage of these methods is that they inevitably increase the computational cost while introducing depth information. Unlike these methods, our model chooses to learn the feature representation under the guidance of depth maps, instead of integrating them. Accordingly, the proposed model not only introduces rich depth cues but also maintains high efficiency.
+![](images/5dde672b08f0974af574d0d2d4a04bd72aaacc58bf342a1fc846bdc6022a2a76.jpg)
+Figure 2: Visualization of the sparse LiDAR maps (left) and the dense LiDAR maps (right).
+Knowledge distillation. Knowledge distillation (KD) is initially proposed by Hinton et al. for model compression, and the main idea of this mechanism is transferring the learned knowledge from large CNN models to the small one. This strategy has been proved in many computer vision tasks, such as 2D object detection (Dai et al., 2021; Chen et al., 2017; Gupta et al., 2016), semantic semantic segmentation (Hou et al., 2020; Liu et al., 2019). However, few work explore it in monocular 3D detection. In this work, we design a KD-based paradigm to efficiently introduce depth cues for monocular 3D detectors.
+LIGA Stereo. We found a recent work LIGA Stereo (Guo et al., 2021) (submitted to arXiv on 18 Aug. 2021) discusses the application of KD for stereo 3D detection under the guidance of LiDAR signals. Here we discuss the main differences of LIGA Stereo and our work. First, the tasks and underlying data are different (monocular vs. stereo), which leads to different conclusions. For example, Guo et al. (2021) concludes that using the predictions of teacher net as ‘soft label’ can not bring benefits. However, our experimental results show the effectiveness of this design. Even more, in our task, supervising the student net in the result space is more effective than feature space. Second, they use an off-the-shelf LiDAR-based model to provide guidance to their model. However, we project the LiDAR signals into image plane and use the resulting data to train the teacher net. Except for the input data, the teacher net and student net are completely aligned, including network architecture, hyper-parameters, and training schedule. Third, to ensure the consistent shape of features, LIGA Stereo need to generate the cost volume from stereo images, which is time-consuming (it need about $3 5 0 \mathrm { m s }$ to estimate 3D boxes from a KITTI image) and hard to achieve for monocular images. In contrast, our method align the feature representations by adjusting the LiDAR-based model, instead of the target model. This design makes our method more efficient (about $3 5 \mathrm { m s }$ per image) and can generalize to all kinds of image-based models in theory.
+# 3 METHOD
+# 3.1 OVERVIEW
+Figure 3 presents the framework of the proposed MonoDistill, which mainly has three components: a monocular 3D detector, an aligned LiDAR-based detector, and several side branches which build
+![](images/21d6ed0b66761b9e1a051bfa6abcdee0ef3180df8a9a46cb1f79def5f9635a05.jpg)
+Figure 3: Illustration of the proposed MonoDistill. We first generate the ‘image-like’ LiDAR maps from the LiDAR signals and then train a teacher model using an identical network to the student model. Finally, we propose three distillation schemes to train the student model under the guidance of the well-trained teacher net. In the inference phase, only the student net is used.
+the bridge to provide guidance from the LiDAR-based detector to our monocular 3D detector. We will introduce the how to build these parts one by one in the rest of this section.
+# 3.2 BASELINE MODEL
+Student model. We use the one-stage monocular 3D detector MonoDLE (Ma et al., 2021) as our baseline model. Particularly, this baseline model adopts DLA-34 (Yu et al., 2017) as the backbone and uses several parallel heads to predict the required items for 3D object detection. Due to this clean and compact design, this model achieves good performance with high efficiency. Besides, we further normalize the confidence of each predicted object using the estimated depth uncertainty (see Appendix A.1 for more details), which brings about 1 AP improvement
+Teacher model. Existing LiDAR-based models are mainly based on the 3D CNN or point-wise CNN. To align the gap between the feature representations of the monocular detector and the LiDAR-based detector, we project the LiDAR points into the image plane to generate the sparse depth map. Further, we also use the interpolation algorithm (Ku et al., 2018) to generate the dense depth, and see Figure 2 for the visualization of generated data. Then, we use these ‘image-like LiDAR maps’ to train a LiDAR-based detector using the identical network with our student model.
+# 3.3 MONODISTILL
+In order to transfer the spatial cues from the well-trained teacher model to the student model, we design three complementary distillation schemes to provide additional guidance to the baseline model.
+Scene-level distillation in the feature space. First, we think directly enforcing the image-based model learns the feature representations of the LiDAR-based models is sub-optimal, caused by the different modalities. The scene level knowledge can help the monocular 3D detectors build a highlevel understanding for the given image by encoding the relative relations of the features, keeping the knowledge structure and alleviating the modality gap. Therefore, we train our student model under the guidance of the high-level semantic features provided by the backbone of the teacher model. To better model the structured cues, we choose to learn the affinity map (Hou et al., 2020) of high-level features, instead of the features themselves. Specifically, we first generate the affinity map, which encodes the similarity of each feature vector pair, for both the teacher and student network, and each element $\mathrm { A } _ { \mathrm { i , j } }$ in this affinity map can be computed by:
+$$
+\mathrm { { A } _ { i , j } = \frac { \mathbf { f _ { i } ^ { T } } \mathbf { f _ { j } } } { | | \mathbf { f _ { i } } | | _ { 2 } \cdot | | \mathbf { f _ { j } } | | _ { 2 } } , }
+$$
+where $\mathbf { f _ { j } }$ and $\mathbf { f _ { j } }$ denote the $\mathbf { i } ^ { t h }$ and $\mathbf { j } ^ { t h }$ feature vector. After that, we use the L1 norm to enforce the student net to learn the structured information from the teacher net:
+$$
+\mathcal { L } _ { \mathbf { s f } } = \frac { 1 } { \mathrm { K } \times \mathrm { K } } \sum _ { i = 1 } ^ { K } \sum _ { j = 1 } ^ { K } | | \mathrm { A } _ { \mathbf { i } , \mathbf { j } } ^ { \mathbf { t } } - \mathrm { A } _ { \mathbf { i } , \mathbf { j } } ^ { \mathbf { s } } | | _ { 1 } ,
+$$
+where K is the number of the feature vectors. Note the computational/storage complexity is quadratically related to K. To reduce the cost, we group all features into several local regions and generate the affinity map using the features of local regions. This makes the training of the proposed model more efficient, and we did not observe any performance drop caused by this strategy.
+Object-level distillation in the feature space. Second, except for the affinity map, directly using the features from teacher net as guidance may also provide valuable cues to the student. However, there is much noise in feature maps, since the background occupies most of the area and is less informative. Distilling knowledge from these regions may make the network deviate from the right optimization direction. To make the knowledge distillation more focused, limiting the distillation area is necessary. Particularly, the regions in the ground-truth 2D bounding boxes are used for knowledge transfer to mitigate the effects of noise. Specifically, given the feature maps of the teacher model and student model $\{ \mathbf { F ^ { t } } , \mathbf { F ^ { s } } \}$ , our second distillation loss can be formulated as.
+$$
+\mathcal { L } _ { \mathbf { o f } } = \frac { 1 } { \mathrm { N } _ { \mathrm { p o s } } } | | \mathrm { M } _ { \mathbf { o f } } ( \mathrm { F } _ { \mathbf { s } } - \mathrm { F } _ { \mathbf { t } } ) | | _ { 2 } ^ { 2 } ,
+$$
+where $\mathrm { M _ { o f } }$ is the mask generated from the center point and the size of 2D bounding box and $\mathrm { N _ { p o s } }$ is the number of valid feature vectors.
+Object-level distillation in the result space. Third, similar to the traditional KD, we use the predictions from the teacher net as extra ‘soft label’ for the student net. Note that in this scheme, only the predictions on the foreground region should be used, because the predictions on the background region are usually false detection. As for the definition of the ‘foreground regions’, inspired by CenterNet (Zhou et al., 2019), a simple baseline is regarding the center point as the foreground region. Furtherly, we find that the quality of the predicted value of the teacher net near the center point is good enough to guide the student net. Therefore, we generate a Gaussian-like mask (Tian et al., 2019; Wang et al., 2021b) based on the position of the center point and the size of 2D bounding box and the pixels whose response values surpass a predefined threshold are sampled, and then we train these samples with equal weights (see Figure 4 for the visualization). After that, our third distillation loss can be formulated as:
+$$
+\mathcal { L } _ { \mathbf { o r } } = \sum _ { k = 1 } ^ { N } | | \mathbf { M _ { o r } ( y _ { k } ^ { s } - y _ { k } ^ { t } ) } | | _ { 1 } ,
+$$
+where $\mathrm { M } _ { \mathbf { o r } }$ is the mask which represents positive and negative samples, $\mathbf { y _ { k } }$ is the output of the $k ^ { t h }$ detection head and $N$ is the number of detection heads.
+Additional strategies. We further propose some strategies for our method. First, for the distillation schemes in the feature space (i.e. $\mathcal { L } _ { \mathrm { s f } }$ and $\mathcal { L } _ { \mathbf { o f } } ^ { \mathrm { ~ ~ } }$ ), we only perform them on the last three blocks of the backbone. The main motivation of this strategy is: The first block usually is rich in the low-level features (such as edges, textures, etc.). The expression forms of the low-level features for LiDAR and image data may be completely different, and enforcing the student net to learn these features in a modality-across manner may mislead it. Second, in order to better guide the student to learn spatial-aware feature representations, we apply the attention based fusion module (FF in Table 1) proposed by Chen et al. (2021b) in our distillation schemes in the feature space ( i.e. $\mathcal { L } _ { \mathrm { s f } }$ and $\mathcal { L } _ { \mathbf { o f } }$ ).
+Loss function. We train our model in an end-to-end manner using the following loss function:
+$$
+{ \mathcal { L } } = { \mathcal { L } } _ { \mathbf { s r c } } + \lambda _ { 1 } \cdot { \mathcal { L } } _ { \mathbf { s f } } + \lambda _ { 2 } \cdot { \mathcal { L } } _ { \mathbf { o f } } + \lambda _ { 3 } \cdot { \mathcal { L } } _ { \mathbf { o r } } ,
+$$
+where $\mathcal { L } _ { \mathrm { s r c } }$ denotes the loss function used in the MonoDLE (Ma et al., 2021). $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 }$ are the hyper-parameters to balance each loss. For the teacher net, only $\mathcal { L } _ { \mathrm { s r c } }$ is adopted.
+![](images/f461b5ed95d7a231a3d707af2b5eec44ac7197d01ed736d51fa6c546069e9f8a.jpg)
+Figure 4: Left: Regard the center point as the foreground region. Right: Generate foreground region from the center point and the size of bounding box. Besides, the 2D bounding boxes are used as the foreground region for $\mathcal { L } _ { \mathbf { o f } }$ .
+# 4 EXPERIMENTS
+# 4.1 SETUP
+Dataset and metrics. We conduct our experiments on the KITTI (Geiger et al., 2012), which is most commonly used dataset in 3D detection task. Specifically, this dataset provides 7,481 training samples and 7,518 testing samples, and we further divide the training data into a train set (3,712 samples) and a validation set (3,769 samples), following prior works (Chen et al., 2015). Both 3D detection and Bird’s Eye View (BEV) detection are evaluated using $\mathrm { { A P } | _ { R _ { 4 0 } } }$ (Simonelli et al., 2019) as metric. We report our final results on the testing set, while the ablation studies are conducted on the validation set. Besides, we mainly focus on the Car category, while also present the performances of Pedestrian and Cyclist in Appendix A.2 for reference.
+Implementation. We provide the implementation details in Appendix A.1. Besides, our code will be open-sourced for the reproducibility.
+# 4.2 MAIN RESULTS
+Ablation studies. Table 1 shows the ablation studies of the proposed methods. Specifically, we found that all three distillation schemes can improve the accuracy of the baseline model, and the improvements of them are complementary. Besides, the feature fusion strategy can also boost the accuracy. Compared with the baseline, our full model improves 3D detection performance by 3.34, 5.02, 2.98 and improve BEV performance by 5.16, 6.62, 3.87 on the moderate, easy and hard settings respectively.
+Table 1: Ablation studies on the KITTI validation set. SF, OF, and OR denote the scene-level distillation in feature space, the object-level distillation in feature space, and the object-level distillation in result space, respectively. Besides, FF means the attention based feature fusion strategy.
+<table><tr><td rowspan="3"></td><td rowspan="3">SF</td><td rowspan="3">OF</td><td rowspan="3">OR</td><td rowspan="3">FF</td><td colspan="3">3D@IOU=0.7</td><td colspan="3">BEV@IOU=0.7</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>15.13</td><td>19.29</td><td>12.78</td><td>20.24</td><td>26.47</td><td>18.29</td></tr><tr><td>a. b.</td><td>√</td><td></td><td></td><td></td><td>16.96</td><td>21.99</td><td>14.42</td><td>22.79</td><td>29.76</td><td>19.78</td></tr><tr><td>c.</td><td></td><td>√</td><td></td><td></td><td>16.85</td><td>21.76</td><td>14.36</td><td>22.30</td><td>28.93</td><td>19.31</td></tr><tr><td>d.</td><td></td><td></td><td>√</td><td></td><td>17.24</td><td>21.63</td><td>14.71</td><td>23.47</td><td>30.52</td><td>20.33</td></tr><tr><td>e.</td><td></td><td>√</td><td></td><td></td><td>17.33</td><td>22.34</td><td>14.63</td><td>22.90</td><td>30.02</td><td>19.84</td></tr><tr><td>f.</td><td>广</td><td></td><td>√</td><td></td><td>17.70</td><td>22.59</td><td>15.17</td><td>23.59</td><td>31.07</td><td></td></tr><tr><td></td><td></td><td>！</td><td>√</td><td></td><td>17.98</td><td>22.58</td><td>15.26</td><td>23.76</td><td>30.98</td><td>20.46 20.52</td></tr><tr><td>g. h.</td><td></td><td>1</td><td></td><td></td><td>18.24</td><td>23.82</td><td>15.49</td><td>25.06</td><td>32.66</td><td>21.88</td></tr><tr><td>i.</td><td></td><td>厂</td><td>厂</td><td>！</td><td>18.47</td><td>24.31</td><td>15.76</td><td>25.40</td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>33.09</td><td>22.16</td></tr></table>
+Detailed design choice. We provide additional experiments in Table 2 for our method. First, as for object-level distillation in the feature space, we investigate the different effects of applying distillation on the whole image and foreground regions. Due to the noise in the background, guiding the foreground regions is more effective than the whole image, which improves the accuracy by 0.72 on the moderate settings in 3D detection. Second, as for object-level distillation in the result space, we compare the different effects of point label and region label. It can be observed that the generated region can significantly increase performance while guiding only in sparse point label brings limited improvements. Our proposed label diffusion strategy can increase the number of positive samples for supervision, thus improving performance.
+Table 2: Evaluation on the KITTI validation set for detailed design choice. OF and OR represent the object-level distillation in feature space and the object-level distillation in result space.
+<table><tr><td rowspan="2">Guidance</td><td rowspan="2">Choice</td><td colspan="3">3D@I0U=0.7</td><td colspan="3">BEV@IOU=0.7</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td rowspan="2">OF</td><td>full</td><td>16.13</td><td>21.52</td><td>14.18</td><td>22.04</td><td>27.85</td><td>19.04</td></tr><tr><td>foreground</td><td>16.85</td><td>21.76</td><td>14.36</td><td>22.30</td><td>28.93</td><td>19.31</td></tr><tr><td rowspan="2">OR</td><td>sparse label</td><td>15.51</td><td>20.58</td><td>13.70</td><td>21.47</td><td>27.16</td><td>18.60</td></tr><tr><td>diffused label</td><td>17.24</td><td>21.63</td><td>14.71</td><td>23.47</td><td>30.52</td><td>20.33</td></tr></table>
+Comparison with state-of-the-art methods. Table 3 and Table 4 compare the proposed method with other state-of-the-art methods on the KITTI test and validation sets. On the test set, the proposed method outperforms existing methods in all metrics. We note that, compared with previous best results, we can obtain 1.83, 0.50, 1.53 improvements on the moderate, easy and hard settings in 3D detection. Furthermore, our method achieves more significant improvements in BEV detection, increasing upon the prior work by 2.51, 1.21, 2.46 on the moderate, easy and hard settings. Moreover, compared with the depth-based methods, our method outperforms them in performance by a margin and is superior to theirs in the inference speed. By contrast, our method only takes 40ms to process a KITTI image, tested on a single NVIDIA GTX 1080Ti, while the Fastest of the depth-based methods (Ma et al., 2019; 2020; Ding et al., 2020; Wang et al., 2021a; Reading et al., 2021) need $1 8 0 \mathrm { m s }$ . On the validation set, the proposed also performs best, both for the $0 . 7 \ \mathrm { I o U }$ threshold and 0.5 IoU threshold. Besides, we also present the performance of the baseline model to better show the effectiveness of the proposed method. Note that we do not report the performances of some depth-based methods (Ma et al., 2019; 2020; Ding et al., 2020; Wang et al., 2021a) due to the data leakage problem \*
+# 4.3 MORE DISCUSSIONS
+What has the student model learned from the teacher model? To locate the source of improvement, we use the items predicted from the baseline model to replace that from our full model, and Table 5 summarizes the results of the cross-model evaluation. From these results, we can see that the teacher model provides effective guidance to the location estimation $( \mathsf { b {  } f } )$ , and improvement of dimension part is also considerable $( \mathrm { c } \to \mathrm { f } )$ . Relatively, the teacher model provides limited valuable cues to the classification and orientation part. This phenomenon suggests the proposed methods boost the performance of the baseline model mainly by introducing the spatial-related information, which is consistent with our initial motivation. Besides, we also show the errors of depth estimation, see Appendix A.3 for the results.
+Is the effectiveness of our method related to the performance of the teacher model? An intuitive conjecture is the student can learn more if the teacher network has better performance. To explore this problem, we also use the sparse LiDAR maps to train a teacher net to provide guidance to the student model (see Figure 2 for the comparison of the sparse and dense data). As shown in Table 6, the performance of the teacher model trained from the sparse LiDAR maps is largely behind by that from dense LiDAR maps (drop to $2 2 . 0 5 \%$ from $4 2 . 4 5 \%$ , moderate setting), while both of them provides comparable benefits to the student model. Therefore, for our task, the performance of the teacher model is not directly related to the performance improvement, while the more critical factor is whether the teacher network contains complementary information to the student network.
+Do we need depth estimation as an intermediate task? As shown in Figure 1, most previous methods choose to estimate the depth maps to provide depth information for monocular 3D detection (information flow: LiDAR data estimated depth map $ 3 \mathrm { D }$ detector). Compared with this scheme, our method directly learns the depth cues from LiDAR-based methods (information flow: LiDAR data $ 3 \mathrm { D }$ detector), avoiding the information loss in the depth estimation step. Here we quantitatively show the information loss in depth estimation using a simple experiment. Specifically, we use DORN $\mathrm { F u }$ et al., 2018) (same as most previous depth augmented methods) to generate the depth maps, and then use them to train the teacher net. Table 7 shows the results of this experiment. Note that, compared with setting c, setting b’s teacher net is trained from a larger training set (23,488 vs. 3,712) with ground-truth depth maps (ground truth depth maps vs. noisy depth maps). Nevertheless, this scheme still lags behind our original method, which means that there is serious information loss in monocular depth estimation (stereo image performs better, which is discussed in Appendix A.4).
+Table 3: Comparison of state-of-the-art methods on the KITTI test set. Methods are ranked by moderate setting. We highlight the best results in bold and the second place in underlined. Only RGB images are required as input in the inference phase for all listed methods. \*: need dense depth maps or LiDAR signals for training. $^ \dagger$ : our baseline model without confidence normalization.
+<table><tr><td rowspan="2">Method</td><td colspan="3">3D@IOU=0.7</td><td colspan="3">BEV@IOU=0.7</td><td rowspan="2">Runtime</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>M3D-RPN (Brazil &amp; Liu, 2019)</td><td>9.71</td><td>14.76</td><td>7.42</td><td>13.67</td><td>21.02</td><td>10.23</td><td>160 ms</td></tr><tr><td>SMOKE (Liu et al., 2020)</td><td>9.76</td><td>14.03</td><td>7.84</td><td>14.49</td><td>20.83</td><td>12.75</td><td>30 ms</td></tr><tr><td>MonoPair (Chen et al., 2020b)</td><td>9.99</td><td>13.04</td><td>8.65</td><td>14.83</td><td>19.28</td><td>12.89</td><td>60 ms</td></tr><tr><td>RTM3D (Li et al., 2020)</td><td>10.34</td><td>14.41</td><td>8.77</td><td>14.20</td><td>19.17</td><td>11.99</td><td>50 ms</td></tr><tr><td>AM3D* (Ma et al., 2019)</td><td>10.74</td><td>16.50</td><td>9.52</td><td>17.32</td><td>25.03</td><td>14.91</td><td>400 ms</td></tr><tr><td>PatchNet* (Ma et al., 2020)</td><td>11.12</td><td>15.68</td><td>10.17</td><td>16.86</td><td>22.97</td><td>14.97</td><td>400 ms</td></tr><tr><td>D4LCN* (Ding et al., 2020)</td><td>11.72</td><td>16.65</td><td>9.51</td><td>16.02</td><td>22.51</td><td>12.55</td><td>200 ms</td></tr><tr><td>MonoDLE† (Ma et al., 2021)</td><td>12.26</td><td>17.23</td><td>10.29</td><td>18.89</td><td>24.79</td><td>16.00</td><td>40 ms</td></tr><tr><td>MonoRUn*(Chen et al., 2021a)</td><td>12.30</td><td>19.65</td><td>10.58</td><td>17.34</td><td>27.94</td><td>15.24</td><td>70 ms</td></tr><tr><td>GrooMeD-NMS (Kumar et al., 2021)</td><td>12.32</td><td>18.10</td><td>9.65</td><td>18.27</td><td>16.19</td><td>14.05</td><td>120 ms</td></tr><tr><td>DDMP-3D* (Wang et al., 2021a)</td><td>12.78</td><td>19.71</td><td>9.80</td><td>17.89</td><td>28.08</td><td>13.44</td><td>180 ms</td></tr><tr><td>CaDDN* (Reading et al., 2021)</td><td>13.41</td><td>19.17</td><td>11.46</td><td>18.91</td><td>27.94</td><td>17.19</td><td>630 ms</td></tr><tr><td>MonoEF (Zhou et al., 2021)</td><td>13.87</td><td>21.29</td><td>11.71</td><td>19.70</td><td>29.03</td><td>17.26</td><td>30 ms</td></tr><tr><td>MonoFlex (Zhang et al., 2021b)</td><td>13.89</td><td>19.94</td><td>12.07</td><td>19.75</td><td>28.23</td><td>16.89</td><td>30 ms</td></tr><tr><td>Autoshape (Liu et al., 2021)</td><td>14.17</td><td>22.47</td><td>11.36</td><td>20.08</td><td>30.66</td><td>15.59</td><td>50 ms</td></tr><tr><td>GUPNet (Lu et al., 2021)</td><td>14.20</td><td>20.11</td><td>11.77</td><td>1</td><td>1</td><td>1</td><td>35ms</td></tr><tr><td>Ours*</td><td>16.03</td><td>22.97</td><td>13.60</td><td>22.59</td><td>31.87</td><td>19.72</td><td>40 ms</td></tr><tr><td>Improvements</td><td>+1.83</td><td>+0.50</td><td>+1.53</td><td>+2.51</td><td>+1.21</td><td>+2.46</td><td>-</td></tr></table>
+Table 4: Performance of the Car category on the KITTI validation set. We highlight the best results in bold and the second place in underlined. †: our baseline model without confidence normalization.
+<table><tr><td rowspan="2">Method</td><td colspan="3">3D@IOU=0.7</td><td colspan="3">BEV@IOU=0.7</td><td colspan="3">3D@IOU=0.5</td><td colspan="3">BEV@IOU=0.5</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>M3D-RPN</td><td>11.07</td><td>14.53</td><td>8.65</td><td>15.62</td><td>20.85</td><td>11.88</td><td>35.94</td><td>48.53</td><td>28.59</td><td>39.60</td><td>53.35</td><td>31.76</td></tr><tr><td>MonoPair</td><td>12.30</td><td>16.28</td><td>10.42</td><td>18.17</td><td>24.12</td><td>15.76</td><td>42.39</td><td>55.38</td><td>37.99</td><td>47.63</td><td>61.06</td><td>41.92</td></tr><tr><td>MonoDLEt</td><td>13.66</td><td>17.45</td><td>11.68</td><td>19.33</td><td>24.97</td><td>17.01</td><td>43.42</td><td>55.41</td><td>37.81</td><td>46.87</td><td>60.73</td><td>41.89</td></tr><tr><td>GrooMeD-NMS</td><td>14.32</td><td>19.67</td><td>11.27</td><td>19.75</td><td>27.38</td><td>15.92</td><td>41.07</td><td>55.62</td><td>32.89</td><td>44.98</td><td>61.83</td><td>36.29</td></tr><tr><td>MonoRUn</td><td>14.65</td><td>20.02</td><td>12.61</td><td>-</td><td>=</td><td>■</td><td>43.39</td><td>59.71</td><td>38.44</td><td>-</td><td>■</td><td>■</td></tr><tr><td>GUPNet</td><td>16.46</td><td>22.76</td><td>13.72</td><td>22.94</td><td>31.07</td><td>19.75</td><td>42.33</td><td>57.62</td><td>37.59</td><td>47.06</td><td>61.78</td><td>40.88</td></tr><tr><td>MonoFlex</td><td>17.51</td><td>23.64</td><td>14.83</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Baseline</td><td>15.13</td><td>19.29</td><td>12.78</td><td>20.24</td><td>26.47</td><td>18.29</td><td>43.54</td><td>57.43</td><td>39.22</td><td>48.49</td><td>63.56</td><td>42.81</td></tr><tr><td>Ours</td><td>18.47</td><td>24.31</td><td>15.76</td><td>25.40</td><td>33.09</td><td>22.16</td><td>49.35</td><td>65.69</td><td>43.49</td><td>53.11</td><td>71.45</td><td>46.94</td></tr></table>
+Table 5: Cross-model evaluation on the KITTI validation set. We extract each required item (location, dimension, orientation, and confidence) from the baseline model (B) and the full model (O), and evaluate them in a cross-model manner.
+<table><tr><td rowspan="3"></td><td rowspan="3">loc.</td><td rowspan="3">dim.</td><td rowspan="3">ori.</td><td rowspan="3">con.</td><td colspan="3">3D@IOU=0.7</td><td colspan="3">BEV@IOU=0.7</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>a.</td><td>B</td><td>B</td><td>B</td><td>B</td><td>15.13</td><td>19.29</td><td>12.78</td><td>20.24</td><td>26.47</td><td>18.29</td></tr><tr><td>b.</td><td>B</td><td>0</td><td>0</td><td>0</td><td>16.05</td><td>20.07</td><td>13.47</td><td>21.31</td><td>27.77</td><td>19.14</td></tr><tr><td>c.</td><td>0</td><td>B</td><td>0</td><td>0</td><td>17.91</td><td>22.87</td><td>15.29</td><td>25.09</td><td>32.78</td><td>21.93</td></tr><tr><td>d.</td><td>0</td><td>0</td><td>B</td><td>0</td><td>18.12</td><td>24.02</td><td>15.34</td><td>25.02</td><td>32.85</td><td>21.84</td></tr><tr><td>e.</td><td>0</td><td>0</td><td>0</td><td>B</td><td>18.41</td><td>24.27</td><td>15.55</td><td>24.98</td><td>32.78</td><td>21.81</td></tr><tr><td>f.</td><td>0</td><td>0</td><td>0</td><td>0</td><td>18.47</td><td>24.31</td><td>15.76</td><td>25.40</td><td>33.09</td><td>22.16</td></tr></table>
+Table 6: Performance of the student model under the guidance of different teacher models. Metric is the $\mathrm { A P } | _ { 4 0 }$ for the 3D detection task on the KITTI validation set. We also show the performance improvements of the student model to the baseline model for better comparison.
+<table><tr><td rowspan="2"></td><td colspan="2">Teacher Model</td><td colspan="2">Student Model</td><td colspan="4">Improvement</td></tr><tr><td>Mod.</td><td>Easy Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>sparse maps</td><td>22.05</td><td>31.67</td><td>18.72 18.07</td><td>23.61</td><td>15.36</td><td>+2.94</td><td>+4.32</td><td>+2.58</td></tr><tr><td>dense maps</td><td>42.57</td><td>58.06</td><td>37.07 18.47</td><td>24.31</td><td>15.76</td><td>+3.34</td><td>+5.02</td><td>+2.98</td></tr></table>
+Table 7: Comparison of using depth estimation as intermediate task or not. Setting a. and c. denote the baseline model and our full model. Setting b. uses the depth maps generated from DORN (Fu et al., 2018) to train the teacher model. Experiments are conducted on the KITTI validation set.
+<table><tr><td></td><td colspan="3">3D@IOU=0.7</td><td colspan="3">BEV @IOU=0.7</td><td colspan="2">AOS@IOU=0.7</td><td colspan="3">2D@IOU=0.7</td></tr><tr><td></td><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod. Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>a.</td><td>15.13</td><td>19.29</td><td>12.78</td><td>20.24</td><td>26.47</td><td>18.29</td><td>90.95 97.46</td><td>83.02</td><td>92.18</td><td>98.37</td><td>85.05</td></tr><tr><td>b.</td><td>17.70</td><td>23.21</td><td>15.02</td><td>23.34</td><td>31.20</td><td>20.40</td><td>91.50 97.77</td><td>83.49</td><td>92.51</td><td>98.54</td><td>85.38</td></tr><tr><td>c.</td><td>18.47</td><td>24.31</td><td>15.76</td><td>25.40</td><td>33.09</td><td>22.16</td><td>91.67 97.88</td><td>83.59</td><td>92.71</td><td>98.58</td><td>85.56</td></tr></table>
+# 4.4 QUALITATIVE RESULTS
+In Figure 5, we show the qualitative comparison of detection results. We can see that the proposed method shows better localization accuracy than the baseline model. See Appendix A.7 for more detailed qualitative results.
+![](images/887a97c282d6cc6b7c869d15ef95deb2c6188122430cf03d43e8ce8b70feba9b.jpg)
+Figure 5: Qualitative results. We use green, blue and red boxes to denote the results from baseline, our method, and ground truth. Besides, we use red circle to highlight the main differences.
+# 5 CONCLUSION
+In this work, we propose the MonoDistill, which introduces spatial cues to the monocular 3D detector based on the knowledge distillation mechanism. Compared with previous schemes, which share the same motivation, our method avoids any modifications on the target model and directly learns the spatial features from the model rich in these features. This design makes the proposed method perform well in both performance and efficiency. To show an all-around display of our model, extensive experiments are conducted on the KITTI dataset, where the proposed method ranks $1 ^ { s t }$ at 25 FPS among all monocular 3D detectors.
+# ACKNOWLEDGEMENTS
+This work was supported in part by the National Natual Science Foundation of China (NSFC) under Grants No.61932020, 61976038, U1908210 and 61772108. Wanli Ouyang was supported by the Australian Research Council Grant DP200103223, FT210100228, and Australian Medical Research Future Fund MRFAI000085.
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+# A APPENDIX
+# A.1 MORE DETAILS OF THE BASELINE MODEL
+Network architecture. The baseline network is extended from the anchor-free 2D object detection framework, which consists of a feature extraction network and seven detection subheads. We employ DLA-34 (Yu et al., 2017) without deformable convolutions as our backbone. The feature maps are downsampled by 4 times and then we take the image features as input and use 3x3 convolution, ReLU, and 1x1 convolution to output predictions for each detection head. Detection head branches include three for 2D components and four for 3D components. Specifically, 2D detection heads include heatmap, offset between the 2D key-point and the 2D box center, and size of 2D box. 3D components include offset between the 2D key-point and the projected 3D object center, depth, dimensions, and orientations. As for objective functions, we train the heatmap with focal loss. The other loss items adopt L1 losses except for depth and orientation. The depth branch employs a modified L1 loss with the assist of heteroscedastic aleatoric uncertainty. Common MultiBin loss is used for the orientation branch. Besides, we propose a strategy to improve the accuracy of baseline. Inspire by (Lu et al., 2021), estimated depth uncertainty can provide confidence for each projection depth. Therefore, we normalize the confidence of each predicted box using depth uncertainty. In this way, the score has capability of indicating the uncertainty of depth.
+Training details. Our model is trained on 2 NVIDIA 1080Ti GPUs in an end-to-end manner for 150 epochs. We employ the common Adam optimizer with initial learning rate $1 . 2 5 e ^ { - 4 }$ , and decay it by ten times at 90 and 120 epochs. To stabilize the training process, we also applied the warm-up strategy (5 epochs). As for data augmentations, only random random flip and center crop are applied. Same as the common knowledge distillation scheme, we first train teacher network in advance, and then fix the teacher network. As for student network, we simply train the detection model to give a suitable initialization. We implemented our method using PyTorch. And our code is based on Ma et al. (2021).
+# A.2 PEDESTRIAN/CYCLIST DETECTION.
+Due to the small sizes, non-rigid structures, and limited training samples, the pedestrians and cyclists are much more challenging to detect than cars. We first report the detection results on test set in Table 8. It can be seen that our proposed method is also competitive with current state-of-the-art methods on the KITTI test set, which increases $0 . 6 9 \mathrm { A P }$ on hard difficulty level of pedestrian category. Note that, the accuracy of these difficult categories fluctuates greatly compared with Car detection due to insufficient training samples (see Table 10 for the details). Due the access to the test server is limited, we conduct more experiments for pedestrian/cyclist on the validation set for general conclusions (we run the proposed method three times with different random seeds), and the experimental results are summarized in Table 9. According to these results, we can find that the proposed method can effectively boost the accuracy of the baseline model for pedestrian/cyclist detection.
+Table 8: Performance of Pedestrian/Cyclist detection on the KITTI test set. We highlight the best results in bold and the second place in underlined.
+<table><tr><td rowspan="2">Method</td><td colspan="3">Pedestrian</td><td colspan="3">Cyclist</td></tr><tr><td>Easy</td><td>Mod.</td><td>Hard</td><td>Easy</td><td>Mod.</td><td>Hard</td></tr><tr><td>M3D-RPN D4LCN</td><td>4.92 4.55</td><td>3.48 3.42</td><td>2.94 2.83</td><td>0.94 2.45</td><td>0.65 1.67</td><td>0.47 1.36</td></tr><tr><td>MonoPair</td><td>10.02</td><td>6.68</td><td>5.53</td><td>3.79</td><td>2.21</td><td>1.83</td></tr><tr><td>MonoFlex</td><td>9.43</td><td>6.31</td><td>5.26</td><td>4.17</td><td>2.35</td><td>2.04</td></tr><tr><td>MonoDLE</td><td>9.64</td><td>6.55</td><td>5.44</td><td>4.59</td><td>2.66</td><td>2.45</td></tr><tr><td>CaDDN</td><td>12.87</td><td>8.14</td><td>6.76</td><td>7.00</td><td>3.14</td><td>3.30</td></tr><tr><td>DDMP-3D</td><td>4.93</td><td>3.55</td><td>3.01</td><td>4.18</td><td>2.50</td><td>2.32</td></tr><tr><td>AutoShape</td><td>5.46</td><td>3.74</td><td>3.03</td><td>5.99</td><td>3.06</td><td>2.70</td></tr><tr><td>Ours</td><td>12.79</td><td>8.17</td><td>7.45</td><td>5.53</td><td>2.81</td><td>2.40</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+Table 9: Performance of Pedestrian/Cyclist detection on the KITTI validation set. Both 0.25 and $0 . 5 \mathrm { I o U }$ thresholds are considered. We report the mean of several experiments for the proposed methods. $\pm$ captures the standard deviation over random seeds.
+<table><tr><td rowspan="2"></td><td rowspan="2">Method</td><td colspan="3">3D@IoU=0.25</td><td colspan="3">3D@IoU=0.5</td></tr><tr><td>Easy</td><td>Mod.</td><td>Hard</td><td>Easy</td><td>Mod.</td><td>Hard</td></tr><tr><td rowspan="2">Pedestrian</td><td>Baseline</td><td>29.07±0.21</td><td>23.77±0.15</td><td>19.85±0.14</td><td>6.8±0.28</td><td>5.17±0.08</td><td>4.37±0.15</td></tr><tr><td>Ours</td><td>32.09±0.71</td><td>25.53±0.55</td><td>21.15±0.79</td><td>8.95±1.26</td><td>6.84±0.81</td><td>5.32±0.75</td></tr><tr><td rowspan="2">Cyclist</td><td>Baseline</td><td>21.06±0.46</td><td>11.87±0.19</td><td>10.77±0.02</td><td>3.71±0.49</td><td>1.88±0.23</td><td>1.64±0.04</td></tr><tr><td>Ours</td><td>24.26±1.29</td><td>13.04±0.44</td><td>12.08±0.68</td><td>5.38±0.91</td><td>2.67±0.40</td><td>2.53±0.38</td></tr></table>
+Table 10: Training samples of each category on the KITTI training set.
+<table><tr><td></td><td>cars</td><td>pedestrians</td><td>cyclists</td></tr><tr><td>#instances</td><td>14,357</td><td>2,207</td><td>734</td></tr></table>
+# A.3 DEPTH ERROR ANALYSIS
+As shown in Figure 6, we compare the depth error between baseline and our method. Specifically, we project all valid samples of the Car category into the image plane to get the corresponding predicted depth values. Then we fit the depth errors between ground truths and predictions as a linear function by least square method. According to the experimental results, we can find that our proposed method can boost the accuracy of depth estimation at different distances.
+![](images/9b43a6ba991c5a0e9121d360ec5db7380b63da90ccd4da60004f2ed0feaac777.jpg)
+Figure 6: Errors of depth estimation. We show the errors of depth estimation as a function of the depth ( $\mathbf { X }$ -axis) for the baseline model (left) and our full model (right).
+# A.4 THE EFFECTS OF STEREO DEPTH
+We also explored the changes in performance under the guidance of estimated stereo depth (Chang & Chen, 2018), and show the results in Table 11. Stereo depth estimation exploits geometric constraints in stereo images to obtain the absolute depth value through pixel-wise matching, which is more accurate compared with monocular depth estimation. Therefore, under the guidance of stereo depth, the model achieves almost the same accuracy as LiDAR signals guidance at $0 . 5 \ \mathrm { I o U }$ threshold, and there is only a small performance drop at 0.7 IoU threshold.
+# A.5 GENERALIZATION OF THE PROPOSED METHOD
+In the main paper, we introduced the proposed method based on MonoDLE (Ma et al., 2021). Here we discuss the generalization ability of the proposed method.
+Generalizing to other baseline models. To show the generalization ability of the proposed method, we apply our method on another monocular detector GUPNet (Lu et al., 2021), which is a two-stage detection method. Experimental results are shown in the Table 12. We can find that the proposed method can also boosts the performances of GUPNet, which confirms the generalization of our method.
+Table 11: Effects of stereo depth estimation. Baseline denotes the baseline model without guidance of teacher network. Stereo Depth and LiDAR Depth denote under the guidance of stereo depth maps and LiDAR signals. Experiments are conducted on the KITTI validation set.
+<table><tr><td rowspan="2"></td><td colspan="3">3D@IOU=0.7</td><td colspan="3">BEV@IOU=0.7</td><td colspan="3">3D@IOU=0.5</td><td colspan="3">BEV@IOU=0.5</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>Baseline</td><td>15.13</td><td>19.29</td><td>12.78</td><td>20.24</td><td>26.47</td><td>18.29</td><td>43.54</td><td>57.43</td><td>39.22</td><td>48.49</td><td>63.56</td><td>42.81</td></tr><tr><td>Stereo Depth</td><td>18.18</td><td>23.54</td><td>15.42</td><td>24.89</td><td>32.26</td><td>21.64</td><td>49.13</td><td>65.18</td><td>43.29</td><td>52.88</td><td>69.47</td><td>46.72</td></tr><tr><td>LiDAR Depth</td><td>18.47</td><td>24.31</td><td>15.76</td><td>25.40</td><td>33.09</td><td>22.16</td><td>49.35</td><td>65.69</td><td>43.49</td><td>53.11</td><td>71.45</td><td>46.94</td></tr></table>
+Table 12: MonoDistill on GUPNet. Experiments are conducted on the KITTI validation set.
+<table><tr><td rowspan="2"></td><td colspan="3">3D@IOU=0.7</td><td colspan="3">3D@IOU=0.5</td></tr><tr><td>Easy</td><td>Mod.</td><td>Hard</td><td>Easy</td><td>Mod.</td><td>Hard</td></tr><tr><td>GUPNet-Baseline</td><td>22.76</td><td>16.46</td><td>13.72</td><td>57.62</td><td>42.33</td><td>37.59</td></tr><tr><td>GUPNet-Ours</td><td>24.43</td><td>16.69</td><td>14.66</td><td>61.72</td><td>44.49</td><td>40.07</td></tr></table>
+Generalizing to sparse LiDAR signals. We also explore the changes in performance under different resolution of LiDAR signals. In particular, following Pseudo-LiDAR $^ { + + }$ (You et al., 2020), we generate the simulated 32-beam/16-beam LiDAR signals and use them to train our teacher model (in the ‘sparse’ setting). We show the experimental results, based on MonoDLE, in the Table 13. We can see that, although the improvement is slightly reduced due to the decrease of the resolution of LiDAR signals, the proposed method significantly boost the performances of baseline model under all setting.
+Table 13: Effects of the resolution of LiDAR signals. Experiments are conducted on the KITTI validation set.
+<table><tr><td rowspan="2"></td><td colspan="3">3D@IOU=0.7</td><td colspan="3">3D@I0U=0.5</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>Baseline</td><td>19.29</td><td>15.13</td><td>12.78</td><td>43.54</td><td>57.43</td><td>39.22</td></tr><tr><td>Ours - 16-beam</td><td>22.49</td><td>17.66</td><td>15.08</td><td>49.39</td><td>65.45</td><td>43.60</td></tr><tr><td>Ours - 32-beam</td><td>23.24</td><td>17.71</td><td>15.19</td><td>49.41</td><td>65.61</td><td>43.46</td></tr><tr><td>Ours - 64-beam</td><td>23.61</td><td>18.07</td><td>15.36</td><td>49.67</td><td>65.97</td><td>43.74</td></tr></table>
+More discussion. Besides, note that the camera parameters of the images on the KITTI test set are different from these of the training/validation set, and the good performance on the test set suggests the proposed method can also generalize to different camera parameters. However, generalizing to the new scenes with different statistical characteristics is a hard task for existing 3D detectors (Yang et al.; Wang et al., 2020b), including the image-based models and LiDAR-based models, and deserves further investigation by future works. We also argue that the proposed method can generalize to the new scenes better than other monocular models because ours model learns the stronger features from the teacher net. These results and analysis will be included in the revised version.
+# A.6 COMPARISON WITH DIRECT DENSE DEPTH SUPERVISION.
+According to the ablation studies in the main paper, we can find that depth cues are the key factor to affect the performance of the monocular 3D models. However, dense depth supervision in the student model without KD may also introduce depth cues to the monocular 3D detectors. Here we conduct the control experiment by adding a new depth estimation branch, which is supervised by the dense LiDAR maps. Note that, this model is trained without KD. Table 14 compares the performances of the baseline model, the new control experiment, and the proposed method. From these results, we can get the following conclusions: (i) additional depth supervision can introduce the spatial cues to the models, thereby improving the overall performance; (ii) the proposed KDbased method significantly performs better than the baseline model and the new control experiment, which demonstrates the effectiveness of our method.
+Table 14: Comparison with direct dense depth supervision. Experiments are conducted on the KITTI validation set.
+<table><tr><td rowspan="3"></td><td colspan="3">3D@IOU=0.7</td><td colspan="3">3D@IOU=0.5</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>Baseline</td><td>15.13</td><td>19.29</td><td>12.78</td><td>43.54</td><td>57.43</td><td>39.22</td></tr><tr><td>Baseline + depth supv.</td><td>17.05</td><td>21.85</td><td>14.54</td><td>46.19</td><td>60.42</td><td>41.88</td></tr><tr><td>Ours</td><td>18.47</td><td>24.31</td><td>15.76</td><td>49.35</td><td>65.69</td><td>43.49</td></tr></table>
+# A.7 MORE QUALITATIVE RESULTS
+In Figure 7, we show more qualitative results on the KITTI dataset. We use orange box, green, and purple boxes for cars, pedestrians, and cyclists, respectively. In Figure 8, we show comparison of detection results in the 3D space. It can be found that our method can significantly improve the accuracy of depth estimation compared with the baseline.
+![](images/9efa6269e04e1f1776fb3a61cd66a2525d62b1b255e0b6b85422f6bdb89b37c6.jpg)
+Figure 7: Qualitative results for multi-class 3D object detection. The boxes’ color of cars, pedestrian, and cyclist are in orange, green, and purple, respectively.
+![](images/ff8ebd28258b37441ea96db3473b8f4a94df501990fefe4bb7ce90d0d8bd18ec.jpg)
+Figure 8: Qualitative results of our method for 3D space. The boxes’ color of ground truth, baseline, and ours are in red, green, and blue, respectively.

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+    {
+        "type": "text",
+        "text": "MONODISTILL: LEARNING SPATIAL FEATURES FOR MONOCULAR 3D OBJECT DETECTION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Zhiyu Chong∗1, Xinzhu $\\mathbf { M _ { a } } { ^ { * 2 } }$ , Hong Zhang1, Yuxin Yue1, Haojie $\\mathbf { L i } ^ { 1 }$ , Zhihui Wang1, and Wanli Ouyang2 ",
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+    {
+        "type": "text",
+        "text": "1Dalian University of Technology 2The University of Sydney {czydlut, jingshui, 22017036}@mail.dlut.edu.cn {xinzhu.ma, wanli.ouyang}@sydney.edu.au {hjli, zhwang}@dlut.edu.cn ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3D object detection is a fundamental and challenging task for 3D scene understanding, and the monocular-based methods can serve as an economical alternative to the stereo-based or LiDAR-based methods. However, accurately detecting objects in the 3D space from a single image is extremely difficult due to the lack of spatial cues. To mitigate this issue, we propose a simple and effective scheme to introduce the spatial information from LiDAR signals to the monocular 3D detectors, without introducing any extra cost in the inference phase. In particular, we first project the LiDAR signals into the image plane and align them with the RGB images. After that, we use the resulting data to train a 3D detector (LiDAR Net) with the same architecture as the baseline model. Finally, this LiDAR Net can serve as the teacher to transfer the learned knowledge to the baseline model. Experimental results show that the proposed method can significantly boost the performance of the baseline model and ranks the $1 ^ { s t }$ place among all monocularbased methods on the KITTI benchmark. Besides, extensive ablation studies are conducted, which further prove the effectiveness of each part of our designs and illustrate what the baseline model has learned from the LiDAR Net. Our code will be released at https://github.com/monster-ghost/MonoDistill. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3D object detection is an indispensable component for 3D scene perception, which has wide applications in the real world, such as autonomous driving and robotic navigation. Although the algorithms with stereo (Li et al., 2019b; Wang et al., 2019; Chen et al., 2020a) or LiDAR sensors (Qi et al., 2018; Shi et al., 2019; 2020) show promising performances, the heavy dependence on the expensive equipment restricts the application of these algorithms. Accordingly, the methods based on the cheaper and more easy-to-deploy monocular cameras (Xu & Chen, 2018; Ma et al., 2019; 2021; Brazil & Liu, 2019; Ding et al., 2020) show great potentials and have attracted lots of attention. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As shown in Figure 1 (a), some prior works (Brazil & Liu, 2019; Simonelli et al., 2019; Chen et al., 2020b) directly estimate the 3D bounding boxes from monocular images. However, because of the lack of depth cues, it is extremely hard to accurately detect the objects in the 3D space, and the localization error is the major issue of these methods (Ma et al., 2021). To mitigate this problem, an intuitive idea is to estimate the depth maps from RGB images, and then use them to augment the input data (Figure 1 (b)) (Xu & Chen, 2018; Ding et al., 2020) or directly use them as the input data (Figure 1 (c)) (Wang et al., 2019; Ma et al., 2019). Although these two strategies have made significant improvement in performance, the drawbacks of them can not be ignored: (1) These methods generally use an off-the-shelf depth estimator to generate the depth maps, which introduce lots of computational cost (e.g. the most commonly used depth estimator (Fu et al., 2018) need about $4 0 0 \\mathrm { m s }$ to process a standard KITTI image). (2) The depth estimator and detector are trained separately, which may lead to a sub-optimal optimization. Recently, Reading et al. (2021) propose an end-to-end framework (Figure 1 (d)) for monocular 3D detection, which can also leverage depth estimator to provide depth cues. Specifically, they introduce a sub-network to estimate the depth distribution and use it to enrich the RGB features. Although this model can be trained end-to-end and achieves better performance, it still suffer from the low inference speed (630ms per image), mainly caused by the depth estimator and the complicated network architecture. Note that the welldesigned monocular detectors, like Ma et al. (2021); Zhang et al. (2021b); Lu et al. (2021), only take about $4 0 \\mathrm { m s }$ per image. ",
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+        "img_path": "images/3fa330cfff050e5dc2ed89024e93f34bd6cce756c45abec7b51ede2082a6377c.jpg",
+        "image_caption": [
+            "Figure 1: Comparison on the high-level paradigms of monocular 3D detectors. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+    {
+        "type": "text",
+        "text": "In this paper, we aim to introduce the depth cues to the monocular 3D detectors without introducing any extra cost in the inference phase. Inspired by the knowledge distillation (Hinton et al.), which can transfer the learned knowledge from a well-trained CNN to another one without any changes in the model design, we propose that the spatial cues may also be transferred by this way from the LiDAR-based models. However, the main problem for this proposal is the difference of the feature representations used in these two kinds of methods (2D images features vs. 3D voxel features). To bridge this gap, we propose to project the LiDAR signals into the image plane and use the 2D CNN, instead of the commonly used 3D CNN or point-wise CNN, to train a ‘image-version’ LiDAR-based model. After this alignment, the knowledge distillation can be friendly applied to enrich the features of our monocular detector. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Based on the above-mentioned motivation and strategy, we propose the distillation based monocular 3D detector (MonoDistill): We first train a teacher net using the projected LiDAR maps (used as the ground truths of the depth estimator in previous works), and then train our monocular 3D detector under the guidance of the teacher net. We argue that, compared with previous works, the proposed method has the following two advantages: First, our method directly learn the spatial cues from the teacher net, instead of the estimated depth maps. This design performs better by avoiding the information loss in the proxy task. Second, our method does not change the network architecture of the baseline model, and thus no extra computational cost is introduced. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Experimental results on the most commonly used KITTI benchmark, where we rank the $1 ^ { s t }$ place among all monocular based models by applying the proposed method on a simple baseline, demonstrate the effectiveness of our approach. Besides, we also conduct extensive ablation studies to present each design of our method in detail. More importantly, these experiments clearly illustrate the improvements are achieved by the introduction of spatial cues, instead of other unaccountable factors in CNN. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            758,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3D detection from only monocular images. 3D detection from only monocular image data is challenging due to the lack of reliable depth information. To alleviate this problem, lots of scholars propose their solutions in different ways, including but not limited to network design (Roddick et al., 2019; Brazil & Liu, 2019; Zhou et al., 2019; Liu et al., 2020; Luo et al., 2021), loss formulation (Simonelli et al., 2019; Ma et al., 2021), 3D prior (Brazil & Liu, 2019), geometric constraint (Mousavian et al., 2017; Qin et al., 2019; Li et al., 2019a; Chen et al., 2020b), or perspective modeling (Zhang et al., 2021a; Lu et al., 2021; Shi et al., 2021). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Depth augmented monocular 3D detection. To provide the depth information to the 3D detectors, several works choose to estimate the depth maps from RGB images. According to the usage of the estimated depth maps, these methods can be briefly divided into three categories. The first class of these methods use the estimated depth maps to augment the RGB images (Figure 1 (b)). In particular, Xu & Chen (2018) propose three fusion strategies of the RGB images and depth maps, while Ding et al. (2020) and Wang et al. (2021a) focus on how to enrich the RGB features with depth maps in the latent feature space. Besides, Wang et al. (2019); Ma et al. (2019) propose another pipeline (Figure 1 (c)): They back-project the depth maps into the 3D space, and then train a LiDAR-based model and use the resulting data (pseudo-LiDAR signal) to predict the 3D boxes. This framework shows promising performance, and lots of works (Weng & Kitani, 2019; Cai et al., 2020; Wang et al., 2020a; Chu et al., 2021) are built on this solid foundation. Recently, Reading et al. (2021) propose another way to leverage the depth cues for monocular 3D detection (Figure 1 (d)). Particularly, they first estimate the depth distribution using a sub-network, and then use it to lift the 2D features into 3D features, which is used to generate the final results. Compared with the previous two families, this model can be trained in the end-to-end manner, avoiding the sub-optimal optimization. However, a common disadvantage of these methods is that they inevitably increase the computational cost while introducing depth information. Unlike these methods, our model chooses to learn the feature representation under the guidance of depth maps, instead of integrating them. Accordingly, the proposed model not only introduces rich depth cues but also maintains high efficiency. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/5dde672b08f0974af574d0d2d4a04bd72aaacc58bf342a1fc846bdc6022a2a76.jpg",
+        "image_caption": [
+            "Figure 2: Visualization of the sparse LiDAR maps (left) and the dense LiDAR maps (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
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+    {
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+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Knowledge distillation. Knowledge distillation (KD) is initially proposed by Hinton et al. for model compression, and the main idea of this mechanism is transferring the learned knowledge from large CNN models to the small one. This strategy has been proved in many computer vision tasks, such as 2D object detection (Dai et al., 2021; Chen et al., 2017; Gupta et al., 2016), semantic semantic segmentation (Hou et al., 2020; Liu et al., 2019). However, few work explore it in monocular 3D detection. In this work, we design a KD-based paradigm to efficiently introduce depth cues for monocular 3D detectors. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "LIGA Stereo. We found a recent work LIGA Stereo (Guo et al., 2021) (submitted to arXiv on 18 Aug. 2021) discusses the application of KD for stereo 3D detection under the guidance of LiDAR signals. Here we discuss the main differences of LIGA Stereo and our work. First, the tasks and underlying data are different (monocular vs. stereo), which leads to different conclusions. For example, Guo et al. (2021) concludes that using the predictions of teacher net as ‘soft label’ can not bring benefits. However, our experimental results show the effectiveness of this design. Even more, in our task, supervising the student net in the result space is more effective than feature space. Second, they use an off-the-shelf LiDAR-based model to provide guidance to their model. However, we project the LiDAR signals into image plane and use the resulting data to train the teacher net. Except for the input data, the teacher net and student net are completely aligned, including network architecture, hyper-parameters, and training schedule. Third, to ensure the consistent shape of features, LIGA Stereo need to generate the cost volume from stereo images, which is time-consuming (it need about $3 5 0 \\mathrm { m s }$ to estimate 3D boxes from a KITTI image) and hard to achieve for monocular images. In contrast, our method align the feature representations by adjusting the LiDAR-based model, instead of the target model. This design makes our method more efficient (about $3 5 \\mathrm { m s }$ per image) and can generalize to all kinds of image-based models in theory. ",
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+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "3.1 OVERVIEW ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Figure 3 presents the framework of the proposed MonoDistill, which mainly has three components: a monocular 3D detector, an aligned LiDAR-based detector, and several side branches which build ",
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+    {
+        "type": "image",
+        "img_path": "images/21d6ed0b66761b9e1a051bfa6abcdee0ef3180df8a9a46cb1f79def5f9635a05.jpg",
+        "image_caption": [
+            "Figure 3: Illustration of the proposed MonoDistill. We first generate the ‘image-like’ LiDAR maps from the LiDAR signals and then train a teacher model using an identical network to the student model. Finally, we propose three distillation schemes to train the student model under the guidance of the well-trained teacher net. In the inference phase, only the student net is used. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "the bridge to provide guidance from the LiDAR-based detector to our monocular 3D detector. We will introduce the how to build these parts one by one in the rest of this section. ",
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+    {
+        "type": "text",
+        "text": "3.2 BASELINE MODEL",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Student model. We use the one-stage monocular 3D detector MonoDLE (Ma et al., 2021) as our baseline model. Particularly, this baseline model adopts DLA-34 (Yu et al., 2017) as the backbone and uses several parallel heads to predict the required items for 3D object detection. Due to this clean and compact design, this model achieves good performance with high efficiency. Besides, we further normalize the confidence of each predicted object using the estimated depth uncertainty (see Appendix A.1 for more details), which brings about 1 AP improvement ",
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+        "type": "text",
+        "text": "Teacher model. Existing LiDAR-based models are mainly based on the 3D CNN or point-wise CNN. To align the gap between the feature representations of the monocular detector and the LiDAR-based detector, we project the LiDAR points into the image plane to generate the sparse depth map. Further, we also use the interpolation algorithm (Ku et al., 2018) to generate the dense depth, and see Figure 2 for the visualization of generated data. Then, we use these ‘image-like LiDAR maps’ to train a LiDAR-based detector using the identical network with our student model. ",
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+    {
+        "type": "text",
+        "text": "3.3 MONODISTILL ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In order to transfer the spatial cues from the well-trained teacher model to the student model, we design three complementary distillation schemes to provide additional guidance to the baseline model. ",
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+    {
+        "type": "text",
+        "text": "Scene-level distillation in the feature space. First, we think directly enforcing the image-based model learns the feature representations of the LiDAR-based models is sub-optimal, caused by the different modalities. The scene level knowledge can help the monocular 3D detectors build a highlevel understanding for the given image by encoding the relative relations of the features, keeping the knowledge structure and alleviating the modality gap. Therefore, we train our student model under the guidance of the high-level semantic features provided by the backbone of the teacher model. To better model the structured cues, we choose to learn the affinity map (Hou et al., 2020) of high-level features, instead of the features themselves. Specifically, we first generate the affinity map, which encodes the similarity of each feature vector pair, for both the teacher and student network, and each element $\\mathrm { A } _ { \\mathrm { i , j } }$ in this affinity map can be computed by: ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/9d1081f2cd7b43241e439566b63df21f293e31476cfeb14a48f016202cc96dc5.jpg",
+        "text": "$$\n\\mathrm { { A } _ { i , j } = \\frac { \\mathbf { f _ { i } ^ { T } } \\mathbf { f _ { j } } } { | | \\mathbf { f _ { i } } | | _ { 2 } \\cdot | | \\mathbf { f _ { j } } | | _ { 2 } } , }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { f _ { j } }$ and $\\mathbf { f _ { j } }$ denote the $\\mathbf { i } ^ { t h }$ and $\\mathbf { j } ^ { t h }$ feature vector. After that, we use the L1 norm to enforce the student net to learn the structured information from the teacher net: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a26ae56f08deff234e99856dd2860a381f5ba94bcd4c278efe99117b7dfa8b79.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathbf { s f } } = \\frac { 1 } { \\mathrm { K } \\times \\mathrm { K } } \\sum _ { i = 1 } ^ { K } \\sum _ { j = 1 } ^ { K } | | \\mathrm { A } _ { \\mathbf { i } , \\mathbf { j } } ^ { \\mathbf { t } } - \\mathrm { A } _ { \\mathbf { i } , \\mathbf { j } } ^ { \\mathbf { s } } | | _ { 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where K is the number of the feature vectors. Note the computational/storage complexity is quadratically related to K. To reduce the cost, we group all features into several local regions and generate the affinity map using the features of local regions. This makes the training of the proposed model more efficient, and we did not observe any performance drop caused by this strategy. ",
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+    {
+        "type": "text",
+        "text": "Object-level distillation in the feature space. Second, except for the affinity map, directly using the features from teacher net as guidance may also provide valuable cues to the student. However, there is much noise in feature maps, since the background occupies most of the area and is less informative. Distilling knowledge from these regions may make the network deviate from the right optimization direction. To make the knowledge distillation more focused, limiting the distillation area is necessary. Particularly, the regions in the ground-truth 2D bounding boxes are used for knowledge transfer to mitigate the effects of noise. Specifically, given the feature maps of the teacher model and student model $\\{ \\mathbf { F ^ { t } } , \\mathbf { F ^ { s } } \\}$ , our second distillation loss can be formulated as. ",
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+    {
+        "type": "equation",
+        "img_path": "images/8641d59123090de874cb834fde2b110c244c08eb904c0a1b6bb4e7bb2451990d.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathbf { o f } } = \\frac { 1 } { \\mathrm { N } _ { \\mathrm { p o s } } } | | \\mathrm { M } _ { \\mathbf { o f } } ( \\mathrm { F } _ { \\mathbf { s } } - \\mathrm { F } _ { \\mathbf { t } } ) | | _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathrm { M _ { o f } }$ is the mask generated from the center point and the size of 2D bounding box and $\\mathrm { N _ { p o s } }$ is the number of valid feature vectors. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Object-level distillation in the result space. Third, similar to the traditional KD, we use the predictions from the teacher net as extra ‘soft label’ for the student net. Note that in this scheme, only the predictions on the foreground region should be used, because the predictions on the background region are usually false detection. As for the definition of the ‘foreground regions’, inspired by CenterNet (Zhou et al., 2019), a simple baseline is regarding the center point as the foreground region. Furtherly, we find that the quality of the predicted value of the teacher net near the center point is good enough to guide the student net. Therefore, we generate a Gaussian-like mask (Tian et al., 2019; Wang et al., 2021b) based on the position of the center point and the size of 2D bounding box and the pixels whose response values surpass a predefined threshold are sampled, and then we train these samples with equal weights (see Figure 4 for the visualization). After that, our third distillation loss can be formulated as: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/eb6271f267f4f82b34d6a0e6572392a44390c7577520f34d92d08832e2294ccc.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathbf { o r } } = \\sum _ { k = 1 } ^ { N } | | \\mathbf { M _ { o r } ( y _ { k } ^ { s } - y _ { k } ^ { t } ) } | | _ { 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathrm { M } _ { \\mathbf { o r } }$ is the mask which represents positive and negative samples, $\\mathbf { y _ { k } }$ is the output of the $k ^ { t h }$ detection head and $N$ is the number of detection heads. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Additional strategies. We further propose some strategies for our method. First, for the distillation schemes in the feature space (i.e. $\\mathcal { L } _ { \\mathrm { s f } }$ and $\\mathcal { L } _ { \\mathbf { o f } } ^ { \\mathrm { ~ ~ } }$ ), we only perform them on the last three blocks of the backbone. The main motivation of this strategy is: The first block usually is rich in the low-level features (such as edges, textures, etc.). The expression forms of the low-level features for LiDAR and image data may be completely different, and enforcing the student net to learn these features in a modality-across manner may mislead it. Second, in order to better guide the student to learn spatial-aware feature representations, we apply the attention based fusion module (FF in Table 1) proposed by Chen et al. (2021b) in our distillation schemes in the feature space ( i.e. $\\mathcal { L } _ { \\mathrm { s f } }$ and $\\mathcal { L } _ { \\mathbf { o f } }$ ). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Loss function. We train our model in an end-to-end manner using the following loss function: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1ce64824fb410dd83e925befd1f715cde0d8b57e96dbf465d705c1a65c8426a6.jpg",
+        "text": "$$\n{ \\mathcal { L } } = { \\mathcal { L } } _ { \\mathbf { s r c } } + \\lambda _ { 1 } \\cdot { \\mathcal { L } } _ { \\mathbf { s f } } + \\lambda _ { 2 } \\cdot { \\mathcal { L } } _ { \\mathbf { o f } } + \\lambda _ { 3 } \\cdot { \\mathcal { L } } _ { \\mathbf { o r } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { L } _ { \\mathrm { s r c } }$ denotes the loss function used in the MonoDLE (Ma et al., 2021). $\\lambda _ { 1 } , \\lambda _ { 2 } , \\lambda _ { 3 }$ are the hyper-parameters to balance each loss. For the teacher net, only $\\mathcal { L } _ { \\mathrm { s r c } }$ is adopted. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/f461b5ed95d7a231a3d707af2b5eec44ac7197d01ed736d51fa6c546069e9f8a.jpg",
+        "image_caption": [
+            "Figure 4: Left: Regard the center point as the foreground region. Right: Generate foreground region from the center point and the size of bounding box. Besides, the 2D bounding boxes are used as the foreground region for $\\mathcal { L } _ { \\mathbf { o f } }$ . "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "4.1 SETUP ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Dataset and metrics. We conduct our experiments on the KITTI (Geiger et al., 2012), which is most commonly used dataset in 3D detection task. Specifically, this dataset provides 7,481 training samples and 7,518 testing samples, and we further divide the training data into a train set (3,712 samples) and a validation set (3,769 samples), following prior works (Chen et al., 2015). Both 3D detection and Bird’s Eye View (BEV) detection are evaluated using $\\mathrm { { A P } | _ { R _ { 4 0 } } }$ (Simonelli et al., 2019) as metric. We report our final results on the testing set, while the ablation studies are conducted on the validation set. Besides, we mainly focus on the Car category, while also present the performances of Pedestrian and Cyclist in Appendix A.2 for reference. ",
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+    {
+        "type": "text",
+        "text": "Implementation. We provide the implementation details in Appendix A.1. Besides, our code will be open-sourced for the reproducibility. ",
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+    {
+        "type": "text",
+        "text": "4.2 MAIN RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Ablation studies. Table 1 shows the ablation studies of the proposed methods. Specifically, we found that all three distillation schemes can improve the accuracy of the baseline model, and the improvements of them are complementary. Besides, the feature fusion strategy can also boost the accuracy. Compared with the baseline, our full model improves 3D detection performance by 3.34, 5.02, 2.98 and improve BEV performance by 5.16, 6.62, 3.87 on the moderate, easy and hard settings respectively. ",
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+    {
+        "type": "table",
+        "img_path": "images/2c87bfc988985a9bf1a4689495be99705aec62c0bce4d2dc07bca9f1abda4035.jpg",
+        "table_caption": [
+            "Table 1: Ablation studies on the KITTI validation set. SF, OF, and OR denote the scene-level distillation in feature space, the object-level distillation in feature space, and the object-level distillation in result space, respectively. Besides, FF means the attention based feature fusion strategy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\"></td><td rowspan=\"3\">SF</td><td rowspan=\"3\">OF</td><td rowspan=\"3\">OR</td><td rowspan=\"3\">FF</td><td colspan=\"3\">3D@IOU=0.7</td><td colspan=\"3\">BEV@IOU=0.7</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>15.13</td><td>19.29</td><td>12.78</td><td>20.24</td><td>26.47</td><td>18.29</td></tr><tr><td>a. b.</td><td>√</td><td></td><td></td><td></td><td>16.96</td><td>21.99</td><td>14.42</td><td>22.79</td><td>29.76</td><td>19.78</td></tr><tr><td>c.</td><td></td><td>√</td><td></td><td></td><td>16.85</td><td>21.76</td><td>14.36</td><td>22.30</td><td>28.93</td><td>19.31</td></tr><tr><td>d.</td><td></td><td></td><td>√</td><td></td><td>17.24</td><td>21.63</td><td>14.71</td><td>23.47</td><td>30.52</td><td>20.33</td></tr><tr><td>e.</td><td></td><td>√</td><td></td><td></td><td>17.33</td><td>22.34</td><td>14.63</td><td>22.90</td><td>30.02</td><td>19.84</td></tr><tr><td>f.</td><td>广</td><td></td><td>√</td><td></td><td>17.70</td><td>22.59</td><td>15.17</td><td>23.59</td><td>31.07</td><td></td></tr><tr><td></td><td></td><td>！</td><td>√</td><td></td><td>17.98</td><td>22.58</td><td>15.26</td><td>23.76</td><td>30.98</td><td>20.46 20.52</td></tr><tr><td>g. h.</td><td></td><td>1</td><td></td><td></td><td>18.24</td><td>23.82</td><td>15.49</td><td>25.06</td><td>32.66</td><td>21.88</td></tr><tr><td>i.</td><td></td><td>厂</td><td>厂</td><td>！</td><td>18.47</td><td>24.31</td><td>15.76</td><td>25.40</td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>33.09</td><td>22.16</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Detailed design choice. We provide additional experiments in Table 2 for our method. First, as for object-level distillation in the feature space, we investigate the different effects of applying distillation on the whole image and foreground regions. Due to the noise in the background, guiding the foreground regions is more effective than the whole image, which improves the accuracy by 0.72 on the moderate settings in 3D detection. Second, as for object-level distillation in the result space, we compare the different effects of point label and region label. It can be observed that the generated region can significantly increase performance while guiding only in sparse point label brings limited improvements. Our proposed label diffusion strategy can increase the number of positive samples for supervision, thus improving performance. ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/63db66e3324930766b2907d715fdd1a9c27794acf76d8763faed4affb9910cc8.jpg",
+        "table_caption": [
+            "Table 2: Evaluation on the KITTI validation set for detailed design choice. OF and OR represent the object-level distillation in feature space and the object-level distillation in result space. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Guidance</td><td rowspan=\"2\">Choice</td><td colspan=\"3\">3D@I0U=0.7</td><td colspan=\"3\">BEV@IOU=0.7</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td rowspan=\"2\">OF</td><td>full</td><td>16.13</td><td>21.52</td><td>14.18</td><td>22.04</td><td>27.85</td><td>19.04</td></tr><tr><td>foreground</td><td>16.85</td><td>21.76</td><td>14.36</td><td>22.30</td><td>28.93</td><td>19.31</td></tr><tr><td rowspan=\"2\">OR</td><td>sparse label</td><td>15.51</td><td>20.58</td><td>13.70</td><td>21.47</td><td>27.16</td><td>18.60</td></tr><tr><td>diffused label</td><td>17.24</td><td>21.63</td><td>14.71</td><td>23.47</td><td>30.52</td><td>20.33</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Comparison with state-of-the-art methods. Table 3 and Table 4 compare the proposed method with other state-of-the-art methods on the KITTI test and validation sets. On the test set, the proposed method outperforms existing methods in all metrics. We note that, compared with previous best results, we can obtain 1.83, 0.50, 1.53 improvements on the moderate, easy and hard settings in 3D detection. Furthermore, our method achieves more significant improvements in BEV detection, increasing upon the prior work by 2.51, 1.21, 2.46 on the moderate, easy and hard settings. Moreover, compared with the depth-based methods, our method outperforms them in performance by a margin and is superior to theirs in the inference speed. By contrast, our method only takes 40ms to process a KITTI image, tested on a single NVIDIA GTX 1080Ti, while the Fastest of the depth-based methods (Ma et al., 2019; 2020; Ding et al., 2020; Wang et al., 2021a; Reading et al., 2021) need $1 8 0 \\mathrm { m s }$ . On the validation set, the proposed also performs best, both for the $0 . 7 \\ \\mathrm { I o U }$ threshold and 0.5 IoU threshold. Besides, we also present the performance of the baseline model to better show the effectiveness of the proposed method. Note that we do not report the performances of some depth-based methods (Ma et al., 2019; 2020; Ding et al., 2020; Wang et al., 2021a) due to the data leakage problem \\* ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 MORE DISCUSSIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "What has the student model learned from the teacher model? To locate the source of improvement, we use the items predicted from the baseline model to replace that from our full model, and Table 5 summarizes the results of the cross-model evaluation. From these results, we can see that the teacher model provides effective guidance to the location estimation $( \\mathsf { b {  } f } )$ , and improvement of dimension part is also considerable $( \\mathrm { c } \\to \\mathrm { f } )$ . Relatively, the teacher model provides limited valuable cues to the classification and orientation part. This phenomenon suggests the proposed methods boost the performance of the baseline model mainly by introducing the spatial-related information, which is consistent with our initial motivation. Besides, we also show the errors of depth estimation, see Appendix A.3 for the results. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Is the effectiveness of our method related to the performance of the teacher model? An intuitive conjecture is the student can learn more if the teacher network has better performance. To explore this problem, we also use the sparse LiDAR maps to train a teacher net to provide guidance to the student model (see Figure 2 for the comparison of the sparse and dense data). As shown in Table 6, the performance of the teacher model trained from the sparse LiDAR maps is largely behind by that from dense LiDAR maps (drop to $2 2 . 0 5 \\%$ from $4 2 . 4 5 \\%$ , moderate setting), while both of them provides comparable benefits to the student model. Therefore, for our task, the performance of the teacher model is not directly related to the performance improvement, while the more critical factor is whether the teacher network contains complementary information to the student network. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Do we need depth estimation as an intermediate task? As shown in Figure 1, most previous methods choose to estimate the depth maps to provide depth information for monocular 3D detection (information flow: LiDAR data estimated depth map $ 3 \\mathrm { D }$ detector). Compared with this scheme, our method directly learns the depth cues from LiDAR-based methods (information flow: LiDAR data $ 3 \\mathrm { D }$ detector), avoiding the information loss in the depth estimation step. Here we quantitatively show the information loss in depth estimation using a simple experiment. Specifically, we use DORN $\\mathrm { F u }$ et al., 2018) (same as most previous depth augmented methods) to generate the depth maps, and then use them to train the teacher net. Table 7 shows the results of this experiment. Note that, compared with setting c, setting b’s teacher net is trained from a larger training set (23,488 vs. 3,712) with ground-truth depth maps (ground truth depth maps vs. noisy depth maps). Nevertheless, this scheme still lags behind our original method, which means that there is serious information loss in monocular depth estimation (stereo image performs better, which is discussed in Appendix A.4). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/482f96d515f8bb415a95d3872567e74933a49225d8bc2d06667614512b79609e.jpg",
+        "table_caption": [
+            "Table 3: Comparison of state-of-the-art methods on the KITTI test set. Methods are ranked by moderate setting. We highlight the best results in bold and the second place in underlined. Only RGB images are required as input in the inference phase for all listed methods. \\*: need dense depth maps or LiDAR signals for training. $^ \\dagger$ : our baseline model without confidence normalization. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">3D@IOU=0.7</td><td colspan=\"3\">BEV@IOU=0.7</td><td rowspan=\"2\">Runtime</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>M3D-RPN (Brazil &amp; Liu, 2019)</td><td>9.71</td><td>14.76</td><td>7.42</td><td>13.67</td><td>21.02</td><td>10.23</td><td>160 ms</td></tr><tr><td>SMOKE (Liu et al., 2020)</td><td>9.76</td><td>14.03</td><td>7.84</td><td>14.49</td><td>20.83</td><td>12.75</td><td>30 ms</td></tr><tr><td>MonoPair (Chen et al., 2020b)</td><td>9.99</td><td>13.04</td><td>8.65</td><td>14.83</td><td>19.28</td><td>12.89</td><td>60 ms</td></tr><tr><td>RTM3D (Li et al., 2020)</td><td>10.34</td><td>14.41</td><td>8.77</td><td>14.20</td><td>19.17</td><td>11.99</td><td>50 ms</td></tr><tr><td>AM3D* (Ma et al., 2019)</td><td>10.74</td><td>16.50</td><td>9.52</td><td>17.32</td><td>25.03</td><td>14.91</td><td>400 ms</td></tr><tr><td>PatchNet* (Ma et al., 2020)</td><td>11.12</td><td>15.68</td><td>10.17</td><td>16.86</td><td>22.97</td><td>14.97</td><td>400 ms</td></tr><tr><td>D4LCN* (Ding et al., 2020)</td><td>11.72</td><td>16.65</td><td>9.51</td><td>16.02</td><td>22.51</td><td>12.55</td><td>200 ms</td></tr><tr><td>MonoDLE† (Ma et al., 2021)</td><td>12.26</td><td>17.23</td><td>10.29</td><td>18.89</td><td>24.79</td><td>16.00</td><td>40 ms</td></tr><tr><td>MonoRUn*(Chen et al., 2021a)</td><td>12.30</td><td>19.65</td><td>10.58</td><td>17.34</td><td>27.94</td><td>15.24</td><td>70 ms</td></tr><tr><td>GrooMeD-NMS (Kumar et al., 2021)</td><td>12.32</td><td>18.10</td><td>9.65</td><td>18.27</td><td>16.19</td><td>14.05</td><td>120 ms</td></tr><tr><td>DDMP-3D* (Wang et al., 2021a)</td><td>12.78</td><td>19.71</td><td>9.80</td><td>17.89</td><td>28.08</td><td>13.44</td><td>180 ms</td></tr><tr><td>CaDDN* (Reading et al., 2021)</td><td>13.41</td><td>19.17</td><td>11.46</td><td>18.91</td><td>27.94</td><td>17.19</td><td>630 ms</td></tr><tr><td>MonoEF (Zhou et al., 2021)</td><td>13.87</td><td>21.29</td><td>11.71</td><td>19.70</td><td>29.03</td><td>17.26</td><td>30 ms</td></tr><tr><td>MonoFlex (Zhang et al., 2021b)</td><td>13.89</td><td>19.94</td><td>12.07</td><td>19.75</td><td>28.23</td><td>16.89</td><td>30 ms</td></tr><tr><td>Autoshape (Liu et al., 2021)</td><td>14.17</td><td>22.47</td><td>11.36</td><td>20.08</td><td>30.66</td><td>15.59</td><td>50 ms</td></tr><tr><td>GUPNet (Lu et al., 2021)</td><td>14.20</td><td>20.11</td><td>11.77</td><td>1</td><td>1</td><td>1</td><td>35ms</td></tr><tr><td>Ours*</td><td>16.03</td><td>22.97</td><td>13.60</td><td>22.59</td><td>31.87</td><td>19.72</td><td>40 ms</td></tr><tr><td>Improvements</td><td>+1.83</td><td>+0.50</td><td>+1.53</td><td>+2.51</td><td>+1.21</td><td>+2.46</td><td>-</td></tr></table>",
+        "bbox": [
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+            169,
+            821,
+            436
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/b2fe2b2cc4e8bfcdc6108656e9fd1021da7fd51bbe0cb0e6dc3883cd79e9069e.jpg",
+        "table_caption": [
+            "Table 4: Performance of the Car category on the KITTI validation set. We highlight the best results in bold and the second place in underlined. †: our baseline model without confidence normalization. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">3D@IOU=0.7</td><td colspan=\"3\">BEV@IOU=0.7</td><td colspan=\"3\">3D@IOU=0.5</td><td colspan=\"3\">BEV@IOU=0.5</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>M3D-RPN</td><td>11.07</td><td>14.53</td><td>8.65</td><td>15.62</td><td>20.85</td><td>11.88</td><td>35.94</td><td>48.53</td><td>28.59</td><td>39.60</td><td>53.35</td><td>31.76</td></tr><tr><td>MonoPair</td><td>12.30</td><td>16.28</td><td>10.42</td><td>18.17</td><td>24.12</td><td>15.76</td><td>42.39</td><td>55.38</td><td>37.99</td><td>47.63</td><td>61.06</td><td>41.92</td></tr><tr><td>MonoDLEt</td><td>13.66</td><td>17.45</td><td>11.68</td><td>19.33</td><td>24.97</td><td>17.01</td><td>43.42</td><td>55.41</td><td>37.81</td><td>46.87</td><td>60.73</td><td>41.89</td></tr><tr><td>GrooMeD-NMS</td><td>14.32</td><td>19.67</td><td>11.27</td><td>19.75</td><td>27.38</td><td>15.92</td><td>41.07</td><td>55.62</td><td>32.89</td><td>44.98</td><td>61.83</td><td>36.29</td></tr><tr><td>MonoRUn</td><td>14.65</td><td>20.02</td><td>12.61</td><td>-</td><td>=</td><td>■</td><td>43.39</td><td>59.71</td><td>38.44</td><td>-</td><td>■</td><td>■</td></tr><tr><td>GUPNet</td><td>16.46</td><td>22.76</td><td>13.72</td><td>22.94</td><td>31.07</td><td>19.75</td><td>42.33</td><td>57.62</td><td>37.59</td><td>47.06</td><td>61.78</td><td>40.88</td></tr><tr><td>MonoFlex</td><td>17.51</td><td>23.64</td><td>14.83</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Baseline</td><td>15.13</td><td>19.29</td><td>12.78</td><td>20.24</td><td>26.47</td><td>18.29</td><td>43.54</td><td>57.43</td><td>39.22</td><td>48.49</td><td>63.56</td><td>42.81</td></tr><tr><td>Ours</td><td>18.47</td><td>24.31</td><td>15.76</td><td>25.40</td><td>33.09</td><td>22.16</td><td>49.35</td><td>65.69</td><td>43.49</td><td>53.11</td><td>71.45</td><td>46.94</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/7f16922b9be719c54e70a5997450fe646fa85d3d108cf9f6c73b84f32ec0ec39.jpg",
+        "table_caption": [
+            "Table 5: Cross-model evaluation on the KITTI validation set. We extract each required item (location, dimension, orientation, and confidence) from the baseline model (B) and the full model (O), and evaluate them in a cross-model manner. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\"></td><td rowspan=\"3\">loc.</td><td rowspan=\"3\">dim.</td><td rowspan=\"3\">ori.</td><td rowspan=\"3\">con.</td><td colspan=\"3\">3D@IOU=0.7</td><td colspan=\"3\">BEV@IOU=0.7</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>a.</td><td>B</td><td>B</td><td>B</td><td>B</td><td>15.13</td><td>19.29</td><td>12.78</td><td>20.24</td><td>26.47</td><td>18.29</td></tr><tr><td>b.</td><td>B</td><td>0</td><td>0</td><td>0</td><td>16.05</td><td>20.07</td><td>13.47</td><td>21.31</td><td>27.77</td><td>19.14</td></tr><tr><td>c.</td><td>0</td><td>B</td><td>0</td><td>0</td><td>17.91</td><td>22.87</td><td>15.29</td><td>25.09</td><td>32.78</td><td>21.93</td></tr><tr><td>d.</td><td>0</td><td>0</td><td>B</td><td>0</td><td>18.12</td><td>24.02</td><td>15.34</td><td>25.02</td><td>32.85</td><td>21.84</td></tr><tr><td>e.</td><td>0</td><td>0</td><td>0</td><td>B</td><td>18.41</td><td>24.27</td><td>15.55</td><td>24.98</td><td>32.78</td><td>21.81</td></tr><tr><td>f.</td><td>0</td><td>0</td><td>0</td><td>0</td><td>18.47</td><td>24.31</td><td>15.76</td><td>25.40</td><td>33.09</td><td>22.16</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            853,
+            825,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/8e1c5de6bf44ac1f4141381049c61fe1fb9e7eb63e5d1e94fce5d7675c1acfee.jpg",
+        "table_caption": [
+            "Table 6: Performance of the student model under the guidance of different teacher models. Metric is the $\\mathrm { A P } | _ { 4 0 }$ for the 3D detection task on the KITTI validation set. We also show the performance improvements of the student model to the baseline model for better comparison. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">Teacher Model</td><td colspan=\"2\">Student Model</td><td colspan=\"4\">Improvement</td></tr><tr><td>Mod.</td><td>Easy Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>sparse maps</td><td>22.05</td><td>31.67</td><td>18.72 18.07</td><td>23.61</td><td>15.36</td><td>+2.94</td><td>+4.32</td><td>+2.58</td></tr><tr><td>dense maps</td><td>42.57</td><td>58.06</td><td>37.07 18.47</td><td>24.31</td><td>15.76</td><td>+3.34</td><td>+5.02</td><td>+2.98</td></tr></table>",
+        "bbox": [
+            189,
+            155,
+            803,
+            219
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            246,
+            825,
+            315
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/272c42d35b7a4be1c247b211f84135dc06e6e0b35ad5735f1c1f667ee5cfcdf9.jpg",
+        "table_caption": [
+            "Table 7: Comparison of using depth estimation as intermediate task or not. Setting a. and c. denote the baseline model and our full model. Setting b. uses the depth maps generated from DORN (Fu et al., 2018) to train the teacher model. Experiments are conducted on the KITTI validation set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">3D@IOU=0.7</td><td colspan=\"3\">BEV @IOU=0.7</td><td colspan=\"2\">AOS@IOU=0.7</td><td colspan=\"3\">2D@IOU=0.7</td></tr><tr><td></td><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod. Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>a.</td><td>15.13</td><td>19.29</td><td>12.78</td><td>20.24</td><td>26.47</td><td>18.29</td><td>90.95 97.46</td><td>83.02</td><td>92.18</td><td>98.37</td><td>85.05</td></tr><tr><td>b.</td><td>17.70</td><td>23.21</td><td>15.02</td><td>23.34</td><td>31.20</td><td>20.40</td><td>91.50 97.77</td><td>83.49</td><td>92.51</td><td>98.54</td><td>85.38</td></tr><tr><td>c.</td><td>18.47</td><td>24.31</td><td>15.76</td><td>25.40</td><td>33.09</td><td>22.16</td><td>91.67 97.88</td><td>83.59</td><td>92.71</td><td>98.58</td><td>85.56</td></tr></table>",
+        "bbox": [
+            178,
+            382,
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+            455
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.4 QUALITATIVE RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            479,
+            379,
+            494
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In Figure 5, we show the qualitative comparison of detection results. We can see that the proposed method shows better localization accuracy than the baseline model. See Appendix A.7 for more detailed qualitative results. ",
+        "bbox": [
+            173,
+            505,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/887a97c282d6cc6b7c869d15ef95deb2c6188122430cf03d43e8ce8b70feba9b.jpg",
+        "image_caption": [
+            "Figure 5: Qualitative results. We use green, blue and red boxes to denote the results from baseline, our method, and ground truth. Besides, we use red circle to highlight the main differences. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            560,
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+            722
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            795,
+            318,
+            810
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose the MonoDistill, which introduces spatial cues to the monocular 3D detector based on the knowledge distillation mechanism. Compared with previous schemes, which share the same motivation, our method avoids any modifications on the target model and directly learns the spatial features from the model rich in these features. This design makes the proposed method perform well in both performance and efficiency. To show an all-around display of our model, extensive experiments are conducted on the KITTI dataset, where the proposed method ranks $1 ^ { s t }$ at 25 FPS among all monocular 3D detectors. ",
+        "bbox": [
+            173,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            367,
+            117
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This work was supported in part by the National Natual Science Foundation of China (NSFC) under Grants No.61932020, 61976038, U1908210 and 61772108. Wanli Ouyang was supported by the Australian Research Council Grant DP200103223, FT210100228, and Australian Medical Research Future Fund MRFAI000085. ",
+        "bbox": [
+            174,
+            132,
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+            189
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+        "bbox": [
+            174,
+            210,
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+        ],
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+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 12
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+    {
+        "type": "text",
+        "text": "A.1 MORE DETAILS OF THE BASELINE MODEL",
+        "text_level": 1,
+        "bbox": [
+            174,
+            138,
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+        ],
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+    {
+        "type": "text",
+        "text": "Network architecture. The baseline network is extended from the anchor-free 2D object detection framework, which consists of a feature extraction network and seven detection subheads. We employ DLA-34 (Yu et al., 2017) without deformable convolutions as our backbone. The feature maps are downsampled by 4 times and then we take the image features as input and use 3x3 convolution, ReLU, and 1x1 convolution to output predictions for each detection head. Detection head branches include three for 2D components and four for 3D components. Specifically, 2D detection heads include heatmap, offset between the 2D key-point and the 2D box center, and size of 2D box. 3D components include offset between the 2D key-point and the projected 3D object center, depth, dimensions, and orientations. As for objective functions, we train the heatmap with focal loss. The other loss items adopt L1 losses except for depth and orientation. The depth branch employs a modified L1 loss with the assist of heteroscedastic aleatoric uncertainty. Common MultiBin loss is used for the orientation branch. Besides, we propose a strategy to improve the accuracy of baseline. Inspire by (Lu et al., 2021), estimated depth uncertainty can provide confidence for each projection depth. Therefore, we normalize the confidence of each predicted box using depth uncertainty. In this way, the score has capability of indicating the uncertainty of depth. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Training details. Our model is trained on 2 NVIDIA 1080Ti GPUs in an end-to-end manner for 150 epochs. We employ the common Adam optimizer with initial learning rate $1 . 2 5 e ^ { - 4 }$ , and decay it by ten times at 90 and 120 epochs. To stabilize the training process, we also applied the warm-up strategy (5 epochs). As for data augmentations, only random random flip and center crop are applied. Same as the common knowledge distillation scheme, we first train teacher network in advance, and then fix the teacher network. As for student network, we simply train the detection model to give a suitable initialization. We implemented our method using PyTorch. And our code is based on Ma et al. (2021). ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 PEDESTRIAN/CYCLIST DETECTION. ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            517,
+            462,
+            531
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Due to the small sizes, non-rigid structures, and limited training samples, the pedestrians and cyclists are much more challenging to detect than cars. We first report the detection results on test set in Table 8. It can be seen that our proposed method is also competitive with current state-of-the-art methods on the KITTI test set, which increases $0 . 6 9 \\mathrm { A P }$ on hard difficulty level of pedestrian category. Note that, the accuracy of these difficult categories fluctuates greatly compared with Car detection due to insufficient training samples (see Table 10 for the details). Due the access to the test server is limited, we conduct more experiments for pedestrian/cyclist on the validation set for general conclusions (we run the proposed method three times with different random seeds), and the experimental results are summarized in Table 9. According to these results, we can find that the proposed method can effectively boost the accuracy of the baseline model for pedestrian/cyclist detection. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/24a99eabe6a30bd5172cb7bc0dd4acf35a84f2dda4217ad6eca0351c014c9c12.jpg",
+        "table_caption": [
+            "Table 8: Performance of Pedestrian/Cyclist detection on the KITTI test set. We highlight the best results in bold and the second place in underlined. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">Pedestrian</td><td colspan=\"3\">Cyclist</td></tr><tr><td>Easy</td><td>Mod.</td><td>Hard</td><td>Easy</td><td>Mod.</td><td>Hard</td></tr><tr><td>M3D-RPN D4LCN</td><td>4.92 4.55</td><td>3.48 3.42</td><td>2.94 2.83</td><td>0.94 2.45</td><td>0.65 1.67</td><td>0.47 1.36</td></tr><tr><td>MonoPair</td><td>10.02</td><td>6.68</td><td>5.53</td><td>3.79</td><td>2.21</td><td>1.83</td></tr><tr><td>MonoFlex</td><td>9.43</td><td>6.31</td><td>5.26</td><td>4.17</td><td>2.35</td><td>2.04</td></tr><tr><td>MonoDLE</td><td>9.64</td><td>6.55</td><td>5.44</td><td>4.59</td><td>2.66</td><td>2.45</td></tr><tr><td>CaDDN</td><td>12.87</td><td>8.14</td><td>6.76</td><td>7.00</td><td>3.14</td><td>3.30</td></tr><tr><td>DDMP-3D</td><td>4.93</td><td>3.55</td><td>3.01</td><td>4.18</td><td>2.50</td><td>2.32</td></tr><tr><td>AutoShape</td><td>5.46</td><td>3.74</td><td>3.03</td><td>5.99</td><td>3.06</td><td>2.70</td></tr><tr><td>Ours</td><td>12.79</td><td>8.17</td><td>7.45</td><td>5.53</td><td>2.81</td><td>2.40</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/0f083cc57c772fe2fb8ffb096a26cfc9f5221c12ae99bbeb4572cb0b797bfd6d.jpg",
+        "table_caption": [
+            "Table 9: Performance of Pedestrian/Cyclist detection on the KITTI validation set. Both 0.25 and $0 . 5 \\mathrm { I o U }$ thresholds are considered. We report the mean of several experiments for the proposed methods. $\\pm$ captures the standard deviation over random seeds. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Method</td><td colspan=\"3\">3D@IoU=0.25</td><td colspan=\"3\">3D@IoU=0.5</td></tr><tr><td>Easy</td><td>Mod.</td><td>Hard</td><td>Easy</td><td>Mod.</td><td>Hard</td></tr><tr><td rowspan=\"2\">Pedestrian</td><td>Baseline</td><td>29.07±0.21</td><td>23.77±0.15</td><td>19.85±0.14</td><td>6.8±0.28</td><td>5.17±0.08</td><td>4.37±0.15</td></tr><tr><td>Ours</td><td>32.09±0.71</td><td>25.53±0.55</td><td>21.15±0.79</td><td>8.95±1.26</td><td>6.84±0.81</td><td>5.32±0.75</td></tr><tr><td rowspan=\"2\">Cyclist</td><td>Baseline</td><td>21.06±0.46</td><td>11.87±0.19</td><td>10.77±0.02</td><td>3.71±0.49</td><td>1.88±0.23</td><td>1.64±0.04</td></tr><tr><td>Ours</td><td>24.26±1.29</td><td>13.04±0.44</td><td>12.08±0.68</td><td>5.38±0.91</td><td>2.67±0.40</td><td>2.53±0.38</td></tr></table>",
+        "bbox": [
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+            155,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/5a5fb34c6f535070daad0128e7b444596125012c6a27c7887f8ea5a8b9f0003d.jpg",
+        "table_caption": [
+            "Table 10: Training samples of each category on the KITTI training set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>cars</td><td>pedestrians</td><td>cyclists</td></tr><tr><td>#instances</td><td>14,357</td><td>2,207</td><td>734</td></tr></table>",
+        "bbox": [
+            336,
+            282,
+            656,
+            314
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.3 DEPTH ERROR ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            340,
+            397,
+            354
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 6, we compare the depth error between baseline and our method. Specifically, we project all valid samples of the Car category into the image plane to get the corresponding predicted depth values. Then we fit the depth errors between ground truths and predictions as a linear function by least square method. According to the experimental results, we can find that our proposed method can boost the accuracy of depth estimation at different distances. ",
+        "bbox": [
+            173,
+            367,
+            825,
+            438
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/9b43a6ba991c5a0e9121d360ec5db7380b63da90ccd4da60004f2ed0feaac777.jpg",
+        "image_caption": [
+            "Figure 6: Errors of depth estimation. We show the errors of depth estimation as a function of the depth ( $\\mathbf { X }$ -axis) for the baseline model (left) and our full model (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            458,
+            821,
+            616
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.4 THE EFFECTS OF STEREO DEPTH ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            690,
+            447,
+            704
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We also explored the changes in performance under the guidance of estimated stereo depth (Chang & Chen, 2018), and show the results in Table 11. Stereo depth estimation exploits geometric constraints in stereo images to obtain the absolute depth value through pixel-wise matching, which is more accurate compared with monocular depth estimation. Therefore, under the guidance of stereo depth, the model achieves almost the same accuracy as LiDAR signals guidance at $0 . 5 \\ \\mathrm { I o U }$ threshold, and there is only a small performance drop at 0.7 IoU threshold. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A.5 GENERALIZATION OF THE PROPOSED METHOD ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 13
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+    {
+        "type": "text",
+        "text": "In the main paper, we introduced the proposed method based on MonoDLE (Ma et al., 2021). Here we discuss the generalization ability of the proposed method. ",
+        "bbox": [
+            176,
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+    },
+    {
+        "type": "text",
+        "text": "Generalizing to other baseline models. To show the generalization ability of the proposed method, we apply our method on another monocular detector GUPNet (Lu et al., 2021), which is a two-stage detection method. Experimental results are shown in the Table 12. We can find that the proposed method can also boosts the performances of GUPNet, which confirms the generalization of our method. ",
+        "bbox": [
+            176,
+            882,
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+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/b891346fdda689e4dff1d4fd80e3c1c252f0ff5285035e396a37a922e76a9af4.jpg",
+        "table_caption": [
+            "Table 11: Effects of stereo depth estimation. Baseline denotes the baseline model without guidance of teacher network. Stereo Depth and LiDAR Depth denote under the guidance of stereo depth maps and LiDAR signals. Experiments are conducted on the KITTI validation set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">3D@IOU=0.7</td><td colspan=\"3\">BEV@IOU=0.7</td><td colspan=\"3\">3D@IOU=0.5</td><td colspan=\"3\">BEV@IOU=0.5</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>Baseline</td><td>15.13</td><td>19.29</td><td>12.78</td><td>20.24</td><td>26.47</td><td>18.29</td><td>43.54</td><td>57.43</td><td>39.22</td><td>48.49</td><td>63.56</td><td>42.81</td></tr><tr><td>Stereo Depth</td><td>18.18</td><td>23.54</td><td>15.42</td><td>24.89</td><td>32.26</td><td>21.64</td><td>49.13</td><td>65.18</td><td>43.29</td><td>52.88</td><td>69.47</td><td>46.72</td></tr><tr><td>LiDAR Depth</td><td>18.47</td><td>24.31</td><td>15.76</td><td>25.40</td><td>33.09</td><td>22.16</td><td>49.35</td><td>65.69</td><td>43.49</td><td>53.11</td><td>71.45</td><td>46.94</td></tr></table>",
+        "bbox": [
+            176,
+            155,
+            825,
+            220
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            178,
+            248,
+            820,
+            277
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/e9ab6bb8b35835a6ed93e079cd8a5123197b3f76797c541f62f83033b6cac0ea.jpg",
+        "table_caption": [
+            "Table 12: MonoDistill on GUPNet. Experiments are conducted on the KITTI validation set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">3D@IOU=0.7</td><td colspan=\"3\">3D@IOU=0.5</td></tr><tr><td>Easy</td><td>Mod.</td><td>Hard</td><td>Easy</td><td>Mod.</td><td>Hard</td></tr><tr><td>GUPNet-Baseline</td><td>22.76</td><td>16.46</td><td>13.72</td><td>57.62</td><td>42.33</td><td>37.59</td></tr><tr><td>GUPNet-Ours</td><td>24.43</td><td>16.69</td><td>14.66</td><td>61.72</td><td>44.49</td><td>40.07</td></tr></table>",
+        "bbox": [
+            258,
+            315,
+            736,
+            376
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Generalizing to sparse LiDAR signals. We also explore the changes in performance under different resolution of LiDAR signals. In particular, following Pseudo-LiDAR $^ { + + }$ (You et al., 2020), we generate the simulated 32-beam/16-beam LiDAR signals and use them to train our teacher model (in the ‘sparse’ setting). We show the experimental results, based on MonoDLE, in the Table 13. We can see that, although the improvement is slightly reduced due to the decrease of the resolution of LiDAR signals, the proposed method significantly boost the performances of baseline model under all setting. ",
+        "bbox": [
+            173,
+            393,
+            825,
+            492
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/558bf4b6f5bf771252a1940a16dccaa8be8fd1f016cfdef80a54c61e0721f075.jpg",
+        "table_caption": [
+            "Table 13: Effects of the resolution of LiDAR signals. Experiments are conducted on the KITTI validation set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">3D@IOU=0.7</td><td colspan=\"3\">3D@I0U=0.5</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>Baseline</td><td>19.29</td><td>15.13</td><td>12.78</td><td>43.54</td><td>57.43</td><td>39.22</td></tr><tr><td>Ours - 16-beam</td><td>22.49</td><td>17.66</td><td>15.08</td><td>49.39</td><td>65.45</td><td>43.60</td></tr><tr><td>Ours - 32-beam</td><td>23.24</td><td>17.71</td><td>15.19</td><td>49.41</td><td>65.61</td><td>43.46</td></tr><tr><td>Ours - 64-beam</td><td>23.61</td><td>18.07</td><td>15.36</td><td>49.67</td><td>65.97</td><td>43.74</td></tr></table>",
+        "bbox": [
+            264,
+            545,
+            728,
+            640
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "More discussion. Besides, note that the camera parameters of the images on the KITTI test set are different from these of the training/validation set, and the good performance on the test set suggests the proposed method can also generalize to different camera parameters. However, generalizing to the new scenes with different statistical characteristics is a hard task for existing 3D detectors (Yang et al.; Wang et al., 2020b), including the image-based models and LiDAR-based models, and deserves further investigation by future works. We also argue that the proposed method can generalize to the new scenes better than other monocular models because ours model learns the stronger features from the teacher net. These results and analysis will be included in the revised version. ",
+        "bbox": [
+            173,
+            655,
+            825,
+            781
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.6 COMPARISON WITH DIRECT DENSE DEPTH SUPERVISION. ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            799,
+            619,
+            814
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "According to the ablation studies in the main paper, we can find that depth cues are the key factor to affect the performance of the monocular 3D models. However, dense depth supervision in the student model without KD may also introduce depth cues to the monocular 3D detectors. Here we conduct the control experiment by adding a new depth estimation branch, which is supervised by the dense LiDAR maps. Note that, this model is trained without KD. Table 14 compares the performances of the baseline model, the new control experiment, and the proposed method. From these results, we can get the following conclusions: (i) additional depth supervision can introduce the spatial cues to the models, thereby improving the overall performance; (ii) the proposed KDbased method significantly performs better than the baseline model and the new control experiment, which demonstrates the effectiveness of our method. ",
+        "bbox": [
+            174,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/2742bdbf0478f78a08244885c52dcf3dc2ced02cd97bc7baf78a64801357eb86.jpg",
+        "table_caption": [
+            "Table 14: Comparison with direct dense depth supervision. Experiments are conducted on the KITTI validation set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\"></td><td colspan=\"3\">3D@IOU=0.7</td><td colspan=\"3\">3D@IOU=0.5</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>Baseline</td><td>15.13</td><td>19.29</td><td>12.78</td><td>43.54</td><td>57.43</td><td>39.22</td></tr><tr><td>Baseline + depth supv.</td><td>17.05</td><td>21.85</td><td>14.54</td><td>46.19</td><td>60.42</td><td>41.88</td></tr><tr><td>Ours</td><td>18.47</td><td>24.31</td><td>15.76</td><td>49.35</td><td>65.69</td><td>43.49</td></tr></table>",
+        "bbox": [
+            241,
+            196,
+            751,
+            276
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.7 MORE QUALITATIVE RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            303,
+            429,
+            318
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In Figure 7, we show more qualitative results on the KITTI dataset. We use orange box, green, and purple boxes for cars, pedestrians, and cyclists, respectively. In Figure 8, we show comparison of detection results in the 3D space. It can be found that our method can significantly improve the accuracy of depth estimation compared with the baseline. ",
+        "bbox": [
+            173,
+            329,
+            825,
+            385
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/9efa6269e04e1f1776fb3a61cd66a2525d62b1b255e0b6b85422f6bdb89b37c6.jpg",
+        "image_caption": [
+            "Figure 7: Qualitative results for multi-class 3D object detection. The boxes’ color of cars, pedestrian, and cyclist are in orange, green, and purple, respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            398,
+            823,
+            631
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/ff8ebd28258b37441ea96db3473b8f4a94df501990fefe4bb7ce90d0d8bd18ec.jpg",
+        "image_caption": [
+            "Figure 8: Qualitative results of our method for 3D space. The boxes’ color of ground truth, baseline, and ours are in red, green, and blue, respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            325,
+            735,
+            651
+        ],
+        "page_idx": 16
+    }
+]

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+# Decoupling Features in Hierarchical Propagation for Video Object Segmentation
+Zongxin Yang1,2, Yi Yang1†
+1 CCAI, College of Computer Science and Technology, Zhejiang University 2 Baidu Research {yangzongxin, yangyics}@zju.edu.cn
+# Abstract
+This paper focuses on developing a more effective method of hierarchical propagation for semi-supervised Video Object Segmentation (VOS). Based on vision transformers, the recently-developed Associating Objects with Transformers (AOT) approach introduces hierarchical propagation into VOS and has shown promising results. The hierarchical propagation can gradually propagate information from past frames to the current frame and transfer the current frame feature from object-agnostic to object-specific. However, the increase of object-specific information will inevitably lead to the loss of object-agnostic visual information in deep propagation layers. To solve such a problem and further facilitate the learning of visual embeddings, this paper proposes a Decoupling Features in Hierarchical Propagation (DeAOT) approach. Firstly, DeAOT decouples the hierarchical propagation of object-agnostic and object-specific embeddings by handling them in two independent branches. Secondly, to compensate for the additional computation from dual-branch propagation, we propose an efficient module for constructing hierarchical propagation, i.e., Gated Propagation Module, which is carefully designed with single-head attention. Extensive experiments show that DeAOT significantly outperforms AOT in both accuracy and efficiency. On YouTube-VOS, DeAOT can achieve $8 6 . 0 \%$ at 22.4fps and $8 2 . 0 \%$ at 53.4fps. Without test-time augmentations, we achieve new state-of-the-art performance on four benchmarks, i.e., YouTubeVOS $( 8 6 . 2 \% )$ , DAVIS 2017 $( 8 6 . 2 \% )$ , DAVIS 2016 $( 9 2 . 9 \% )$ , and VOT 2020 (0.622). Project page: https://github.com/z-x-yang/AOT.
+# 1 Introduction
+Video Object Segmentation (VOS), which aims at recognizing and segmenting one or multiple objects of interest in a given video, has attracted much attention as a fundamental task of video understanding. This paper focuses on semi-supervised VOS, which requires algorithms to track and segment objects throughout a video sequence given objects’ annotated masks at one or several frames.
+Early VOS methods are mainly based on finetuning segmentation networks on the annotated frames [7, 32, 51] or constructing pixel-wise matching maps [10, 50]. Based on the advance of attention mechanisms [5,48,53], many attention-based VOS algorithms have been proposed in recent years and achieved significant improvement. STM [34] and the following works [11, 43, 44] leverage a memory network to store and read the target features of predicted past frames and apply a non-local attention mechanism to match the target in the current frame. Furthermore, AOT [61, 63, 65] introduces hierarchical propagation into VOS based on transformers [8, 48] and can associate multiple objects collaboratively by utilizing the IDentification (ID) mechanism [63]. The hierarchical propagation can gradually propagate ID information from past frames to the current frame and has shown promising VOS performance with remarkable scalability.
+![](images/7279e4da2ad23e730d4becab10d7af16fa0af683220cb0e802e708d751058ce5.jpg)
+Figure 1: (a) AOT [63] hierarchically propagates (Prop) object-specific information (i.e., specific to the given object(s)) into the object-agnostic visual embedding. (b) By contrast, DeAOT decouples the propagation of visual and ID embeddings in two branches. (c) Speed-accuracy comparison. All the results were fairly recorded on the same device, 1 Tesla V100 GPU.
+Fig. 1a shows that AOT’s hierarchical propagation can transfer the current frame feature from an object-agnostic visual embedding to an object-specific ID embedding by hierarchically propagating the reference information into the current frame. The hierarchical structure enables AOT to be structurally scalable between state-of-the-art performance and real-time efficiency. Intuitively, the increase of ID information will inevitably lead to the loss of initial visual information since the dimension of features is limited. However, matching objects’ visual features, the only clues provided by the current frame, is crucial for attention-based VOS solutions. To avoid the loss of visual information in deeper propagation layers and facilitate the learning of visual embeddings, a desirable manner (Fig. 1b) is to decouple object-agnostic and object-specific embeddings in the propagation.
+Based on the above motivation, this paper proposes a novel hierarchical propagation approach for VOS, i.e., Decoupling Features in Hierarchical Propagation (DeAOT). Unlike AOT, which shares the embedding space for visual (object-agnostic) and ID (object-specific) embeddings, DeAOT decouples them into different branches using individual propagation processes while sharing their attention maps. To compensate for the additional computation from the dual-branch propagation, we propose a more efficient module for constructing hierarchical propagation, i.e., Gated Propagation Module (GPM). By carefully designing GPM for VOS, we are able to use single-head attention to match objects and propagate information instead of the stronger multi-head attention [48], which we found to be an efficiency bottleneck of AOT [63].
+To evaluate the proposed DeAOT approach, a series of experiments are conducted on three VOS benchmarks (YouTube-VOS [57], DAVIS 2017 [39], and DAVIS 2016 [38]) and one Visual Object Tracking (VOT) benchmark (VOT 2020 [24]). On the large-scale VOS benchmark, YouTube-VOS, the DeAOT variant networks remarkably outperform AOT counterparts in both accuracy and run-time speed as shown in Fig. 1c. Particularly, our R50-DeAOT-L can achieve $8 6 . 0 \%$ at a nearly real-time speed, 22.4fps, and our DeAOT-T can achieve $8 2 . 0 \%$ at 53.4fps, which is superior compared to AOTT [63] $8 0 . 2 \%$ , 41.0fps). Without any test-time augmentations, our SwinB-DeAOT-L achieves topranked performance on four VOS/VOT benchmarks, i.e., YouTube-VOS 2018/2019 $( 8 6 . 2 \% / 8 6 . 1 \%$ ), DAVIS 2017 Val/Test $( 8 6 . 2 \% / 8 2 . 8 \% )$ ), DAVIS 2016 $( 9 2 . 9 \% )$ , and VOT 2020 (0.622 EAO).
+Overall, our contributions are summarized below:
+• We propose a highly-effective VOS framework, DeAOT, by decoupling object-agnostic and objectspecific features in hierarchical propagation. DeAOT achieves top-ranked performance and efficiency on four VOS/VOT benchmarks [24, 38, 39, 57].
+• We design an efficient module, GPM, for constructing hierarchical matching and propagation. By using GPM, DeAOT variants are consistently faster than AOT counterparts, although DeAOT’s propagation processes are twice as AOT’s.
+# 2 Related Work
+Semi-supervised Video Object Segmentation. Given a video with one or several annotated frames (the first frame in general), semi-supervised VOS [52] requires algorithms to propagate the mask annotations to the entire video. Traditional methods often solve an optimization problem with an energy defined over a graph structure [2, 4, 49]. Based on deep neural networks (DNN), deep learning based VOS methods have achieved significant progress and dominated the field in recent years.
+Finetuning-based Methods. Early DNN-based methods rely on fine-tuning pre-trained segmentation networks at test time to make the networks focus on the given object. Among them, OSVOS [7] and MoNet [56] propose to fine-tune pre-trained networks on the first-frame annotation. OnAVOS [51] extends the first-frame fine-tuning by introducing an online adaptation mechanism. Following these approaches, MaskTrack [37] and PReM [32] further utilize optical flow to help propagate the segmentation mask from one frame to the next.
+Template-based Methods. To avoid using the test-time fine-tuning, many researchers regard the annotated frames as templates and investigate how to match with them. For example, OSMN [60] employs a network to extract object embedding and another one to predict segmentation based on the embedding. PML [10] learns pixel-wise embedding with the nearest neighbor classifier, and VideoMatch [22] uses a matching layer to map the pixels of the current frame to the annotated frame in a learned embedding space. Following these methods, FEELVOS [50] and $\mathrm { C F B I ( + ) }$ [62, 64] extend the pixel-level matching mechanism by additionally doing local matching with the previous frame, and RPCM [58] proposes a correction module to improve the reliability of pixel-level matching. Instead of using matching mechanisms, LWL [6] proposes to use an online few-shot learner to learn to decode object segmentation.
+Attention-based Methods. Based on the advance of attention mechanisms [5,48,53], STM [34] and the following works (e.g., KMN [43] and STCN [11]) leverage a memory network to embed past-frame predictions into memory and apply a non-local attention mechanism on the memory to propagate mask information to the current frame. Differently, SST [17] proposes to calculate pixel-level matching maps based on the attention maps of transformer blocks [48]. Recently, AOT [61, 63, 65] introduces hierarchical propagation into VOS and can associate multiple objects collaboratively with the proposed ID mechanism.
+Visual Transformers. Transformers [48] was initially proposed to build hierarchical attention-based networks for natural language processing (NLP). Compared to RNNs, transformer networks model global correlation or attention in parallel, leading to better memory efficiency, and thus have been widely used in NLP tasks [15, 40, 46]. Similar to Non-local Neural Networks [53], transformer blocks compute correlation with all the input elements and aggregate their information by using attention mechanisms [5]. Recently, transformer blocks were introduced to computer vision and have shown promising performance in many tasks, such as image classification [16, 30, 47], object detection [8]/segmentation [25, 35, 54, 66], image generation [36], and video understanding [1, 26, 31].
+Based on transformers, AOT [63] proposes a Long Short-Term Transformer (LSTT) structure for constructing hierarchical propagation. By hierarchically propagating object information, AOT variants [63] have shown promising performance with remarkable scalability. Unlike AOT, which shares the embedding space for object-agnostic and object-specific embeddings, we propose to decouple them into different branches using individual propagation processes. Such a dual-branch paradigm avoids the loss of object-agnostic information and achieves significant improvement. Besides, a more efficient structure, GPM, is proposed for hierarchical propagation.
+# 3 Rethinking Hierarchical Propagation for VOS
+Attention-based VOS methods [11, 34, 43, 63] are dominating the field of VOS. In these methods, STM [34] and following algorithms [11, 43] uses a single attention layer to propagate mask information from memorized frames to the current frame. The use of only a single attention layer restricts the scalability of algorithms. Hence, AOT [63] introduces hierarchical propagation into VOS by proposing the Long Short-term Transformer (LSTT) structure, which can propagate the mask information in a hierarchical coarse-to-fine manner. By adjusting the layer number of LSTT, AOT variants can be ranged from state-of-the-art performance to real-time run-time speed.
+Let $Q \in \mathbb { R } ^ { H W \times C }$ and $K , V \in \mathbb { R } ^ { T H W \times C }$ denote the query embedding of the current frame, the key embedding, and the value embedding of the memorized frames respectively, where $T , H$ , $W$ , $C$ represent the temporal, height, width, and channel dimensions. The formula of a common attention-based VOS propagation is,
+$$
+A t t ( Q , K , V ) = C o r r ( Q , K ) V = s o f t m a x ( \frac { Q K ^ { t r } } { \sqrt { C } } ) V ,
+$$
+where the matching (or attention) map is calculated by the correlation function, $C o r r ( * , * )$ .
+To formulate a hierarchical propagation with $L$ layers, we further define $X _ { l } ^ { t } \in \mathbb { R } ^ { H W \times C }$ as the input feature embedding of $l$ -th propagation layer $( l \in \{ 1 , 2 , . . . , L \} )$ at $t$ frame. Moreover, $X _ { l } ^ { \mathbf { m } } =$ $\bar { C o n c a t } ( X _ { l } ^ { m _ { 1 } } , . . . , X _ { l } ^ { m _ { T } } )$ and $Y ^ { \mathbf { m } } = { C o n c a t ( Y ^ { m _ { 1 } } , . . . , Y ^ { m _ { T } } ) }$ stands for the feature embeddings and object masks in the memorized frames with indices $\mathbf { m } = \{ m _ { 1 } , . . . , m _ { T } \}$ . Then, the formulation of $l$ -th propagation layer in AOT’s hierarchical propagation can be simplified as,
+$$
+\widetilde { X } _ { l } ^ { t } = A t t ( X _ { l } ^ { t } W _ { l } ^ { K } , X _ { l } ^ { \mathbf { m } } W _ { l } ^ { K } , X _ { l } ^ { \mathbf { m } } W _ { l } ^ { V } + I D ( Y ^ { \mathbf { m } } ) ) ,
+$$
+where $I D ( * )$ denotes the IDentification (ID) embedding [63] function used to encode masks. Besides, $W _ { l } ^ { K } \in \mathbb { R } ^ { \tilde { C } \times C _ { k } }$ and $W _ { l } ^ { V } \in \mathbb { R } ^ { C \times C _ { v } }$ are trainable parameters for projecting features into matching space and propagation space, respectively. For simplicity, the formulation keeps only the parts related to mask propagation in LSTT.
+Obviously, before all the propagation layers, the current frame feature, $X _ { 1 } ^ { t }$ , is an object-agnostic feature extracted from an image encoder (e.g., ResNet-50 [21]). Nevertheless, the mask information $I D ( Y ^ { \mathbf { m } } )$ will be gradually and hierarchically propagated into the current frame, and the output feature, $\smash { \widetilde { X } _ { L } ^ { t } }$ , will become object-specific and can be decoded into the ID/mask prediction by a decoder network (e.g., FPN [27]). In other words, step by step, the hierarchical propagation transfers the current frame feature, $X _ { l } ^ { t }$ , from an object-agnostic visual embedding to an object-specific ID embedding, as demonstrated in Fig. 1a.
+Intuitively, the absorption of object-specific ID information will inevitably lead to the oblivion of object-agnostic visual information within $X _ { 1 } ^ { t }$ since the channel dimension of $X _ { l } ^ { t }$ is limited. Such a phenomenon can also be observed by increasing the ID information directly. As shown in Fig. 2, the performance of AOT heavily drops as we increase the information amount of $I D ( Y ^ { \mathbf { m } } )$ by containing more IDs inside. On the other hand, the significant progress of VOS in recent years is mainly based on matching object-agnostic visual embeddings (e.g., pixel-level matching methods [58, 62, 64] and single-layer attention-based methods [11, 34, 43] mentioned above). Hence, we argue that the loss of visual information in deeper propagation layers limits the performance of hierarchical propagation.
+How to design a hierarchical propagation structure which can keep or even refine the initial object-agnostic visual information? Fig. 1b shows a simple, straightforward, and desirable approach, i.e., propagating object-agnostic and object-specific information in two different branches (Visual Branch and ID Branch). The object-agnostic branch is responsible for gathering visual information, refining visual features, and matching objects. By contrast, the object-specific branch is responsible for absorbing ID information propagated from memorized frames. These two branches share the attention maps used to match objects and propagate features. Compared to the singlebranch LSTT, our dual-branch approach can keep and further refine visual features in the hierarchical propagation and thus can further facilitate the learning of visual embeddings.
+![](images/21db22f8b0e573ba55b2535d32bb447a29475416a0a891369472f8fd92b4fcd1.jpg)
+Figure 2: The performance of AOT [63] will be degraded by increasing ID’s maximum number.
+# 4 Decoupling Features in Hierarchical Propagation
+This section will introduce a new framework, Decoupling Features in Hierarchical Propagation (DeAOT), for solving semi-supervised video object segmentation. We show an overview of DeAOT in Fig. 3a. Given a video with a reference frame annotation, DeAOT propagates the annotation to the entire video frame-by-frame. The multi-object annotation is encoded by the IDentification (ID) mechanism [63]. Different from AOT, DeAOT decouples the hierarchical propagation of visual embedding and ID embedding, i.e., DeAOT propagates these two embeddings in two branches. Furthermore, DeAOT constructs the hierarchical propagation by using the proposed Gated Propagation Module (GPM), which is more efficient and effective than the LSTT block used in AOT.
+![](images/6e1cf58d895dd54d6318244e298b354bd91f5db091f3236699bfcce74b6a985d.jpg)
+Figure 3: (a) Overview. Decoupling Features in Hierarchical Propagation (DeAOT) decouples the propagation of visual embedding and IDentification (ID) embedding [63] in two branches, i.e., Visual Branch and ID Branch. The propagation module is the proposed efficient GPM module. (b) A demonstration of the Gated Propagation Module (GPM) in both Visual and ID branches. LN: Layer Normalization [3]. (c) We propose to use the Gated Propagation (GP) function to construct GPM. DW-Conv: depth-wise convolution. Mul: matrix multiplication.
+# 4.1 Hierarchical Dual-branch Propagation
+Different from the previous attention-based VOS methods [34,43, 44, 63], DeAOT propagates objects’ visual features and mask features in two parallel branches. In detail, the visual branch is responsible for matching objects, gathering past visual information, and refining object features. To re-identify the objects, the ID branch reuses the matching maps (attention maps) calculated by the visual branch to propagate the ID embedding (encoded by the ID mechanism [63]) from past frames to the current frame. Both the branches share the same hierarchical structure with $L$ propagation layers.
+Visual Branch is responsible for matching objects by calculating attention maps on patch-wise visual embeddings. The visual embeddings in the memorized frames will be propagated to the current frame regarding the attention maps. Since the propagation is not directly related to the object-specific ID embedding, the visual branch can learn to refine visual embeddings to be more contrastive but avoid being biased toward the given object-specific information. Let $I$ denote visual embeddings, we modify Eq. 2 into a layer of object-agnostic visual propagation,
+$$
+\begin{array} { r } { \widetilde { I } _ { l } ^ { t } = A t t ( I _ { l } ^ { t } { W } _ { l } ^ { K } , { I } _ { l } ^ { \mathbf { m } } { W } _ { l } ^ { K } , { I } _ { l } ^ { \mathbf { m } } { W } _ { l } ^ { V } ) } \\ { = C o r r ( I _ { l } ^ { t } { W } _ { l } ^ { K } , { I } _ { l } ^ { \mathbf { m } } { W } _ { l } ^ { K } ) { I } _ { l } ^ { \mathbf { m } } { W } _ { l } ^ { I } , } \end{array}
+$$
+which doesn’t leverage the object-specific ID embedding, $I D ( Y ^ { \mathbf { m } } )$ . Thus, the visual branch can learn to keep and refine the visual embedding in the hierarchical propagation.
+ID Branch is designed for propagating the object-specific information from past frames to the current frame. The prediction of object-specific segmentation is essential for VOS and can not be processed by the above object-agnostic visual propagation branch. Let $M$ denote the object-specific embeddings in our identification branch, the formulation of our object-specific ID propagation is,
+$$
+\begin{array} { r l } & { \widetilde { M } _ { l } ^ { t } = A t t ( I _ { l } ^ { t } W _ { l } ^ { K } , I _ { l } ^ { \mathbf { m } } W _ { l } ^ { K } , M _ { l } ^ { \mathbf { m } } W _ { l } ^ { \overline { { V } } } + I D ( Y ^ { \mathbf { m } } ) ) } \\ & { \qquad = C o r r ( I _ { l } ^ { t } W _ { l } ^ { K } , I _ { l } ^ { \mathbf { m } } W _ { l } ^ { K } ) ( M _ { l } ^ { \mathbf { m } } W _ { l } ^ { \overline { { V } } } + I D ( Y ^ { \mathbf { m } } ) ) , } \end{array}
+$$
+where $W _ { l } ^ { \overline { { V } } } \in \mathbb { R } ^ { C \times C _ { v } }$ is a trainable projection matrix for the identification propagation. Particularly, the identification propagation shares the same attention maps, $C o r r ( I _ { l } ^ { t } W _ { l } ^ { K } , I _ { l } ^ { \mathbf { m } } W _ { l } ^ { K } )$ , from the visual branch, since the identification of objects is mainly based on objects’ visual features instead of their ID indices. Without the visual information, the tracking of objects is inapplicable.
+# 4.2 Gated Propagation Module
+Instead of using the LSTT block [63], which employs multi-head attention in propagation, we stack the hierarchical propagation based on the proposed Gated Propagation Module (GPM), which is designed based on more efficient single-head attention.
+LSTT Block [63] includes four parts, i.e., a long-term attention responsible for propagating information from the memorized frames (in m), a short-term attention responsible for propagating information from a spatial neighborhood in the previous $( t - 1 )$ frame, a self-attention module for associating objects in the current $\mathbf { \rho } ( t )$ frame, and a feed-forward module. The three kinds of attention modules are built on the multi-head [48] extension of Eq. 1 or Eq. 2. According to the experiments in Table 3b, reducing the head number from multiple heads (8 heads in default) to a single head will decrease the performance of AOT but can significantly improve the run-time speed, which means the multi-head attention is an efficiency bottleneck of LSTT. Concretely, the computational complexity of long-term attention is $\mathcal { O } ( N T \bar { H ^ { 2 } } W ^ { 2 } )$ , which is proportional to the head number $N$ since each head contains a correlation function, $C o r r ( Q , K )$ .
+Gated Propagation Function. To avoid using multiple attention heads but not decrease the network performance, we redesign the attention-based VOS propagation defined in Eq. 1 and propose a gated propagation function as demonstrated in Fig. 3c. Let $\overset { \cdot } { U } \in \mathbb { R } ^ { H W \times C }$ denotes a gating embedding, the function is
+$$
+G P ( U , Q , K , V ) = { \mathcal { F } } _ { d w } ( \sigma ( U ) \odot C o r r ( Q , K ) V ) W ^ { O } ,
+$$
+where $\sigma$ is a non-linear gating function, $\odot$ denotes element-wise multiplication, $\mathcal { F } _ { d w } ( * )$ stands for a depth-wise 2D convolution layer [13], and $W ^ { O } \in \mathbb { R } ^ { C _ { v } \times C }$ is the trainable weight of output projection. Firstly, we augment the attention-based propagation (Eq. 1) by using a conditional gate, $\sigma ( U )$ , which we empirically found to be effective in VOS. Notably, the presence of gating in weak attention mechanisms (e.g., single-head attention) is also beneficial in some transformer-based methods [23,29] for NLP. Moreover, we leverage a depth-wise convolution $\mathcal { F } _ { d w } ( * )$ to enhance the modeling of local spatial context in a lightweight manner.
+Gated Propagation Module consists of three kinds of gated propagation, self-propagation, long-term propagation, and short-term propagation. Compared with LSTT, GPM removes the feed-forward module for further saving computation and parameters. All the propagation processes employ the gated propagation function defined in Eq. 5. In DeAOT, both the propagation branches (i.e., visual branch and identification branch) are stacked by GPM as shown in Fig. 3b.
+Based on the formulation of visual propagation (Eq. 3) and ID propagation (Eq. 4), the Long-term Propagation can be formulated as
+$$
+G P _ { l t } ^ { v i s } ( I _ { l } ^ { t } , I _ { l } ^ { t } , I _ { l } ^ { \mathbf { m } } , I _ { l } ^ { \mathbf { m } } ) = G P ( I _ { l } ^ { t } W _ { l } ^ { U } , I _ { l } ^ { t } W _ { l } ^ { K } , I _ { l } ^ { \mathbf { m } } W _ { l } ^ { K } , I _ { l } ^ { \mathbf { m } } W _ { l } ^ { V } ) ,
+$$
+$$
+G P _ { l t } ^ { i d } ( M _ { l } ^ { t } , I _ { l } ^ { t } , I _ { l } ^ { \mathbf { m } } , M _ { l } ^ { \mathbf { m } } , Y ^ { \mathbf { m } } ) = G P ( M _ { l } ^ { t } W _ { l } ^ { \overline { { U } } } , I _ { l } ^ { t } W _ { l } ^ { K } , I _ { l } ^ { \mathbf { m } } W _ { l } ^ { K } , M _ { l } ^ { \mathbf { m } } W _ { l } ^ { \overline { { V } } } + I D ( Y ^ { \mathbf { m } } ) )
+$$
+for the visual branch and ID branch, respectively. The ID propagation reuses the attention maps of the visual propagation as discussed in Eq. 4. Based on the long-term propagation, we can formulate the Short-term Propagation at spatial location $p$ to be
+$$
+G P _ { s t } ^ { v i s } ( I _ { l } ^ { t } , I _ { l } ^ { t } , I _ { l } ^ { t - 1 } , I _ { l } ^ { t - 1 } | p ) = G P _ { l t } ^ { v i s } ( I _ { l , p } ^ { t } , I _ { l , p } ^ { t } , I _ { l , N ( p ) } ^ { t - 1 } , I _ { l , N ( p ) } ^ { t - 1 } ) ,
+$$
+$$
+G P _ { s t } ^ { i d } ( M _ { l } ^ { t } , I _ { l } ^ { t } , I _ { l } ^ { t - 1 } , M _ { l } ^ { t - 1 } , Y ^ { t - 1 } | p ) = G P _ { l t } ^ { i d } ( M _ { l , p } ^ { t } , I _ { l , p } ^ { t } , I _ { l , N ( p ) } ^ { t - 1 } , M _ { l , N ( p ) } ^ { t - 1 } | Y _ { N ( p ) } ^ { t - 1 } ) ,
+$$
+where $I _ { l , p } ^ { t } , M _ { l , p } ^ { t } \in \mathbb { R } ^ { 1 \times C }$ are the feature of $I _ { l } ^ { t } , M _ { l } ^ { t }$ at location $p$ respectively, and $\mathcal { N } ( \boldsymbol { p } )$ stands for a $\lambda \times \lambda$ n o $p$ rt-term propagat of the previous each location  frame. Since $p$ is he restricted in its spatial neighbourhood (It−1l,N (p) $M _ { l , \mathcal { N } ( p ) } ^ { t - 1 } )$ $( t - 1 )$ object motions across several contiguous video frames are always smooth, non-local propagation processes becomes inefficient and not necessary in short-term information propagation [62].
+Finally, the Self-Propagation can also be formulated similar to the long-term propagation, i.e.,
+$$
+G P _ { s e l f } ^ { v i s } ( I _ { l } ^ { t } | M _ { l } ^ { t } ) = G P ( I _ { l } ^ { t } W _ { l } ^ { U } , ( I _ { l } ^ { t } \oplus M _ { l } ^ { t } ) W _ { l } ^ { K } , ( I _ { l } ^ { t } \oplus M _ { l } ^ { t } ) W _ { l } ^ { K } , I _ { l } ^ { t } W _ { l } ^ { V } ) ,
+$$
+$$
+G P _ { s e l f } ^ { i d } ( M _ { l } ^ { t } | I _ { l } ^ { t } ) = G P ( M _ { l } ^ { t } W _ { l } ^ { \overline { { U } } } , ( I _ { l } ^ { t } \oplus M _ { l } ^ { t } ) W _ { l } ^ { K } , ( I _ { l } ^ { t } \oplus M _ { l } ^ { t } ) W _ { l } ^ { K } , M _ { l } ^ { t } W _ { l } ^ { \overline { { V } } } ) ,
+$$
+![](images/3456830cdbf2b5c330f65a6caf78aa7bc5b6c0c0a41f390209f71d29ddae1182.jpg)
+Figure 4: Qualitative results. (top) DeAOT performs better than AOT [63] on tiny or scale-changing objects. (bottom) DeAOT fails to track highly similar objects when serious occlusion happens.
+where $\oplus$ is a concatenation process on the channel dimension. In the self-propagations, both the visual embedding $I _ { l } ^ { t }$ and ID embedding $M _ { l } ^ { t }$ are used in the calculation of attention maps (i.e., $C o r r ( Q , K ) )$ . Here, the object-specific $M _ { l } ^ { t }$ performs like a positional embedding [48] additional to the visual embedding $I _ { l } ^ { \bar { t } }$ . We found that such a process can help associate the objects in the current frame more effectively. Apart from this, the current frame segmentation $Y ^ { t }$ is unavailable before being decoded and is not used in the ID self-propagation $G P _ { s e l f } ^ { i d }$ . For simplicity, we reuse the parameter symbols in Eq. 6 and 7, but the trainable parameters are not shared with long-term propagation.
+# 5 Implementation Details
+Network Details: Consistent with AOT [63], three kinds of encoders are used in our experiments, i.e., MobileNet-V2 [42] (in default), ResNet-50 (R50) [21], and Swin-B [30]. The decoder is the same FPN [27] network. Besides, the spatial neighborhood size $\lambda$ is set to 15, and the maximum object number within the ID embedding is 10. In our GPM module, the channel dimension $C$ of visual and ID embeddings is 256, the matching features’ dimension $C _ { k }$ is 128, and the propagation features’ dimension $C _ { v }$ is 512. Moreover, the kernel size of $\mathcal { F } _ { d w }$ is 5, and the gating function $\sigma ( * )$ is SiLU/Swish [18, 41].
+To make fair comparisons with AOT’s variants [63], we build corresponding DeAOT variants with different GPM number $L$ or long-term memory size m. The hyper-parameters of these variants are: DeAOT-T: $L = 1$ , $\mathbf { m } = \{ 1 \}$ ; DeAOT-S: $L = 2$ , $\mathbf { m } = \{ 1 \}$ ; DeAOT-B: $L = 3$ , $\mathbf { m } = \{ 1 \}$ ; DeAOT-L: $L = 3$ , ${ \bf m } = \{ 1 , 1 + \delta , 1 + \bar { 2 } \delta , \ldots \}$ . DeAOT-T/S/B considers only the reference frame as the long-term memory, leading to consistent run-time speeds. DeAOT-L updates the long-term memory per $\delta$ (set to 2/5 for training/testing) frames as AOT-L [63].
+Training Details: Following [34, 43, 44, 55, 63], we first pre-train DeAOT on synthetic video sequence generated from static image datasets [12, 19, 20, 28, 45] by randomly applying multiple image augmentations [55]. Then, we do main training on the VOS benchmarks [39, 57] by randomly applying video augmentations [62, 63]. Besides, we keep our optimization strategies and related hyper-parameters the same as AOT. More details are supplied in Supplementary.
+# 6 Experimental Results
+We conduct experiments on three popular VOS benchmarks (YouTube-VOS [57], DAVIS 2017 [39], and DAVIS 2016 [38]) and one challenging Visual Object Tracking (VOT) benchmark (VOT 2020 [24]), which gives segmentation annotations and can be used to evaluate VOS algorithms.
+To validate DeAOT’s generalization ability, all the benchmarks share the same model parameters. When evaluating YouTube-VOS, we use the default 6fps videos, which are restricted to be smaller than $1 . 3 \times 4 8 0 p$ resolution. On DAVIS, the default 480p 24fps videos are used. For evaluating VOT 2020, more details can be found in the supplementary material.
+The evaluation metrics for VOS benchmarks include the $\mathcal { I }$ score (calculated as the average IoU score between the prediction and the ground truth mask), the $\mathcal { F }$ score (calculated as an average boundary similarity measure between the boundary of the prediction and the ground truth), and their mean value (denoted as $\mathcal { I } \& \mathcal { F } )$ . As to VOT 2020, we use the official EAO criteria [24]. We evaluate all the results on official evaluation servers or with official tools.
+Table 1: The quantitative evaluation on multi-object benchmarks, YouTube-VOS [57] and DAVIS 2017 [39]. ${ \mathcal { I } } _ { S } / { \mathcal { F } } _ { S } / { \mathcal { I } } _ { U } / { \mathcal { F } } _ { U }$ : $\mathcal { T } / \mathcal { F }$ on seen/unseen classes. $^ \ddag$ : timing extrapolated from single-object speed assuming linear scaling in the number of objects. $\star$ : recorded on our device.
+<table><tr><td></td><td colspan="5">YouTube-VOS 2018 Val</td><td colspan="5">YouTube-VOS 2019 Val</td><td colspan="4">DAVIS-17 Val</td><td colspan="4">DAVIS-17 Test</td></tr><tr><td>Method</td><td>Avg</td><td>Js</td><td>Fs</td><td>JuFu</td><td></td><td></td><td></td><td>AvgJsFs</td><td>Ju</td><td>Fu</td><td>fps</td><td>Avg</td><td>J</td><td>F</td><td>Avg</td><td>J</td><td>F</td><td>fps</td></tr><tr><td>KMN[ECCV20][43]</td><td>81.4</td><td>81.4</td><td>85.6</td><td>75.3</td><td>83.3</td><td>-</td><td></td><td>-</td><td>-</td><td>=</td><td></td><td>82.8</td><td>80.0</td><td>85.6</td><td>77.2</td><td>74.1 80.3</td><td></td><td>1</td></tr><tr><td>CFBI[ECCV20][62]</td><td>81.4</td><td>81.1</td><td>85.875.3</td><td></td><td>83.4</td><td></td><td></td><td>81.0 80.685.1</td><td>75.283.0</td><td></td><td>3.4</td><td>81.9</td><td>79.3</td><td>84.5</td><td></td><td>76.673.080.1</td><td></td><td>2.9</td></tr><tr><td>SST[CVPR21][17]</td><td>81.7</td><td>81.2</td><td>-</td><td>76.0</td><td>，</td><td></td><td>81.880.9</td><td>1</td><td>76.6</td><td>-</td><td>1</td><td>82.5</td><td>79.9</td><td>85.1</td><td>-</td><td>1</td><td>-</td><td>，</td></tr><tr><td>HMMN[ICCV21] [44]</td><td>82.682.1</td><td></td><td>87.076.8</td><td></td><td>84.6</td><td></td><td>82.581.7</td><td></td><td>86.1 77.3 85.0</td><td></td><td>1</td><td>84.7</td><td>81.9</td><td>87.5</td><td></td><td>78.674.7</td><td>82.5</td><td>3.4</td></tr><tr><td>CFBI+[TPAMI21][64]</td><td>82.881.8</td><td></td><td>86.677.1</td><td></td><td>85.6</td><td></td><td>82.681.7</td><td>86.2</td><td>77.1</td><td>85.2</td><td>4.0</td><td>82.9</td><td>80.1</td><td>85.7</td><td>78.0</td><td>74.481.6</td><td></td><td>3.4</td></tr><tr><td>STCN[NeurIPS21] [11]</td><td>83.0</td><td>81.9</td><td></td><td>86.577.9</td><td>85.7</td><td>82.7</td><td>81.1</td><td>85.4</td><td>78.285.9</td><td></td><td>8.4*</td><td>85.4</td><td>82.2</td><td>88.6</td><td>76.1</td><td>72.779.6</td><td></td><td>19.5*</td></tr><tr><td>RPCM[AAAI22] [58]</td><td>84.0</td><td>83.1</td><td>87.7</td><td>78.5</td><td>86.7</td><td>83.9</td><td>82.6</td><td>86.9</td><td>79.1</td><td>87.1</td><td>-</td><td>83.7</td><td>81.3</td><td>86.0</td><td>79.2</td><td>75.882.6</td><td></td><td>-</td></tr><tr><td>AOT-T[63]</td><td>80.2</td><td>80.1</td><td>84.5</td><td></td><td>82.2</td><td>79.7</td><td>79.6</td><td>83.8</td><td>73.7</td><td>81.8</td><td>41.0</td><td>79.9</td><td>77.4</td><td>82.3</td><td>72.0</td><td></td><td></td><td>51.4</td></tr><tr><td>DeAOT-T</td><td>82.0</td><td>81.6</td><td>86.3</td><td>74.0 75.8</td><td>84.2</td><td>82.0</td><td>81.2</td><td>85.6</td><td>76.4</td><td>84.7</td><td>53.4</td><td>80.5</td><td>77.7</td><td>83.3</td><td>73.7</td><td>68.3 70.077.3</td><td>75.7</td><td>63.5</td></tr><tr><td>AOT-S [63]</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>78.7</td><td></td><td></td><td></td><td></td><td>40.0</td></tr><tr><td>DeAOT-S</td><td>82.6</td><td>82.0 83.3</td><td>86.7 88.3</td><td>76.6</td><td>85.0</td><td>82.2</td><td>81.3</td><td>85.9</td><td>76.6</td><td>84.9</td><td>27.1</td><td>81.3</td><td></td><td>83.9</td><td>73.9</td><td>70.3</td><td>77.5</td><td>49.2</td></tr><tr><td>AOT-B [63]</td><td>84.0</td><td></td><td></td><td>77.9</td><td>86.6</td><td>83.8</td><td>82.8</td><td>87.5</td><td>78.1</td><td>86.8</td><td>38.7</td><td>80.8</td><td>77.8</td><td>83.8</td><td>75.4</td><td>71.9</td><td>79.0</td><td></td></tr><tr><td></td><td>83.5</td><td>82.6</td><td>87.5</td><td>77.7</td><td>86.0</td><td>83.3</td><td>82.4</td><td>87.1</td><td>77.8</td><td>86.0</td><td>20.5</td><td>82.5</td><td>79.7</td><td>85.2</td><td>75.5</td><td>71.6</td><td>79.3</td><td>29.6</td></tr><tr><td>DeAOT-B</td><td>84.6</td><td>83.9</td><td>88.9</td><td>78.5</td><td>87.0</td><td>84.6</td><td>83.5</td><td>88.3</td><td>79.1</td><td>87.5</td><td>30.4</td><td>82.2</td><td>79.2</td><td>85.1</td><td>76.2</td><td>72.5</td><td>79.9</td><td>40.9</td></tr><tr><td>AOT-L [63]</td><td>83.8</td><td>82.9</td><td>87.9</td><td>77.7</td><td>86.5</td><td>83.7</td><td>82.8</td><td>87.5</td><td>78.0</td><td>86.7</td><td>16.0</td><td>83.8</td><td>81.1</td><td>86.4</td><td>78.3</td><td>74.3</td><td>82.3</td><td>18.7</td></tr><tr><td>DeAOT-L</td><td>84.8</td><td>84.2</td><td>89.4</td><td>78.6</td><td>87.0</td><td>84.7</td><td>83.8</td><td>88.8</td><td>79.0</td><td>87.2</td><td>24.7</td><td>84.1</td><td>81.0</td><td>87.1</td><td>77.9</td><td>74.1</td><td>81.7</td><td>28.5</td></tr><tr><td>R50-AOT-L [63]</td><td>84.1</td><td>83.7</td><td>88.5</td><td>78.1</td><td>86.1</td><td>84.1</td><td>83.5</td><td>88.1</td><td>78.4</td><td>86.3</td><td>14.9</td><td>84.9</td><td>82.3</td><td>87.5</td><td>79.6</td><td>75.983.3</td><td></td><td>18.0</td></tr><tr><td>R50-DeAOT-L</td><td>86.0</td><td>84.9</td><td>89.9</td><td>80.4</td><td>88.7</td><td>85.9</td><td>84.6</td><td>89.4</td><td>80.8</td><td>88.9</td><td>22.4</td><td></td><td>85.282.288.2</td><td></td><td>80.7</td><td>76.9</td><td>84.5</td><td>27.0</td></tr><tr><td>SwinB-AOT-L [63]</td><td>84.5</td><td>84.3</td><td>89.3</td><td>77.9</td><td>86.4</td><td>84.5</td><td>84.0</td><td>88.8</td><td>78.4</td><td>86.7</td><td>9.3</td><td>85.4</td><td>82.4</td><td>88.4</td><td>81.2</td><td>77.3 82.878.9 86.7</td><td>85.1</td><td>12.1</td></tr><tr><td>SwinB-DeAOT-L</td><td>86.2 85.6 90.6 80.0 88.4</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>86.1 85.3 90.2 80.4</td><td>88.6</td><td>11.9</td><td></td><td>86.2 83.1 89.2</td><td></td><td></td><td></td><td></td><td>15.4</td></tr></table>
+# 6.1 Compare with the State-of-the-art Methods
+YouTube-VOS [57] is a large-scale multi-object VOS benchmark, which contains 3471 videos in the training split with 65 categories and 474/507 videos in the Validation 2018/2019 split with additional 26 unseen categories. Table 1 shows that DeAOT variants remarkably outperforms AOT counterparts in both accuracy and run-time speed on YouTube-VOS 2018/2019. For example, our R50-DeAOT-L achieves $\mathbf { 8 6 . 0 \% / 8 5 . 9 \% }$ $( \mathcal { I } \& \mathcal { F } )$ at 22.4fps, which is superior compared to R50-AOTL [63] $( 8 4 . 1 \% / 8 4 . 1 \%$ at 14.9fps). Particularly, our SwinB-DeAOT-L achieves new state-of-the-art performance $( 8 6 . 2 \% / 8 6 . 1 \% )$ ), surpassing previous methods by more than $1 . 7 \% / 1 . 6 \%$ . In addition, our smallest variant, DeAOT-T, precedes SST [17] $( 8 2 . 0 \% / 8 2 . \dot { 0 } \%$ vs $8 1 . 7 \% / 8 1 . 8 \% )$ ) and runs about $1 5 \times$ faster than CFBI [62] (53.4fps vs 3.4fps).
+DAVIS 2017 [39] is a multi-object extension of DAVIS 2016. The training/validation split consists of 60/30 videos with 138/59 objects, and the test split contains 30 more challenging videos with 89 objects. As shown in Table 1, DeAOT variants can generalize to DAVIS 2017 well. R50- DeAOT-L achieves $8 5 . 2 \% / 8 0 . 7 \%$ on the validation/test split at a real-time speed (27fps), surpassing R50-AOT-L in accuracy and efficiency. Also, SwinB-DeAOT-L achieves the top-ranked performance on DAVIS 2017 $( 8 6 . 2 \% / 8 2 . 8 \% )$ ).
+DAVIS 2016 [38] is a single-object benchmark containing 20 videos in the validation split, and we show related experiments in Table 2. Although AOT-like methods focus on multi-object scenarios, our DeAOT-L is faster and more robust than STCN [11], whose architecture was designed for single-object VOS. Besides, SwinBDeAOT-L achieves $9 2 . 9 \%$ and outperforms all the VOS methods as well.
+Table 2: The quantitative evaluation on the singleobject benchmarks, DAVIS 2016 [38] and VOT 2020 [24]. $\mathrm { E A O } ^ { R T }$ : real-time EAO metric [24].
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>DAVIS 2016</td><td rowspan=1 colspan=2>VOT 2020</td></tr><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=2>AvgJ F fps</td><td rowspan=1 colspan=2>EAOEAORT</td></tr><tr><td rowspan=1 colspan=1>CFBI+ [64]RPCM [58]HMMN [44]STCN[11]</td><td rowspan=1 colspan=1>89.988.791.190.687.194.090.889.692.091.690.892.5</td><td rowspan=1 colspan=1>5.95.810.027.2*</td><td rowspan=1 colspan=1>“---</td><td rowspan=1 colspan=1>-=--</td></tr><tr><td rowspan=1 colspan=1>AlphaRef [59]RPT[33]MixFormer-L [14]</td><td rowspan=1 colspan=1>-  1  ·-   1   11   -   1</td><td rowspan=1 colspan=1>--1</td><td rowspan=1 colspan=1>0.4820.5300.555</td><td rowspan=1 colspan=1>0.4860.290-</td></tr><tr><td rowspan=1 colspan=1>AOT-T [63]DeAOT-T</td><td rowspan=1 colspan=1>86.886.187.488.987.889.9</td><td rowspan=1 colspan=1>51.463.5</td><td rowspan=1 colspan=1>0.4350.472</td><td rowspan=1 colspan=1>0.4330.463</td></tr><tr><td rowspan=1 colspan=1>AOT-S [63]DeAOT-S</td><td rowspan=1 colspan=1>89.488.690.289.387.690.9</td><td rowspan=1 colspan=1>40.049.2</td><td rowspan=1 colspan=1>0.5120.593</td><td rowspan=1 colspan=1>0.4990.559</td></tr><tr><td rowspan=1 colspan=1>AOT-B [63]DeAOT-B</td><td rowspan=1 colspan=1>89.988.791.191.089.492.5</td><td rowspan=1 colspan=1>29.640.9</td><td rowspan=1 colspan=1>0.5410.571</td><td rowspan=1 colspan=1>0.5330.542</td></tr><tr><td rowspan=1 colspan=1>AOT-L [63]DeAOT-L</td><td rowspan=1 colspan=1>90.489.691.192.090.393.7</td><td rowspan=1 colspan=1>18.728.5</td><td rowspan=1 colspan=1>0.5740.591</td><td rowspan=1 colspan=1>0.5600.554</td></tr><tr><td rowspan=1 colspan=1>R50-AOT-L [63]R50-DeAOT-L</td><td rowspan=1 colspan=1>91.190.192.192.390.594.0</td><td rowspan=1 colspan=1>18.027.0</td><td rowspan=1 colspan=1>0.5690.613</td><td rowspan=1 colspan=1>0.5400.571</td></tr><tr><td rowspan=1 colspan=1>SwinB-AOT-L[63]SwinB-DeAOT-L</td><td rowspan=1 colspan=1>92.090.793.392.991.194.7</td><td rowspan=1 colspan=1>12.115.4</td><td rowspan=1 colspan=1>0.5860.622</td><td rowspan=1 colspan=1>0.5230.559</td></tr></table>
+Table 3: Ablation study. The experiments are conducted on YouTube-VOS 2018 [57] and based on DeAOT-S without pre-training on static images. De: decoupling features. $C$ : the channel dimension. Prop: propagation type. LT/ST: long-term/short-term. $k s$ : kernel size.
+<table><tr><td colspan="5">(a)Propagation module</td><td colspan="9">(b) Head number (Nh)</td><td colspan="5"></td><td colspan="5">(d) ks of Fdw</td></tr><tr><td>Module|</td><td>IC|J&amp;F</td><td></td><td></td><td>Js Ju</td><td>Model|Nh|J&amp;F</td><td></td><td></td><td></td><td>JsJulfps</td><td></td><td></td><td></td><td>Prop</td><td>Vis ID|J&amp;F Js Ju</td><td></td><td></td><td></td><td></td><td>ks|J&amp;F Js Ju</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>GPM</td><td>256</td><td>82.5</td><td></td><td>82.3 76.1</td><td>DeAOT</td><td></td><td>1</td><td>82.5</td><td></td><td>82.3 76.1</td><td></td><td>38.7</td><td>LT/ST</td><td>√</td><td></td><td>82.5</td><td></td><td>82.3 76.1 5</td><td></td><td></td><td></td><td>82.5</td><td></td><td>82.3 76.1</td></tr><tr><td>w/oDe</td><td>256</td><td>81.5</td><td></td><td>81.4 75.0</td><td>DeAOT</td><td></td><td>8</td><td>82.5</td><td>82.3 75.8</td><td></td><td></td><td>24.7</td><td>LT/ST</td><td>&lt;√</td><td></td><td>82.1</td><td>82.2 75.7</td><td></td><td>0</td><td></td><td>81.1</td><td></td><td>81.574.2</td><td></td></tr><tr><td>w/o De</td><td>512</td><td>82.0</td><td></td><td>82.1 75.4</td><td>AOT</td><td></td><td>1</td><td>79.6</td><td>80.1 72.6</td><td></td><td></td><td>44.6</td><td>Self</td><td></td><td></td><td>82.5</td><td></td><td>82.3 76.1</td><td>3</td><td></td><td>82.2</td><td>82.4</td><td>82.2 76.1</td><td>82.2 75.8</td></tr><tr><td>LSTT</td><td>[256</td><td>80.3</td><td></td><td>80.673.7</td><td>AOT</td><td></td><td></td><td>80.3</td><td>80.6 73.7</td><td></td><td></td><td>27.1</td><td>Self</td><td></td><td></td><td>82.2</td><td>82.1 75.7</td><td></td><td>9</td><td></td><td></td><td></td><td></td><td></td></tr></table>
+VOT 2020 [24] consists of 60 single-object videos with challenging scenarios including fast motion, occlusion, etc. The average frame number of VOT 2020 is 327, which is much longer than the maximum video length of the above VOS benchmarks. DeAOT shows superior performance on VOT 2020 in Table 2. The DeAOT variants larger than DeAOT-T outperform MixFormer-L [14] (the state-of-the-art tracker), RPT [33] (VOT 2020 short-term challenge winner), and AlphaRef [59] (VOT 2020 real-time challenge winner) in both EAO and real-time EAO scores. Specifically, SwinBDeAOT-L achieves 0.622 EAO, outstandingly exceeding MixFormer-L by 0.067, and R50-DeAOT-L achieves 0.571 EAO under a real-time requirement, impressively overtaking AlphaRef by 0.085.
+Qualitative results: Fig. 4 give qualitative comparisons to AOT. By introducing the dual-branch propagation, R50-DeAOT-L performs better than R50-AOT-L on tiny or scale-changing objects $s k i$ poles or ski board). Nevertheless, R50-DeAOT-L still may fails to track multiple highly similar objects (dancer and cow) when serious occlusion happens.
+# 6.2 Ablation Study
+This section analyzes the necessity of dual-branch propagation and GPM of DeAOT in Table 3.
+Propagation module: Table 3a shows that the performance of DeAOT drops from $8 2 . 5 \%$ to $8 1 . 5 \%$ by coupling the propagation of visual and ID embeddings (w/o De) like AOT. Furthermore, doubling the channel dimensions only partially relieves the performance loss. Moreover, the performance will be seriously degraded to $8 0 . 3 \%$ by replacing our GPM with the LSTT module of AOT. In conclusion, the dual-branch propagation approach and the GPM module are crucial in improving VOS performance.
+Head number: According to the results in Table 3b, the head number $( N _ { h } )$ of attention-based modules is negatively correlated with the efficiency of AOT/De-AOT. The single-head AOT (44.6fps) runs much faster than the default AOT $N _ { h } { = } 8$ , 27.1fps) but loses $0 . 7 \%$ accuracy. By contrast, DeAOT is robust to the head number by using our proposed GPM module.
+Attention map: Our DeAOT shares the attention maps between two propagation branches. Table 3c shows the study of different kinds of attention maps. Concretely, visual embeddings are essential in building attention maps in the long-term/short-term propagation, whose attention maps are used to match objects. Introducing ID embeddings does not help learn better visual embeddings and will decrease the performance $8 2 . 5 \%$ vs $8 2 . 1 \%$ ). In the self-propagation, however, utilizing the ID embedding as a positional embedding will facilitate the association of objects $8 2 . 2 \%$ vs $8 2 . 5 \%$ ) in the current frame.
+Kernel size of $\mathcal { F } _ { d w }$ : Large receptive fields have been proved to be critical in segmentation-related tasks [9]. The depth-wise convolution, $\mathcal { F } _ { d w }$ , is an important part of GPM for enlarging the receptive fields. Without $\mathcal { F } _ { d w }$ , the performance of DeAOT drops from $8 2 . 5 \%$ to $8 1 . 1 \%$ , as shown in Table 3d. We empirically found the best kernel size of $\mathcal { F } _ { d w }$ is 5 among $\{ 3 , 5 , 9 \}$ .
+# 7 Conclusion
+This paper proposes a highly effective and efficient framework, Decoupling Features in Hierarchical Propagation (DeAOT), for video object segmentation. Based on the rethinking of AOT-like hierarchical propagation, we propose to decouple the propagation of visual and ID embeddings into two network branches and thus avoid the loss of visual information in deep propagation layers. Besides, we propose the Gated Propagation Module (GPM), an efficient module for constructing hierarchical VOS propagation. Applying GPM to the dual-branch propagation, our DeAOT variant networks achieve new state-of-the-art performance on four VOS/VOT benchmarks with superior run-time speed compared to previous solutions.
+Acknowledgements. This work is partly supported by the Fundamental Research Funds for the Central Universities (No. 226-2022-00051).
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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] see the end of Sec. 1
+(b) Did you describe the limitations of your work? [Yes] we discuss the failure cases in Sec. 6, and demonstrate them in Fig. 4.
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] see the supplementary material.
+(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] the paper does not contain any theoretical assumptions.
+(b) Did you include complete proofs of all theoretical results? [N/A] the paper does not contain any theoretical proofs.
+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] the instructions and details needed to reproduce the main results are supplied in Sec. 5 and the supplementary material.
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] see Sec. 5 and the supplementary materials.
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A] we follow the usual format used in previous state-ofthe-art methods [6, 11, 34, 43, 62, 63] to report and compare the results. Besides, all the networks are simultaneously evaluated on four benchmarks without re-training or checkpoint selection.
+(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] Sec. 5 and the supplementary materials.
+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] we cite the creators or original papers of all the related code, data, and models used in this paper.
+(b) Did you mention the license of the assets? [No] all the assets are free for research study and widely used in previous related works.
+(c) Did you include any new assets either in the supplemental material or as a URL? [No] but the code of our proposed approach will be made publicly available as soon as the paper is accepted.
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No] all the datasets are publicly available, free for research study, and commonly used in previous related works.
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] all the datasets are commonly used in previous research works.
+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] we didn’t use crowdsourcing or conduct research with human subjects.
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] we didn’t use crowdsourcing or conduct research with human subjects.
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] we didn’t use crowdsourcing or conduct research with human subjects.

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