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+# DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps
+
+Cheng $\mathbf { L } \mathbf { u } ^ { \dagger }$ , Yuhao Zhou†, Fan Bao†, Jianfei $\mathbf { C h e n } ^ { \dagger * }$ , Chongxuan $\mathbf { L i } ^ { \dagger }$ , Jun Zhu†∗ †Dept. of Comp. Sci. & Tech., Institute for AI, BNRist Center, THBI Lab †Tsinghua-Bosch Joint ML Center, Tsinghua University, Beijing, 100084 China ‡Gaoling School of Artificial Intelligence, Renmin University of China, ‡Beijing Key Laboratory of Big Data Management and Analysis Methods, Beijing, China {lucheng.lc15, yuhaoz.cs}@gmail.com; bf19@mails.tsinghua.edu.cn chongxuanli@ruc.edu.cn; {jianfeic, dcszj}@tsinghua.edu.cn
+
+# Abstract
+
+Diffusion probabilistic models (DPMs) are emerging powerful generative models. Despite their high-quality generation performance, DPMs still suffer from their slow sampling as they generally need hundreds or thousands of sequential function evaluations (steps) of large neural networks to draw a sample. Sampling from DPMs can be viewed alternatively as solving the corresponding diffusion ordinary differential equations (ODEs). In this work, we propose an exact formulation of the solution of diffusion ODEs. The formulation analytically computes the linear part of the solution, rather than leaving all terms to black-box ODE solvers as adopted in previous works. By applying change-of-variable, the solution can be equivalently simplified to an exponentially weighted integral of the neural network. Based on our formulation, we propose DPM-Solver, a fast dedicated high-order solver for diffusion ODEs with the convergence order guarantee. DPM-Solver is suitable for both discrete-time and continuous-time DPMs without any further training. Experimental results show that DPM-Solver can generate high-quality samples in only 10 to 20 function evaluations on various datasets. We achieve $4 . 7 0 \ : \mathrm { F I D }$ in 10 function evaluations and 2.87 FID in 20 function evaluations on the CIFAR10 dataset, and a $4 \sim 1 6 \times$ speedup compared with previous state-of-the-art training-free samplers on various datasets.2
+
+# 1 Introduction
+
+Diffusion probabilistic models (DPMs) [1–3] are emerging powerful generative models with promising performance on many tasks, such as image generation [4, 5], video generation [6], text-to-image generation [7], speech synthesis [8, 9] and lossless compression [10]. DPMs are defined by discretetime random processes [1, 2] or continuous-time stochastic differential equations (SDEs) [3], which learn to gradually remove the noise added to the data points. Compared with the widely-used generative adversarial networks (GANs) [11] and variational auto-encoders (VAEs) [12], DPMs can not only compute exact likelihood [3], but also achieve even better sample quality for image generation [4]. However, to obtain high-quality samples, DPMs usually need hundreds or thousands of sequential steps of large neural network evaluations, thereby resulting in a much slower sampling speed than the single-step GANs or VAEs. Such inefficiency is becoming a critical bottleneck for the adoption of DPMs in downstream tasks, leading to an urgent request to design fast samplers for DPMs.
+
+![](images/4d6dd94407a2a26c38622fe661183fe6ba98b37b4ec1a90af34b958f1acc3ac9.jpg)  
+Figure 1: Samples by DDIM [19] with 10, 15, 20, 100 number of function evaluations (NFE), and DPM-Solver (ours) with only 10 NFE, using the pre-trained DPMs on ImageNet $2 5 6 \times 2 5 6$ with classifier guidance [4].
+
+Existing fast samplers for DPMs can be divided into two categories. The first category includes knowledge distillation [13, 14] and noise level or sample trajectory learning [15–18]. Such methods require a possibly expensive training stage before they can be used for efficient sampling. Furthermore, their applicability and flexibility might be limited. It might require nontrivial effort to adapt the method to different models, datasets, and number of sampling steps. The second category consists of training-free [19–21] samplers, which are suitable for all pre-trained DPMs in a simple plug-andplay manner. Training-free samplers include adopting implicit [19] or analytical [21] generation process, advanced differential equation (DE) solvers [3, 20, 22–24] and dynamic programming [18]. However, these methods still require $\sim 5 0$ function evaluations [21] to generate high-quality samples (comparable to those generated by plain samplers in about 1000 function evaluations), thereby are still time-consuming.
+
+In this work, we bring the efficiency of training-free samplers to a new level to produce high-quality samples in the “few-step sampling” regime, where the sampling can be done within around 10 steps of sequential function evaluations. We tackle the alternative problem of sampling from DPMs as solving the corresponding diffusion ordinary differential equations (ODEs) of DPMs, and carefully examine the structure of diffusion ODEs. Diffusion ODEs have a semi-linear structure — they consist of a linear function of the data variable and a nonlinear function parameterized by neural networks. Such structure is omitted in previous training-free samplers [3, 20], which directly use black-box DE solvers. To utilize the semi-linear structure, we derive an exact formulation of the solutions of diffusion ODEs by analytically computing the linear part of the solutions, avoiding the corresponding discretization error. Furthermore, by applying change-of-variable, the solutions can be equivalently simplified to an exponentially weighted integral of the neural network. Such integral is very special and can be efficiently approximated by the numerical methods for exponential integrators [25].
+
+Based on our formulation of solutions, we propose DPM-Solver, a fast dedicated solver for diffusion ODEs by approximating the above integral. Specifically, we propose first-order, second-order and third-order versions of DPM-Solver with convergence order guarantees. We further propose an adaptive step size schedule for DPM-Solver. In general, DPM-Solver is applicable to both continuoustime and discrete-time DPMs, and also conditional sampling with classifier guidance [4]. Fig. 1 demonstrates the speedup performance of a Denoising Diffusion Implicit Models (DDIM) [19] baseline and DPM-Solver, which shows that DPM-Solver can generate high-quality samples with as few as 10 function evaluations and is much faster than DDIM on the ImageNet 256x256 dataset [26]. Our additional experimental results show that DPM-Solver can greatly improve the sampling speed of both discrete-time and continuous-time DPMs, and it can achieve excellent sample quality in around 10 function evaluations, which is much faster than all previous training-free samplers of DPMs.
+
+# 2 Diffusion Probabilistic Models
+
+We review diffusion probabilistic models and their associated differential equations in this section.
+
+# 2.1 Forward Process and Diffusion SDEs
+
+Assume that we have a $D$ -dimensional random variable $\pmb { x } _ { 0 } \in \mathbb { R } ^ { D }$ with an unknown distribution $q _ { 0 } ( { \pmb x } _ { 0 } )$ . Diffusion Probabilistic Models (DPMs) [1–3, 10] define a forward process $\{ \pmb { x } _ { t } \} _ { t \in [ 0 , T ] }$ with $T > 0$ starting with $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ , such that for any $t \in [ 0 , T ]$ , the distribution of $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ conditioned on $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ satisfies
+
+$$
+\begin{array} { r } { q _ { 0 t } ( \pmb { x } _ { t } | \pmb { x } _ { 0 } ) = \mathcal { N } ( \pmb { x } _ { t } | \alpha ( t ) \pmb { x } _ { 0 } , \sigma ^ { 2 } ( t ) \pmb { I } ) , } \end{array}
+$$
+
+where $\alpha ( t ) , \sigma ( t ) \in \mathbb { R } ^ { + }$ are differentiable functions of $t$ with bounded derivatives, and we denote them as $\alpha _ { t } , \sigma _ { t }$ for simplicity. The choice for $\alpha _ { t }$ and $\sigma _ { t }$ is referred to as the noise schedule of a DPM. Let $q _ { t } ( \pmb { x } _ { t } )$ denote the marginal distribution of $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ , DPMs choose noise schedules to ensure that $q _ { T } ( \pmb { x } _ { T } ) \overset { \cdot } { \approx } \dot { \mathcal { N } } ( \pmb { x } _ { T } | \mathbf { 0 } , \tilde { \sigma } ^ { 2 } \pmb { I } )$ for some $\tilde { \sigma } > 0$ , and the signal-to-noise-ratio (SNR) $\alpha _ { t } ^ { 2 } / \sigma _ { t } ^ { 2 }$ is strictly decreasing w.r.t. $t$ [10]. Moreover, Kingma et al. [10] prove that the following stochastic differential equation (SDE) has the same transition distribution $q _ { 0 t } ( \pmb { x } _ { t } | \pmb { x } _ { 0 } )$ as in Eq. (2.1) for any $t \in [ 0 , T ]$ :
+
+$$
+\mathrm { d } \pmb { x } _ { t } = f ( t ) \pmb { x } _ { t } \mathrm { d } t + g ( t ) \mathrm { d } \pmb { w } _ { t } , \quad \pmb { x } _ { 0 } \sim q _ { 0 } ( \pmb { x } _ { 0 } ) ,
+$$
+
+where ${ \pmb w } _ { t } \in \mathbb { R } ^ { D }$ is the standard Wiener process, and
+
+$$
+f ( t ) = \frac { \mathrm { d } \log \alpha _ { t } } { \mathrm { d } t } , \quad g ^ { 2 } ( t ) = \frac { \mathrm { d } \sigma _ { t } ^ { 2 } } { \mathrm { d } t } - 2 \frac { \mathrm { d } \log \alpha _ { t } } { \mathrm { d } t } \sigma _ { t } ^ { 2 } .
+$$
+
+Under some regularity conditions, Song et al. [3] show that the forward process in Eq. (2.2) has an equivalent reverse process from time $T$ to $0$ , starting with the marginal distribution $q _ { T } ( { \pmb x } _ { T } )$ :
+
+$$
+\mathrm { d } \pmb { x } _ { t } = [ f ( t ) \pmb { x } _ { t } - g ^ { 2 } ( t ) \nabla _ { \pmb { x } } \log q _ { t } ( \pmb { x } _ { t } ) ] \mathrm { d } t + g ( t ) \mathrm { d } \bar { \pmb { w } } _ { t } , \quad \pmb { x } _ { T } \sim q _ { T } ( \pmb { x } _ { T } ) ,
+$$
+
+where $\bar { \mathbf { \nabla } } \bar { \mathbf { \nabla } } \bar { \mathbf { \nabla } } \bar { \mathbf { \nabla } } \bar { \mathbf { \nabla } } \bar { \mathbf { \nabla } } \bar { \mathbf { w } } _ { t }$ is a standard Wiener process in the reverse time. The only unknown term in Eq. (2.4) is the score function $\nabla _ { \pmb { x } } \log q _ { t } ( \pmb { x } _ { t } )$ at each time $t$ . In practice, DPMs use a neural network $\epsilon _ { \theta } ( x _ { t } , t )$ parameterized by $\theta$ to estimate the scaled score function: $- \sigma _ { t } \nabla _ { \pmb { x } } \log q _ { t } ( \pmb { x } _ { t } )$ . The parameter $\theta$ is optimized by minimizing the following objective [2, 3]:
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { L } ( \theta ; \omega ( t ) ) : = \frac { 1 } { 2 } \int _ { 0 } ^ { T } \omega ( t ) \mathbb { E } _ { q _ { t } ( \mathbf { \Delta x } _ { t } ) } \Big [ \| \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) + \sigma _ { t } \nabla _ { \mathbf { x } } \log q _ { t } ( \mathbf { \Delta x } _ { t } ) \| _ { 2 } ^ { 2 } \Big ] \mathrm { d } t } \\ { \displaystyle \qquad = \frac { 1 } { 2 } \int _ { 0 } ^ { T } \omega ( t ) \mathbb { E } _ { q _ { 0 } ( \mathbf { x } _ { 0 } ) } \mathbb { E } _ { q ( \epsilon ) } \Big [ \| \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) - \epsilon \| _ { 2 } ^ { 2 } \Big ] \mathrm { d } t + C , } \end{array}
+$$
+
+where $\omega ( t )$ is a weighting function, $\epsilon \sim q ( \epsilon ) = \mathcal { N } ( \epsilon | \mathbf { 0 } , I )$ , ${ \pmb x } _ { t } = \alpha _ { t } { \pmb x } _ { 0 } + \sigma _ { t } { \pmb \epsilon }$ , and $C$ is a constant independent of $\theta$ . As $\epsilon _ { \theta } ( x _ { t } , t )$ can also be regarded as predicting the Gaussian noise added to $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ , it is usually called the noise prediction model. Since the ground truth of $\epsilon _ { \theta } ( x _ { t } , t )$ is $- \sigma _ { t } \nabla _ { \pmb { x } } \log q _ { t } ( \pmb { x } _ { t } )$ , DPMs replace the score function in Eq. (2.4) by $- \mathbf { \epsilon } \mathbf { \epsilon } \bar { \mathbf { \alpha } } ( \mathbf { x } _ { t } , t ) / \sigma _ { t }$ and define a parameterized reverse process (diffusion $S D E$ ) from time $T$ to $0$ , starting with $\pmb { x } _ { T } \overset { \cdot } { \sim } \mathcal { N } ( \mathbf { 0 } , \tilde { \sigma } ^ { 2 } \pmb { I } )$ :
+
+$$
+\mathrm { d } x _ { t } = \left[ f ( t ) x _ { t } + \frac { g ^ { 2 } ( t ) } { \sigma _ { t } } \epsilon _ { \theta } ( x _ { t } , t ) \right] \mathrm { d } t + g ( t ) \mathrm { d } \bar { w } _ { t } , \quad x _ { T } \sim \mathcal { N } ( \mathbf { 0 } , \tilde { \sigma } ^ { 2 } I ) .
+$$
+
+Samples can be generated from DPMs by solving the diffusion SDE in Eq. (2.5) with numerical solvers, which discretize the SDE from $T$ to 0. Song et al. [3] proved that the traditional ancestral sampling method for DPMs [2] can be viewed as a first-order SDE solver for Eq. (2.5). However, these first-order methods usually need hundreds of or thousands of function evaluations to converge [3], leading to extremely slow sampling speed.
+
+# 2.2 Diffusion (Probability Flow) ODEs
+
+When discretizing SDEs, the step size is limited by the randomness of the Wiener process [27, Chap. 11]. A large step size (small number of steps) often causes non-convergence, especially in high dimensional spaces. For faster sampling, one can consider the associated probability flow ODE [3], which has the same marginal distribution at each time $t$ as that of the SDE. Specifically, for DPMs, Song et al. [3] proved that the probability flow ODE of Eq. (2.4) is
+
+$$
+\frac { \mathrm { d } \pmb { x } _ { t } } { \mathrm { d } t } = f ( t ) \pmb { x } _ { t } - \frac { 1 } { 2 } g ^ { 2 } ( t ) \nabla _ { \pmb { x } } \log q _ { t } ( \pmb { x } _ { t } ) , \quad \pmb { x } _ { T } \sim q _ { T } ( \pmb { x } _ { T } ) ,
+$$
+
+where the marginal distribution of $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ is also $q _ { t } ( \pmb { x } _ { t } )$ . By replacing the score function with the noise prediction model, Song et al. [3] defined the following parameterized ODE (diffusion $O D E$ ):
+
+$$
+\frac { \mathrm { d } \pmb { x } _ { t } } { \mathrm { d } t } = \pmb { h } _ { \theta } ( \pmb { x } _ { t } , t ) : = f ( t ) \pmb { x } _ { t } + \frac { g ^ { 2 } ( t ) } { 2 \sigma _ { t } } \epsilon _ { \theta } ( \pmb { x } _ { t } , t ) , \quad \pmb { x } _ { T } \sim \mathcal { N } ( \mathbf { 0 } , \tilde { \sigma } ^ { 2 } \mathbf { I } ) .
+$$
+
+Samples can be drawn by solving the ODE from $T$ to 0. Comparing with SDEs, ODEs can be solved with larger step sizes as they have no randomness. Furthermore, we can take advantage of efficient numerical ODE solvers to accelerate the sampling. Song et al. [3] used the RK45 ODE solver [28] for the diffusion ODEs, which generates samples in $\sim 6 0$ function evaluations to reach comparable quality with a 1000-step SDE solver for Eq. (2.5) on the CIFAR-10 dataset [29]. However, existing general-purpose ODE solvers still cannot generate satisfactory samples in the few-step $\sim 1 0$ steps) sampling regime. To the best of our knowledge, there is still a lack of training-free samplers for DPMs in the few-step sampling regime, and the sampling speed of DPMs is still a critical issue.
+
+# 3 Customized Fast Solvers for Diffusion ODEs
+
+As highlighted in Sec. 2.2, discretizing SDEs is generally difficult in high dimensions [27, Chap. 11] and it is hard to converge within few steps. In contrast, ODEs are easier to solve, yielding a potential for fast samplers. However, as mentioned in Sec. 2.2, the general black-box ODE solver used in previous work [3] empirically fails to converge in few steps. This motivates us to design a dedicated solver for diffusion ODEs to enable fast and high-quality few-step sampling. We start with a detailed investigation of the specific structure of diffusion ODEs.
+
+# 3.1 Simplified Formulation of Exact Solutions of Diffusion ODEs
+
+The key insight of this work is that given an initial value $\mathbf { \delta } _ { \mathbf { \mathcal { X } } _ { s } }$ at time $s > 0$ , the solution $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ at each time $t < s$ of diffusion ODEs in Eq. (2.7) can be simplified into a very special exact formulation which can be efficiently approximated.
+
+Our first key observation is that a part of the solution $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ can be exactly computed by considering the particular structure of diffusion ODEs. The r.h.s. of diffusion ODEs in Eq. (2.7) consists of two parts: the part $f ( t ) x _ { t }$ is a linear function of $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ , and the other part $\frac { g ^ { 2 } ( t ) } { 2 \sigma _ { t } } \epsilon _ { \theta } ( \pmb { x } _ { t } , t )$ is generally a nonlinear function of $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ because of the neural network $\epsilon _ { \theta } ( x _ { t } , t )$ . This type of ODE is referred to as semi-linear ODE. The black-box ODE solvers adopted by previous work [3] are ignorant of this semi-linear structure as they take the whole $h _ { \theta } ( x _ { t } , \bar  t ) $ in Eq. (2.7) as the input, which causes discretization errors of both the linear and nonlinear term. We note that for semi-linear ODEs, the solution at time $t$ can be exactly formulated by the “variation of constants” formula [30]:
+
+$$
+\pmb { x } _ { t } = e ^ { \int _ { s } ^ { t } f ( \tau ) \mathrm { d } \tau } \pmb { x } _ { s } + \int _ { s } ^ { t } \left( e ^ { \int _ { \tau } ^ { t } f ( r ) \mathrm { d } r } \frac { g ^ { 2 } \big ( \tau \big ) } { 2 \sigma _ { \tau } } \epsilon _ { \theta } ( \pmb { x } _ { \tau } , \tau ) \right) \mathrm { d } \tau .
+$$
+
+This formulation decouples the linear part and the nonlinear part. In contrast to black-box ODE solvers, the linear part is now exactly computed, which eliminates the approximation error of the linear term. However, the integral of the nonlinear part is still complicated because it couples the coefficients about the noise schedule (i.e., $f ( \tau ) , g ( \tau ) \bar { , } \sigma _ { \tau } )$ and the complex neural network $\epsilon _ { \theta }$ , which is still hard to approximate.
+
+Our second key observation is that the integral of the nonlinear part can be greatly simplified by introducing a special variable. Let $\lambda _ { t } : = \log \bar { ( \alpha _ { t } / \sigma _ { t } ) }$ (one half of the log-SNR), then $\lambda _ { t }$ is a strictly decreasing function of $t$ (due to the definition of DPMs as discussed in Sec. 2.1). We can rewrite $g ( t )$ in Eq. (2.3) as
+
+$$
+g ^ { 2 } ( t ) = \frac { \mathrm { d } \sigma _ { t } ^ { 2 } } { \mathrm { d } t } - 2 \frac { \mathrm { d } \log { \alpha _ { t } } } { \mathrm { d } t } \sigma _ { t } ^ { 2 } = 2 \sigma _ { t } ^ { 2 } \left( \frac { \mathrm { d } \log { \sigma _ { t } } } { \mathrm { d } t } - \frac { \mathrm { d } \log { \alpha _ { t } } } { \mathrm { d } t } \right) = - 2 \sigma _ { t } ^ { 2 } \frac { \mathrm { d } \lambda _ { t } } { \mathrm { d } t } .
+$$
+
+Combining with $f ( t ) = \mathrm { d } \log \alpha _ { t } / \mathrm { d } t$ in Eq. (2.3), we can rewrite Eq. (3.1) as
+
+$$
+\pmb { x } _ { t } = \frac { \alpha _ { t } } { \alpha _ { s } } \pmb { x } _ { s } - \alpha _ { t } \int _ { s } ^ { t } \left( \frac { \mathrm { d } \lambda _ { \tau } } { \mathrm { d } \tau } \right) \frac { \sigma _ { \tau } } { \alpha _ { \tau } } \pmb { \epsilon } _ { \theta } ( \pmb { x } _ { \tau } , \tau ) \mathrm { d } \tau .
+$$
+
+As $\lambda ( t ) = \lambda _ { t }$ is a strictly decreasing function of $t$ , it has an inverse function $t _ { \lambda } ( \cdot )$ satisfying $t = t _ { \lambda } ( \lambda ( t ) )$ . We further change the subscripts of $_ { \textbf { \em x } }$ and $\epsilon _ { \theta }$ from $t$ to $\lambda$ and denote $\hat { \pmb x } _ { \lambda } : = \pmb x _ { t _ { \lambda } ( \lambda ) }$ $\hat { \epsilon } _ { \boldsymbol { \theta } } ( \hat { x } _ { \lambda } , \lambda ) : = \epsilon _ { \boldsymbol { \theta } } ( x _ { t _ { \lambda } ( \lambda ) } , t _ { \lambda } ( \lambda ) )$ . Rewrite Eq. (3.3) by “change-of-variable” for $\lambda$ , then we have:
+
+Proposition 3.1 (Exact solution of diffusion ODEs). Given an initial value $\mathbf { \delta } _ { \mathbf { \mathcal { X } } _ { s } }$ at time $s > 0$ , the solution $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ at time $t \in [ 0 , s ]$ of diffusion ODEs in Eq. (2.7) is:
+
+$$
+\pmb { x } _ { t } = \frac { \alpha _ { t } } { \alpha _ { s } } \pmb { x } _ { s } - \alpha _ { t } \int _ { \lambda _ { s } } ^ { \lambda _ { t } } e ^ { - \lambda } \hat { \pmb { \epsilon } } _ { \theta } ( \hat { \pmb { x } } _ { \lambda } , \lambda ) \mathrm { d } \lambda .
+$$
+
+We call the integral $\begin{array} { r } { \int e ^ { - \lambda } \hat { \epsilon } _ { \theta } ( \hat { \pmb x } _ { \lambda } , \lambda ) \mathrm { d } \lambda } \end{array}$ the exponentially weighted integral of $\scriptstyle { \hat { \epsilon } } _ { \theta }$ , which is very special and highly related to the exponential integrators in the literature of ODE solvers [25]. To the best of our knowledge, such formulation has not been revealed in prior work of diffusion models.
+
+Eq. (3.4) provides a new perspective for approximating the solutions of diffusion ODEs. Specifically, given $\mathbf { \delta } _ { \mathbf { \mathcal { X } } _ { s } }$ at time $s$ , According to Eq. (3.4), approximating the solution at time $t$ is equivalent to directly approximating the exponentially weighted integral of $\hat { \epsilon } _ { \theta }$ from $\lambda _ { s }$ to $\lambda _ { t }$ , which avoids the error of the linear terms and is well-studied in the literature of exponential integrators [25, 31]. Based on this insight, we propose fast solvers for diffusion ODEs, as detailed in the following sections.
+
+# 3.2 High-Order Solvers for Diffusion ODEs
+
+In this section, we propose high-order solvers for diffusion ODEs with convergence order guarantee by leveraging our proposed solution formulation Eq. (3.4). The proposed solvers and analysis are highly motivated by the methods of exponential integrators [25, 31] in the ODE literature.
+
+Specifically, given an initial value $\mathbf { \nabla } _ { \mathbf { x } _ { T } }$ at time $T$ and $M + 1$ time steps $\{ t _ { i } \} _ { i = 0 } ^ { M }$ decreasing from $t _ { 0 } = T$ to $t _ { M } = 0$ . Let $\tilde { \mathbf { x } } _ { t _ { 0 } } = \mathbf { x } _ { T }$ be the initial value. The proposed solvers use $M$ steps to iteratively compute a sequence $\{ \tilde { { \pmb { x } } } _ { t _ { i } } \} _ { i = 0 } ^ { M }$ to approximate the true solutions at time steps $\{ t _ { i } \} _ { i = 0 } ^ { M }$ . In particular, the last iterate $\tilde { \boldsymbol { x } } _ { t _ { M } }$ approximates the true solution at time 0.
+
+In order to reduce the approximation error between $\tilde { \pmb { x } } _ { t _ { M } }$ and the true solution at time 0, we need to reduce the approximation error for each $\tilde { \mathbf { x } } _ { t _ { i } }$ at every step [30]. Starting with the previous value $\tilde { \pmb { x } } _ { t _ { i - 1 } }$ at time $t _ { i - 1 }$ , according to Eq. (3.4), the exact solution $\pmb { x } _ { t _ { i - 1 }  t _ { i } }$ at time $t _ { i }$ is given by
+
+$$
+\pmb { x } _ { t _ { i - 1 }  t _ { i } } = \frac { \alpha _ { t _ { i } } } { \alpha _ { t _ { i - 1 } } } \tilde { \pmb { x } } _ { t _ { i - 1 } } - \alpha _ { t _ { i } } \int _ { \lambda _ { t _ { i - 1 } } } ^ { \lambda _ { t _ { i } } } e ^ { - \lambda } \hat { \pmb { \epsilon } } _ { \theta } ( \hat { \pmb { x } } _ { \lambda } , \lambda ) \mathrm { d } \lambda .
+$$
+
+Therefore, to compute the value $\tilde { \boldsymbol { x } } _ { t _ { i } }$ for approximating $\pmb { x } _ { t _ { i - 1 }  t _ { i } }$ , we need to approximate the exponentially weighted integral of $\hat { \epsilon } _ { \theta }$ from $\lambda _ { t _ { i - 1 } }$ to $\lambda _ { t _ { i } }$ . Denote $h _ { i } : = \lambda _ { t _ { i } } - \lambda _ { t _ { i - 1 } }$ , and $\hat { \epsilon } _ { \theta } ^ { ( n ) } ( \hat { { \mathbf x } } _ { \lambda } , \lambda ) \mathrel { \mathop : } =$ dnϵˆθ(xˆλ,λ)dλn as the n-th order total derivative of ϵˆθ(xˆλ, λ) w.r.t. λ. For k ≥ 1, the (k − 1)-th order Taylor expansion of $\hat { \epsilon } _ { \theta } ( \hat { \pmb x } _ { \lambda } , \lambda )$ w.r.t. $\lambda$ at $\lambda _ { t _ { i - 1 } }$ is
+
+$$
+\hat { \epsilon } _ { \theta } ( \hat { x } _ { \lambda } , \lambda ) = \sum _ { n = 0 } ^ { k - 1 } \frac { ( \lambda - \lambda _ { t _ { i - 1 } } ) ^ { n } } { n ! } \hat { \epsilon } _ { \theta } ^ { ( n ) } ( \hat { x } _ { \lambda _ { t _ { i - 1 } } } , \lambda _ { t _ { i - 1 } } ) + \mathcal { O } ( ( \lambda - \lambda _ { t _ { i - 1 } } ) ^ { k } ) ,
+$$
+
+Substituting the above Taylor expansion into Eq. (3.5) yields
+
+$$
+\pmb { x } _ { t _ { i - 1 }  t _ { i } } = \frac { \alpha _ { t _ { i } } } { \alpha _ { t _ { i - 1 } } } \tilde { \pmb { x } } _ { t _ { i - 1 } } - \alpha _ { t _ { i } } \sum _ { n = 0 } ^ { k - 1 } \hat { \pmb { \epsilon } } _ { \theta } ^ { ( n ) } ( \hat { \pmb { x } } _ { { \pmb { \lambda } } _ { t _ { i - 1 } } } , \lambda _ { t _ { i - 1 } } ) \int _ { \lambda _ { t _ { i - 1 } } } ^ { \lambda _ { t _ { i } } } e ^ { - \lambda } \frac { ( \lambda - \lambda _ { t _ { i - 1 } } ) ^ { n } } { n ! } \mathrm { d } \lambda + \mathcal { O } ( h _ { i } ^ { k + 1 } ) ,
+$$
+
+where the integral $\begin{array} { r } { \int e ^ { - \lambda } \frac { ( \lambda - \lambda _ { t _ { i - 1 } } ) ^ { n } } { n ! } \mathrm { d } \lambda } \end{array}$ can be analytically computed by repeatedly applying $n$ times of integration-by-parts (see Appendix B.2). Therefore, to approximate $\pmb { x } _ { t _ { i - 1 }  t _ { i } }$ , we only need to approximate the $n$ -th order total derivatives $\hat { \epsilon } _ { \theta } ^ { ( n ) } ( \hat { \pmb { x } } _ { \lambda } , \lambda )$ for $n \leq k - 1$ , which is a well-studied problem in the ODE literature [31, 32]. By dropping the $\mathcal { O } ( h _ { i } ^ { k + 1 } )$ error term and approximating the first $( k - 1 )$ -th total derivatives with the “stiff order conditions” [31, 32], we can derive $k$ -th-order ODE solvers for diffusion ODEs. We name such solvers as DPM-Solver overall, and DPM-Solver- $k$ for a specific order $k$ . Here we take $k = 1$ for demonstration. In this case, Eq. (3.6) becomes
+
+$$
+\begin{array} { l } { \displaystyle { \boldsymbol { x } } _ { t _ { i - 1 } \to t _ { i } } = \frac { \alpha _ { t _ { i } } } { \alpha _ { t _ { i - 1 } } } \tilde { \boldsymbol { x } } _ { t _ { i - 1 } } - \alpha _ { t _ { i } } \epsilon _ { \theta } ( \tilde { \boldsymbol { x } } _ { t _ { i - 1 } } , t _ { i - 1 } ) \int _ { \lambda _ { t _ { i - 1 } } } ^ { \lambda _ { t _ { i } } } e ^ { - \lambda } \mathrm { d } \lambda + \mathcal { O } ( h _ { i } ^ { 2 } ) } \\ { \displaystyle = \frac { \alpha _ { t _ { i } } } { \alpha _ { t _ { i - 1 } } } \tilde { \boldsymbol { x } } _ { t _ { i - 1 } } - \sigma _ { t _ { i } } ( e ^ { h _ { i } } - 1 ) \epsilon _ { \theta } ( \tilde { \boldsymbol { x } } _ { t _ { i - 1 } } , t _ { i - 1 } ) + \mathcal { O } ( h _ { i } ^ { 2 } ) . } \end{array}
+$$
+
+By dropping the high-order error term $\mathcal { O } ( h _ { i } ^ { 2 } )$ , we can obtain an approximation for $\pmb { x } _ { t _ { i - 1 }  t _ { i } }$ . As $k = 1$ here, we call this solver DPM-Solver- $^ { l }$ , and the detailed algorithm is as following.
+
+DPM-Solver-1. Given an initial value $\mathbf { \nabla } _ { \mathbf { x } _ { T } }$ and $M + 1$ time steps $\{ t _ { i } \} _ { i = 0 } ^ { M }$ decreasing from $t _ { 0 } = T$ to $t _ { M } = 0$ . Starting with $\tilde { \mathbf { x } } _ { t _ { 0 } } = \mathbf { x } _ { T }$ , the sequence $\{ \tilde { { x } } _ { t _ { i } } \} _ { i = 1 } ^ { M }$ is computed iteratively as follows:
+
+$$
+\tilde { \boldsymbol { x } } _ { t _ { i } } = \frac { \alpha _ { t _ { i } } } { \alpha _ { t _ { i - 1 } } } \tilde { \boldsymbol { x } } _ { t _ { i - 1 } } - \sigma _ { t _ { i } } ( e ^ { h _ { i } } - 1 ) \boldsymbol { \epsilon } _ { \boldsymbol { \theta } } ( \tilde { \boldsymbol { x } } _ { t _ { i - 1 } } , t _ { i - 1 } ) , \quad \mathrm { w h e r e ~ } h _ { i } = \lambda _ { t _ { i } } - \lambda _ { t _ { i - 1 } } .
+$$
+
+For $k \geq 2$ , approximating the first $k$ terms of the Taylor expansion needs additional intermediate points between $t$ and $s$ [31]. The derivation is more technical so we defer it to Appendix B. Below we propose algorithms for $k = 2 , 3$ and name them as DPM-Solver-2 and DPM-Solver-3, respectively.
+
+# Algorithm 1 DPM-Solver-2.
+
+Require: initial value $\mathbf { \nabla } _ { \mathbf { \mathcal { X } } T }$ , time steps $\{ t _ { i } \} _ { i = 0 } ^ { M }$ , model $\epsilon _ { \theta }$
+
+# Algorithm 2 DPM-Solver-3.
+
+Require: initial value $\mathbf { \nabla } _ { \mathbf { \mathcal { X } } T }$ , time steps $\{ t _ { i } \} _ { i = 0 } ^ { M }$ , model $\epsilon _ { \theta }$
+
+$$
+\begin{array} { r l } & { \quad _ { s 2 i - 1 }  _ { t _ { \lambda } } ( \overline { { \lambda } } _ { t _ { i - 1 } } + r _ { 1 } h _ { i } ) , \quad s _ { 2 i }  t _ { \lambda } ( \lambda _ { t _ { i - 1 } } + r _ { 2 } h _ { i } ) } \\ & { u _ { 2 i - 1 }  \frac { \alpha _ { s _ { 2 i - 1 } } } { \alpha _ { t _ { i - 1 } } } \tilde { x } _ { t _ { i - 1 } } - \sigma _ { s _ { 2 i - 1 } } ( e ^ { r _ { 1 } h _ { i } } - 1 ) \epsilon _ { \theta } ( \tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) } \\ & { D _ { 2 i - 1 }  \epsilon _ { \theta } ( u _ { 2 i - 1 } , s _ { 2 i - 1 } ) - \epsilon _ { \theta } ( \tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) } \\ & { u _ { 2 i }  \frac { \alpha _ { s _ { 2 i } } } { \alpha _ { t _ { i - 1 } } } \tilde { x } _ { t _ { i - 1 } } - \sigma _ { s _ { 2 i } } ( e ^ { r _ { 2 } h _ { i } } - 1 ) \epsilon _ { \theta } ( \tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) - \frac { \sigma _ { s _ { 2 i } } r _ { 2 } } { r _ { 1 } } ( \frac { e ^ { r _ { 2 } h _ { i } } - 1 } { r _ { 2 } h _ { i } } - 1 ) D _ { 2 i - 1 } } \\ & { D _ { 2 i }  \epsilon _ { \theta } ( u _ { 2 i } , s _ { 2 i } ) - \epsilon _ { \theta } ( \tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) } \\ & { \tilde { x } _ { t _ { i } }  \frac { \alpha _ { t _ { i } } } { \alpha _ { t _ { i - 1 } } } \tilde { x } _ { t _ { i - 1 } } - \sigma _ { t _ { i } } ( e ^ { h _ { i } } - 1 ) \epsilon _ { \theta } ( \tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) - \frac { \sigma _ { t _ { i } } } { r _ { 2 } } ( \frac { e ^ { h _ { i } } - 1 } { h } - 1 ) D _ { 2 i } } \end{array}
+$$
+
+10: return $\tilde { \pmb { x } } _ { t _ { M } }$
+
+Here, $t _ { \lambda } ( \cdot )$ is the inverse function of $\lambda ( t )$ , which has an analytical formulation for the practical noise schedule used in [2, 16], as shown in Appendix D. The chosen intermediate points are $( s _ { i } , \pmb { u } _ { i } )$ for DPM-Solver-2 and $\left( s _ { 2 i - 1 } , { \pmb u } _ { 2 i - 1 } \right)$ and $( s _ { 2 i } , { \pmb u } _ { 2 i } )$ for DPM-Solver-3. As shown in the algorithm, DPM-Solver- $k$ requires $k$ function evaluations per step for $k = 1 , 2 , 3$ . Despite the more expensive steps, higher-order solvers $( k = 2 , 3$ ) are usually more efficient since they require much fewer steps to converge, due to their higher convergence order. We show that DPM-Solver- $k$ is $k$ -th-order solver, as stated in the following theorem. The proof is in Appendix B.
+
+Theorem 3.2 (DPM-Solver- $k$ as a $k$ -th-order solver). Assume $\epsilon _ { \theta } ( x _ { t } , t )$ follows the regularity conditions detailed in Appendix B.1, then for $k = 1 , 2 , 3$ , DPM-Solver- $k$ is a $k$ -th order solver for diffusion ODEs, i.e., for the sequence $\{ \tilde { \pmb { x } } _ { t _ { i } } \} _ { i = 1 } ^ { M }$ computed by DPM-Solver- $k$ , the approximation error at time 0 satisfies $\tilde { \pmb { x } } _ { t _ { M } } - \pmb { x } _ { 0 } = \mathcal { O } ( h _ { \operatorname* { m a x } } ^ { k } )$ , where $h _ { m a x } = \mathrm { m a x } _ { 1 \leq i \leq M } ( \lambda _ { t _ { i } } - \lambda _ { t _ { i - 1 } } )$ .
+
+Finally, solvers with $k \geq 4$ need much more intermediate points as shown by previous work [31, 32] for exponential integrators. Therefore, we only consider $k$ from 1 to 3 in this work, while leaving the solvers with higher $k$ for future study.
+
+# 3.3 Step Size Schedule
+
+The proposed solvers in Sec. 3.2 need to specify the time steps $\{ t _ { i } \} _ { i = 0 } ^ { M }$ in advance. We propose two choices of the time step schedule. One choice is handcrafted, which is to uniformly split the interval $[ \lambda _ { T } , \lambda _ { 0 } ]$ , i.e. $\begin{array} { r } { \lambda _ { t _ { i } } = \dot { \lambda _ { T } } + \frac { i } { M } ( \lambda _ { 0 } - \lambda _ { T } ) } \end{array}$ , $i = 0 , \ldots , M$ . Note that this is different from previous work [2, 3] which chooses uniform steps for $t _ { i }$ . Empirically, DPM-Solver with uniform time steps $\lambda _ { t _ { i } }$ can already generate quite good samples in few steps, where results are listed in Appendix E. As the other choice, we propose an adaptive step size algorithm, which dynamically adjusts the step size by combining different orders of DPM-Solver. The adaptive algorithm is inspired by [20] and we defer its implementation details to Appendix C.
+
+For few-step sampling, we need to use up all the number of function evaluations (NFE). When the NFE is not divisible by 3, we firstly apply DPM-Solver-3 as much as possible, and then add a single step of DPM-Solver-1 or DPM-Solver-2 (dependent on the reminder of $K$ divided by 3), as detailed in Appendix D. In the subsequent experiments, we use such combination of solvers with the uniform step size schedule for $\mathrm { N F E } \leq 2 0$ , and otherwise the adaptive step size schedule.
+
+# 3.4 Sampling from Discrete-Time DPMs
+
+Discrete-time DPMs [2] train the noise prediction model at $N$ fixed time steps $\{ t _ { n } \} _ { n = 1 } ^ { N }$ , and the noise prediction model is parameterized by $\tilde { \epsilon } _ { \theta } ( { \boldsymbol x } _ { n } , n )$ for $n = 0 , \ldots , N - 1$ , where each ${ \pmb x } _ { n }$ is corresponding to the value at time $t _ { n + 1 }$ . We can transform the discrete-time noise prediction model to the continuous version by letting $\begin{array} { r } { \epsilon _ { \theta } ( x , t ) : = \tilde { \epsilon } _ { \theta } ( x , \frac { ( N - 1 ) t } { T } ) } \end{array}$ , for all $\pmb { x } \in \mathbb { R } ^ { d } , t \in [ 0 , T ]$ . Note that the input time of $\tilde { \epsilon } _ { \theta }$ may not be integers, but we find that the noise prediction model can still work well, and we hypothesize that it is because of the smooth time embeddings (e.g., position embeddings [2]). By such reparameterization, the noise prediction model can adopt the continuous-time steps as input, and thus we can also use DPM-Solver for fast sampling.
+
+# 4 Comparison with Existing Fast Sampling Methods
+
+Here, we discuss the relationship and highlight the difference between DPM-Solver and existing ODE-based fast sampling methods for DPMs. We further briefly discuss the advantage of training-free samplers over those training-based ones.
+
+# 4.1 DDIM as DPM-Solver-1
+
+Denoising Diffusion Implicit Models (DDIM) [19] design a deterministic method for fast sampling from DPMs. For two adjacent time steps $t _ { i - 1 }$ and $t _ { i }$ , assume that we have a solution $\tilde { \boldsymbol { x } } _ { t _ { i - 1 } }$ at time $t _ { i - 1 }$ , then a single step of DDIM from time $t _ { i - 1 }$ to time $t _ { i }$ is
+
+$$
+\tilde { \pmb { x } } _ { t _ { i } } = \frac { \alpha _ { t _ { i } } } { \alpha _ { t _ { i - 1 } } } \tilde { \pmb { x } } _ { t _ { i - 1 } } - \alpha _ { t _ { i } } \left( \frac { \sigma _ { t _ { i - 1 } } } { \alpha _ { t _ { i - 1 } } } - \frac { \sigma _ { t _ { i } } } { \alpha _ { t _ { i } } } \right) \epsilon _ { \theta } ( \tilde { \pmb { x } } _ { t _ { i - 1 } } , t _ { i - 1 } ) .
+$$
+
+Although motivated by entirely different perspectives, we show that the updates of DPM-Solver-1 and Denoising Diffusion Implicit Models (DDIM) [19] are identical. By the definition of $\lambda$ , we have $\frac { \sigma _ { t _ { i - 1 } } } { \alpha _ { t _ { i - 1 } } } = e ^ { - \lambda _ { t _ { i - 1 } } ^ { - } }$ and $\begin{array} { r } { \frac { \sigma _ { t _ { i } } } { \alpha _ { t _ { i } } } = e ^ { - \lambda _ { t _ { i } } } } \end{array}$ . Plugging these and $h _ { i } = \lambda _ { t _ { i } } - \lambda _ { t _ { i - 1 } }$ to Eq. (4.1) results in exactly a step of DPM-Solver-1 in Eq. (3.7). However, the semi-linear ODE formulation of DPM-Solver allows for principled generalization to higher-order solvers and convergence order analysis.
+
+Recent work [13] also show that DDIM is a first-order discretization of diffusion ODEs by differentiating both sides of Eq. (4.1). However, they cannot explain the difference between DDIM and the first-order Euler discretization of diffusion ODEs. In contrast, by showing that DDIM is a special case of DPM-Solver, we reveal that DDIM makes full use of the semi-linearity of diffusion ODEs, which explains its superiority over traditional Euler methods.
+
+# 4.2 Comparison with Traditional Runge-Kutta Methods
+
+One can obtain a high-order solver by directly applying traditional explicit Runge-Kutta (RK) methods to the diffusion ODE in Eq. (2.7). Specifically, RK methods write the solution of Eq. (2.7) in the
+
+Table 1: FID ↓ on CIFAR-10 for different orders of Runge-Kutta (RK) methods and DPM-Solvers, varying the number of function evaluations (NFE). For RK methods, we evaluate diffusion ODEs w.r.t. both $t$ (Eq. (2.7)) and $\lambda$ (Eq. (E.1)). We use uniform step size in $t$ for RK (t), and uniform step size in $\lambda$ for RK $( \lambda )$ and DPM-Solvers.   
+
+<table><tr><td>Sampling method\NFE</td><td>12</td><td>18</td><td>24</td><td>30</td><td>36</td><td>42</td><td>48</td></tr><tr><td>RK2 (t)</td><td>16.40</td><td>7.25</td><td>3.90</td><td>3.63</td><td>3.58</td><td>3.59</td><td>3.54</td></tr><tr><td>RK2(入)</td><td>107.81</td><td>42.04</td><td>17.71</td><td>7.65</td><td>4.62</td><td>3.58</td><td>3.17</td></tr><tr><td>DPM-Solver-2</td><td>5.28</td><td>3.43</td><td>3.02</td><td>2.85</td><td>2.78</td><td>2.72</td><td>2.69</td></tr><tr><td>RK3 (t)</td><td>48.75</td><td>21.86</td><td>10.90</td><td>6.96</td><td>5.22</td><td>4.56</td><td>4.12</td></tr><tr><td>RK3 (入)</td><td>34.29</td><td>4.90</td><td>3.50</td><td>3.03</td><td>2.85</td><td>2.74</td><td>2.69</td></tr><tr><td>DPM-Solver-3</td><td>6.03</td><td>2.90</td><td>2.75</td><td>2.70</td><td>2.67</td><td>2.65</td><td>2.65</td></tr></table>
+
+following integral form:
+
+$$
+{ \bf { x } } _ { t } = { \bf { x } } _ { s } + \int _ { s } ^ { t } h _ { \theta } ( { \bf { x } } _ { \tau } , \tau ) \mathrm { { d } } \tau = { \bf { x } } _ { s } + \int _ { s } ^ { t } \left( f ( \tau ) { \bf { x } } _ { \tau } + \frac { g ^ { 2 } ( \tau ) } { 2 \sigma _ { \tau } } \epsilon _ { \theta } ( { \bf { x } } _ { \tau } , \tau ) \right) \mathrm { { d } } \tau ,
+$$
+
+and use some intermediate time steps between $[ t , s ]$ and combine the evaluations of $h _ { \theta }$ at these time steps to approximate the whole integral. The approximation error of explicit RK methods depends on $h _ { \theta }$ , which consists of the error corresponding to both the linear term $f ( \tau ) x _ { \tau }$ and the nonlinear noise prediction model $\epsilon _ { \theta }$ . However, the error of the linear term may increase exponentially because the exact solution of the linear term has an exponential coefficient (as shown in Eq. (3.1)). There are many empirical evidence [25, 31] showing that directly using explicit RK methods for semi-linear ODEs may suffer from unstable numerical issues for large step size. We also demonstrate the empirical difference of the proposed DPM-Solver and the traditional explicit RK methods in Sec. 5.1, which shows that DPM-Solver have smaller discretization errors than the RK methods with the same order.
+
+# 4.3 Training-based Fast Sampling Methods for DPMs
+
+Samplers that need extra training or optimization include knowledge distillation [13, 14], learning the noise level or variance [15, 16, 33], and learning the noise schedule or sample trajectory [17, 18]. Although the progressive distillation method [13] can obtain a fast sampler within 4 steps, it needs further training costs and loses part of the information in the original DPM (e.g., after distillation, the noise prediction model cannot predict the noise (score function) at every time step between $[ 0 , T ] )$ . In contrast, training-free samplers can keep all the information of the original model, and thereby can be directly extended to the conditional sampling by combining the original model and an external classifier [4] (e.g. see Appendix D for the conditional sampling with classifier guidance).
+
+Beyond directly designing fast samplers for DPMs, several works also propose novel types of DPMs which supports faster sampling. For instance, defining a low-dimensional latent variable for DPMs [34]; designing special diffusion processes with bounded score functions [35]; combining GANs with the reverse process of DPMs [36]. The proposed DPM-Solver may also be suitable for accelerating the sampling of these DPMs, and we leave them for future work.
+
+# 5 Experiments
+
+In this section, we show that as a training-free sampler, DPM-Solver can greatly speedup the sampling of existing pre-trained DPMs, including both continuous-time and discrete-time ones, with both linear noise schedule [2, 19] and cosine noise schedule [16]. We vary different number of function evaluations (NFE) which is the number of calls to the noise prediction model $\epsilon _ { \theta } ( x _ { t } , t )$ , and compare the sample quality between DPM-Solver and other methods. For each experiment, We draw 50K samples and use the widely adopted FID score [37] to evaluate the sample quality, where lower FID usually implies better sample quality.
+
+Unless explicitly mentioned, we always use the solver combination with the uniform step size schedule in Sec. 3.3 if the NFE budget is less than 20, and otherwise the DPM-Solver-3 with the adaptive step size schedule in Sec. 3.3. We refer to Appendix D for other implementation details of DPM-Solver and Appendix E for detailed settings.
+
+![](images/4e1bcabdec913d95ec9ff692dca4450fca481c16e793a3c86ffe1cf469e22305.jpg)  
+Figure 2: Sample quality measured by FID $\downarrow$ of different sampling methods for DPMs on CIFAR-10 with both continuous-time and discrete-time models, CelebA 64x64, ImageNet 64x64, ImageNet $1 2 8 \mathrm { x } 1 2 8$ and LSUN bedroom $2 5 6 \times 2 5 6$ with discrete-time models, varying the number of function evaluations (NFE). The method $^ { \dag } { \bf G } { \bf G } { \bf D } { \bf M }$ [18] needs extra training to optimize the sample trajectory, while other methods are training-free. To get the strongest baseline, we use the quadratic step size for DDIM on CelebA, which has a better FID than that of the uniform step size in the original paper [19].
+
+# 5.1 Comparison with Continuous-Time Sampling Methods
+
+We firstly compare DPM-Solver with other continuous-time sampling methods for DPMs. The compared methods include the Euler-Maruyama discretization for diffusion SDEs [3], the adaptive step size solver for diffusion SDEs [20] and the RK methods for diffusion ODEs [3, 28] in Eq. (2.7). We compare these methods for sampling from a pre-trained continuous-time “VP deep” model [3] on the CIFAR-10 dataset [29] with the linear noise schedule.
+
+Fig. 2a shows the efficiency of compared solvers. We use uniform time steps with 50, 200, 1000 NFE for the diffusion SDE with Euler discretization, and vary the tolerance hyperparameter [3, 20] for the adaptive step size SDE solver [20] and RK45 ODE solver [28] to control the NFE. DPM-Solver can generate good sample quality within around 10 NFE, while other solvers have large discretization error even in 50 NFE, which shows that DPM-Solver can achieve ${ \sim } 5 $ speedup of the previous best solver. In particular, we achieve 4.70 FID with 10 NFE, 3.75 FID with 12 NFE, 3.24 FID with 15 NFE, and 2.87 FID with 20 NFE, which is the fastest sampler on CIFAR-10.
+
+As an ablation study, we also compare the second-order and third-order DPM-Solver and RK methods, as shown in Table 1. We compare RK methods for diffusion ODEs w.r.t. both time $t$ in Eq. (2.7) and half-log-SNR $\lambda$ by applying change-of-variable (see detailed formulations in Appendix E.1). The results show that given the same NFE, the sample quality of DPM-Solver is consistently better than RK methods with the same order. The superior efficiency of DPM-Solver is particularly evident in the few-step regime under 15 NFE, where RK methods have rather large discretization errors. This is mainly because DPM-Solver analytically computes the linear term, avoiding the corresponding discretization error. Besides, the higher order DPM-Solver-3 converges faster than DPM-Solver-2, which matches the order analysis in Theorem 3.2.
+
+# 5.2 Comparison with Discrete-Time Sampling Methods
+
+We use the method in Sec. 3.4 for using DPM-Solver in discrete-time DPMs, and then compare DPM-Solver with other discrete-time training-free samplers, including DDPM [2], DDIM [19], Analytic-DDPM [21], Analytic-DDIM [21], PNDM [22], FastDPM [38] and Itô-Taylor [24]. We also compare with GGDM [18], which uses the same pre-trained model but needs further training for the sampling trajectory. We compare the sample quality by varying NFE from 10 to 1000.
+
+Specifically, we use the discrete-time model trained by $L _ { \mathrm { s i m p l e } }$ in [2] on the CIFAR-10 dataset with linear noise schedule; the discrete-time model in [19] on CelebA 64x64 [39] with linear noise schedule; the discrete-time model trained by $L _ { \mathrm { h y b r i d } }$ in [16] on ImageNet 64x64 [26] with cosine noise schedule; the discrete-time model with classifier guidance in [4] on ImageNet 128x128 [26] with linear noise schedule; the discrete-time model in [4] on LSUN bedroom $2 5 6 \times 2 5 6$ [40] with linear noise schedule. For the models trained on ImageNet, we only use their “mean” model and omit the “variance” model. As shown in Fig. 2, on all datasets, DPM-Solver can obtain reasonable samples within 12 steps (FID 4.65 on CIFAR-10, FID 3.71 on CelebA 64x64 and FID 19.97 on ImageNet 64x64, FID 4.08 on ImageNet $1 2 8 \mathbf { x } 1 2 8 _ { \rho }$ ), which is $4 \sim 1 6 \times$ faster than the previous fastest training-free sampler. DPM-Solver even outperforms GGDM, which requires additional training.
+
+# 6 Conclusions
+
+We tackle the problem of fast and training-free sampling from DPMs. We propose DPM-Solver, a fast dedicated training-free solver of diffusion ODEs for fast sampling of DPMs in around 10 steps of function evaluations. DPM-Solver leverages the semi-linearity of diffusion ODEs and it directly approximates a simplified formulation of exact solutions of diffusion ODEs, which consists of an exponentially weighted integral of the noise prediction model. Inspired by numerical methods for exponential integrators, we propose first-order, second-order and third-order DPMSolver to approximate the exponentially weighted integral of noise prediction models with theoretical convergence guarantee. We propose both handcrafted and adaptive step size schedule, and apply DPM-Solver for both continuous-time and discrete-time DPMs. Our experimental results show that DPM-Solver can generate high-quality samples in around 10 function evaluations on various datasets, and it can achieve $4 \sim 1 6 \times$ speedup compared with previous state-of-the-art training-free samplers.
+
+Limitations and broader impact Despite the promising speedup performance, DPM-Solver is designed for fast sampling, which may be not suitable for accelerating the likelihood evaluations of DPMs. Besides, compared to the commonly-used GANs, diffusion models with DPM-Solver are still not fast enough for real-time applications. In addition, like other deep generative models, DPMs may be used to generate adverse fake contents, and the proposed solver may further amplify the potential undesirable influence of deep generative models for malicious applications.
+
+# Acknowledgements
+
+This work was supported by National Key Research and Development Project of China (No. 2021ZD0110502); NSF of China Projects (Nos. 62061136001, 61620106010, 62076145, U19B2034, U1811461, U19A2081, 6197222, 62106120); Beijing NSF Project (No. JQ19016); Beijing Outstanding Young Scientist Program NO. BJJWZYJH012019100020098; a grant from Tsinghua Institute for Guo Qiang; the NVIDIA NVAIL Program with GPU/DGX Acceleration; the High Performance Computing Center, Tsinghua University; the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (22XNKJ13). J.Z is also supported by the XPlorer Prize.
+
+# References
+
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+[2] J. Ho, A. Jain, and P. Abbeel, “Denoising diffusion probabilistic models,” in Advances in Neural Information Processing Systems, vol. 33, 2020, pp. 6840–6851.   
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See section 6.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See section 6.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Appendix B. (b) Did you include complete proofs of all theoretical results? [Yes] See Appendix B.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Code is attached in the supplemental materials, with the appendix.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Our method is training-free. But we also report the hyperparameters for evaluations used in our proposed solver.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We observe that the standard deviation of the FID evaluations of DPM-Solver are rather small (mainly less than 0.01) because the FID is already averaged over 50K samples, following existing work [18, 20, 21]. The small standard deviation does not change the conclusion.   
+(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] The GPU type and amount is detailed in Appendix E.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [Yes] See Appendix E   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We include our code in the supplemental materials.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No] All of the datasets used in the experiments are publicly available.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] We mentioned the human privacy issues of the ImageNet dataset in Appendix E.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/dev/2uAaGwlP_V/2uAaGwlP_V_content_list.json

@@ -0,0 +1,1443 @@
+[
+    {
+        "type": "text",
+        "text": "DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps ",
+        "text_level": 1,
+        "bbox": [
+            199,
+            122,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Cheng $\\mathbf { L } \\mathbf { u } ^ { \\dagger }$ , Yuhao Zhou†, Fan Bao†, Jianfei $\\mathbf { C h e n } ^ { \\dagger * }$ , Chongxuan $\\mathbf { L i } ^ { \\dagger }$ , Jun Zhu†∗ †Dept. of Comp. Sci. & Tech., Institute for AI, BNRist Center, THBI Lab †Tsinghua-Bosch Joint ML Center, Tsinghua University, Beijing, 100084 China ‡Gaoling School of Artificial Intelligence, Renmin University of China, ‡Beijing Key Laboratory of Big Data Management and Analysis Methods, Beijing, China {lucheng.lc15, yuhaoz.cs}@gmail.com; bf19@mails.tsinghua.edu.cn chongxuanli@ruc.edu.cn; {jianfeic, dcszj}@tsinghua.edu.cn ",
+        "bbox": [
+            207,
+            224,
+            790,
+            328
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            363,
+            535,
+            380
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Diffusion probabilistic models (DPMs) are emerging powerful generative models. Despite their high-quality generation performance, DPMs still suffer from their slow sampling as they generally need hundreds or thousands of sequential function evaluations (steps) of large neural networks to draw a sample. Sampling from DPMs can be viewed alternatively as solving the corresponding diffusion ordinary differential equations (ODEs). In this work, we propose an exact formulation of the solution of diffusion ODEs. The formulation analytically computes the linear part of the solution, rather than leaving all terms to black-box ODE solvers as adopted in previous works. By applying change-of-variable, the solution can be equivalently simplified to an exponentially weighted integral of the neural network. Based on our formulation, we propose DPM-Solver, a fast dedicated high-order solver for diffusion ODEs with the convergence order guarantee. DPM-Solver is suitable for both discrete-time and continuous-time DPMs without any further training. Experimental results show that DPM-Solver can generate high-quality samples in only 10 to 20 function evaluations on various datasets. We achieve $4 . 7 0 \\ : \\mathrm { F I D }$ in 10 function evaluations and 2.87 FID in 20 function evaluations on the CIFAR10 dataset, and a $4 \\sim 1 6 \\times$ speedup compared with previous state-of-the-art training-free samplers on various datasets.2 ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            672,
+            310,
+            690
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Diffusion probabilistic models (DPMs) [1–3] are emerging powerful generative models with promising performance on many tasks, such as image generation [4, 5], video generation [6], text-to-image generation [7], speech synthesis [8, 9] and lossless compression [10]. DPMs are defined by discretetime random processes [1, 2] or continuous-time stochastic differential equations (SDEs) [3], which learn to gradually remove the noise added to the data points. Compared with the widely-used generative adversarial networks (GANs) [11] and variational auto-encoders (VAEs) [12], DPMs can not only compute exact likelihood [3], but also achieve even better sample quality for image generation [4]. However, to obtain high-quality samples, DPMs usually need hundreds or thousands of sequential steps of large neural network evaluations, thereby resulting in a much slower sampling speed than the single-step GANs or VAEs. Such inefficiency is becoming a critical bottleneck for the adoption of DPMs in downstream tasks, leading to an urgent request to design fast samplers for DPMs. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/4d6dd94407a2a26c38622fe661183fe6ba98b37b4ec1a90af34b958f1acc3ac9.jpg",
+        "image_caption": [
+            "Figure 1: Samples by DDIM [19] with 10, 15, 20, 100 number of function evaluations (NFE), and DPM-Solver (ours) with only 10 NFE, using the pre-trained DPMs on ImageNet $2 5 6 \\times 2 5 6$ with classifier guidance [4]. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Existing fast samplers for DPMs can be divided into two categories. The first category includes knowledge distillation [13, 14] and noise level or sample trajectory learning [15–18]. Such methods require a possibly expensive training stage before they can be used for efficient sampling. Furthermore, their applicability and flexibility might be limited. It might require nontrivial effort to adapt the method to different models, datasets, and number of sampling steps. The second category consists of training-free [19–21] samplers, which are suitable for all pre-trained DPMs in a simple plug-andplay manner. Training-free samplers include adopting implicit [19] or analytical [21] generation process, advanced differential equation (DE) solvers [3, 20, 22–24] and dynamic programming [18]. However, these methods still require $\\sim 5 0$ function evaluations [21] to generate high-quality samples (comparable to those generated by plain samplers in about 1000 function evaluations), thereby are still time-consuming. ",
+        "bbox": [
+            174,
+            354,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we bring the efficiency of training-free samplers to a new level to produce high-quality samples in the “few-step sampling” regime, where the sampling can be done within around 10 steps of sequential function evaluations. We tackle the alternative problem of sampling from DPMs as solving the corresponding diffusion ordinary differential equations (ODEs) of DPMs, and carefully examine the structure of diffusion ODEs. Diffusion ODEs have a semi-linear structure — they consist of a linear function of the data variable and a nonlinear function parameterized by neural networks. Such structure is omitted in previous training-free samplers [3, 20], which directly use black-box DE solvers. To utilize the semi-linear structure, we derive an exact formulation of the solutions of diffusion ODEs by analytically computing the linear part of the solutions, avoiding the corresponding discretization error. Furthermore, by applying change-of-variable, the solutions can be equivalently simplified to an exponentially weighted integral of the neural network. Such integral is very special and can be efficiently approximated by the numerical methods for exponential integrators [25]. ",
+        "bbox": [
+            174,
+            512,
+            825,
+            678
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Based on our formulation of solutions, we propose DPM-Solver, a fast dedicated solver for diffusion ODEs by approximating the above integral. Specifically, we propose first-order, second-order and third-order versions of DPM-Solver with convergence order guarantees. We further propose an adaptive step size schedule for DPM-Solver. In general, DPM-Solver is applicable to both continuoustime and discrete-time DPMs, and also conditional sampling with classifier guidance [4]. Fig. 1 demonstrates the speedup performance of a Denoising Diffusion Implicit Models (DDIM) [19] baseline and DPM-Solver, which shows that DPM-Solver can generate high-quality samples with as few as 10 function evaluations and is much faster than DDIM on the ImageNet 256x256 dataset [26]. Our additional experimental results show that DPM-Solver can greatly improve the sampling speed of both discrete-time and continuous-time DPMs, and it can achieve excellent sample quality in around 10 function evaluations, which is much faster than all previous training-free samplers of DPMs. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Diffusion Probabilistic Models ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            861,
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+            878
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We review diffusion probabilistic models and their associated differential equations in this section. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 Forward Process and Diffusion SDEs ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Assume that we have a $D$ -dimensional random variable $\\pmb { x } _ { 0 } \\in \\mathbb { R } ^ { D }$ with an unknown distribution $q _ { 0 } ( { \\pmb x } _ { 0 } )$ . Diffusion Probabilistic Models (DPMs) [1–3, 10] define a forward process $\\{ \\pmb { x } _ { t } \\} _ { t \\in [ 0 , T ] }$ with $T > 0$ starting with $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ , such that for any $t \\in [ 0 , T ]$ , the distribution of $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ conditioned on $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ satisfies ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b98f04c1bbffa9531e257210ae6112c502489822e557b5a3afdb274392b83d54.jpg",
+        "text": "$$\n\\begin{array} { r } { q _ { 0 t } ( \\pmb { x } _ { t } | \\pmb { x } _ { 0 } ) = \\mathcal { N } ( \\pmb { x } _ { t } | \\alpha ( t ) \\pmb { x } _ { 0 } , \\sigma ^ { 2 } ( t ) \\pmb { I } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            167,
+            622,
+            188
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha ( t ) , \\sigma ( t ) \\in \\mathbb { R } ^ { + }$ are differentiable functions of $t$ with bounded derivatives, and we denote them as $\\alpha _ { t } , \\sigma _ { t }$ for simplicity. The choice for $\\alpha _ { t }$ and $\\sigma _ { t }$ is referred to as the noise schedule of a DPM. Let $q _ { t } ( \\pmb { x } _ { t } )$ denote the marginal distribution of $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ , DPMs choose noise schedules to ensure that $q _ { T } ( \\pmb { x } _ { T } ) \\overset { \\cdot } { \\approx } \\dot { \\mathcal { N } } ( \\pmb { x } _ { T } | \\mathbf { 0 } , \\tilde { \\sigma } ^ { 2 } \\pmb { I } )$ for some $\\tilde { \\sigma } > 0$ , and the signal-to-noise-ratio (SNR) $\\alpha _ { t } ^ { 2 } / \\sigma _ { t } ^ { 2 }$ is strictly decreasing w.r.t. $t$ [10]. Moreover, Kingma et al. [10] prove that the following stochastic differential equation (SDE) has the same transition distribution $q _ { 0 t } ( \\pmb { x } _ { t } | \\pmb { x } _ { 0 } )$ as in Eq. (2.1) for any $t \\in [ 0 , T ]$ : ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8433318473b1628737ca5404d0a929d6a4570d9919cf8824f87c274bc4a0bbfd.jpg",
+        "text": "$$\n\\mathrm { d } \\pmb { x } _ { t } = f ( t ) \\pmb { x } _ { t } \\mathrm { d } t + g ( t ) \\mathrm { d } \\pmb { w } _ { t } , \\quad \\pmb { x } _ { 0 } \\sim q _ { 0 } ( \\pmb { x } _ { 0 } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            285,
+            648,
+            303
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\pmb w } _ { t } \\in \\mathbb { R } ^ { D }$ is the standard Wiener process, and ",
+        "bbox": [
+            173,
+            309,
+            517,
+            324
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e5d12cd513ddf8ae3fa219a1feddb4594e4cf22e33260b27389e15d81ab87dc7.jpg",
+        "text": "$$\nf ( t ) = \\frac { \\mathrm { d } \\log \\alpha _ { t } } { \\mathrm { d } t } , \\quad g ^ { 2 } ( t ) = \\frac { \\mathrm { d } \\sigma _ { t } ^ { 2 } } { \\mathrm { d } t } - 2 \\frac { \\mathrm { d } \\log \\alpha _ { t } } { \\mathrm { d } t } \\sigma _ { t } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            332,
+            661,
+            364
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Under some regularity conditions, Song et al. [3] show that the forward process in Eq. (2.2) has an equivalent reverse process from time $T$ to $0$ , starting with the marginal distribution $q _ { T } ( { \\pmb x } _ { T } )$ : ",
+        "bbox": [
+            174,
+            371,
+            826,
+            400
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/098b1c09edd91e1e8b539459bb92a3c5f191abb42aa1157755522d3dce62bcae.jpg",
+        "text": "$$\n\\mathrm { d } \\pmb { x } _ { t } = [ f ( t ) \\pmb { x } _ { t } - g ^ { 2 } ( t ) \\nabla _ { \\pmb { x } } \\log q _ { t } ( \\pmb { x } _ { t } ) ] \\mathrm { d } t + g ( t ) \\mathrm { d } \\bar { \\pmb { w } } _ { t } , \\quad \\pmb { x } _ { T } \\sim q _ { T } ( \\pmb { x } _ { T } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            406,
+            728,
+            426
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { w } } _ { t }$ is a standard Wiener process in the reverse time. The only unknown term in Eq. (2.4) is the score function $\\nabla _ { \\pmb { x } } \\log q _ { t } ( \\pmb { x } _ { t } )$ at each time $t$ . In practice, DPMs use a neural network $\\epsilon _ { \\theta } ( x _ { t } , t )$ parameterized by $\\theta$ to estimate the scaled score function: $- \\sigma _ { t } \\nabla _ { \\pmb { x } } \\log q _ { t } ( \\pmb { x } _ { t } )$ . The parameter $\\theta$ is optimized by minimizing the following objective [2, 3]: ",
+        "bbox": [
+            176,
+            433,
+            825,
+            489
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/18da74f3ada9dcef2cf0de53303bacedefeb327430ef720bfe85381fb0de2f45.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathcal { L } ( \\theta ; \\omega ( t ) ) : = \\frac { 1 } { 2 } \\int _ { 0 } ^ { T } \\omega ( t ) \\mathbb { E } _ { q _ { t } ( \\mathbf { \\Delta x } _ { t } ) } \\Big [ \\| \\epsilon _ { \\theta } ( \\mathbf { x } _ { t } , t ) + \\sigma _ { t } \\nabla _ { \\mathbf { x } } \\log q _ { t } ( \\mathbf { \\Delta x } _ { t } ) \\| _ { 2 } ^ { 2 } \\Big ] \\mathrm { d } t } \\\\ { \\displaystyle \\qquad = \\frac { 1 } { 2 } \\int _ { 0 } ^ { T } \\omega ( t ) \\mathbb { E } _ { q _ { 0 } ( \\mathbf { x } _ { 0 } ) } \\mathbb { E } _ { q ( \\epsilon ) } \\Big [ \\| \\epsilon _ { \\theta } ( \\mathbf { x } _ { t } , t ) - \\epsilon \\| _ { 2 } ^ { 2 } \\Big ] \\mathrm { d } t + C , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            496,
+            727,
+            571
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\omega ( t )$ is a weighting function, $\\epsilon \\sim q ( \\epsilon ) = \\mathcal { N } ( \\epsilon | \\mathbf { 0 } , I )$ , ${ \\pmb x } _ { t } = \\alpha _ { t } { \\pmb x } _ { 0 } + \\sigma _ { t } { \\pmb \\epsilon }$ , and $C$ is a constant independent of $\\theta$ . As $\\epsilon _ { \\theta } ( x _ { t } , t )$ can also be regarded as predicting the Gaussian noise added to $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ , it is usually called the noise prediction model. Since the ground truth of $\\epsilon _ { \\theta } ( x _ { t } , t )$ is $- \\sigma _ { t } \\nabla _ { \\pmb { x } } \\log q _ { t } ( \\pmb { x } _ { t } )$ , DPMs replace the score function in Eq. (2.4) by $- \\mathbf { \\epsilon } \\mathbf { \\epsilon } \\bar { \\mathbf { \\alpha } } ( \\mathbf { x } _ { t } , t ) / \\sigma _ { t }$ and define a parameterized reverse process (diffusion $S D E$ ) from time $T$ to $0$ , starting with $\\pmb { x } _ { T } \\overset { \\cdot } { \\sim } \\mathcal { N } ( \\mathbf { 0 } , \\tilde { \\sigma } ^ { 2 } \\pmb { I } )$ : ",
+        "bbox": [
+            173,
+            575,
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+            647
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/65fdde0d1d949853db59a598710255c20a35d6c68ca6401f8b46cd9f514525f2.jpg",
+        "text": "$$\n\\mathrm { d } x _ { t } = \\left[ f ( t ) x _ { t } + \\frac { g ^ { 2 } ( t ) } { \\sigma _ { t } } \\epsilon _ { \\theta } ( x _ { t } , t ) \\right] \\mathrm { d } t + g ( t ) \\mathrm { d } \\bar { w } _ { t } , \\quad x _ { T } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\tilde { \\sigma } ^ { 2 } I ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            654,
+            732,
+            689
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Samples can be generated from DPMs by solving the diffusion SDE in Eq. (2.5) with numerical solvers, which discretize the SDE from $T$ to 0. Song et al. [3] proved that the traditional ancestral sampling method for DPMs [2] can be viewed as a first-order SDE solver for Eq. (2.5). However, these first-order methods usually need hundreds of or thousands of function evaluations to converge [3], leading to extremely slow sampling speed. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Diffusion (Probability Flow) ODEs ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            781,
+            454,
+            797
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "When discretizing SDEs, the step size is limited by the randomness of the Wiener process [27, Chap. 11]. A large step size (small number of steps) often causes non-convergence, especially in high dimensional spaces. For faster sampling, one can consider the associated probability flow ODE [3], which has the same marginal distribution at each time $t$ as that of the SDE. Specifically, for DPMs, Song et al. [3] proved that the probability flow ODE of Eq. (2.4) is ",
+        "bbox": [
+            173,
+            808,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/db0329942038b0f1cd9de0b5b9716a202705a11feeab57d38ae6c43085d15745.jpg",
+        "text": "$$\n\\frac { \\mathrm { d } \\pmb { x } _ { t } } { \\mathrm { d } t } = f ( t ) \\pmb { x } _ { t } - \\frac { 1 } { 2 } g ^ { 2 } ( t ) \\nabla _ { \\pmb { x } } \\log q _ { t } ( \\pmb { x } _ { t } ) , \\quad \\pmb { x } _ { T } \\sim q _ { T } ( \\pmb { x } _ { T } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            886,
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+            916
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the marginal distribution of $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ is also $q _ { t } ( \\pmb { x } _ { t } )$ . By replacing the score function with the noise prediction model, Song et al. [3] defined the following parameterized ODE (diffusion $O D E$ ): ",
+        "bbox": [
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+            90,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/772c489895a83788baac27c0b2cc0da51d29a5ff2c72c378ea8301efa3cfe52d.jpg",
+        "text": "$$\n\\frac { \\mathrm { d } \\pmb { x } _ { t } } { \\mathrm { d } t } = \\pmb { h } _ { \\theta } ( \\pmb { x } _ { t } , t ) : = f ( t ) \\pmb { x } _ { t } + \\frac { g ^ { 2 } ( t ) } { 2 \\sigma _ { t } } \\epsilon _ { \\theta } ( \\pmb { x } _ { t } , t ) , \\quad \\pmb { x } _ { T } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\tilde { \\sigma } ^ { 2 } \\mathbf { I } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            277,
+            121,
+            720,
+            155
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Samples can be drawn by solving the ODE from $T$ to 0. Comparing with SDEs, ODEs can be solved with larger step sizes as they have no randomness. Furthermore, we can take advantage of efficient numerical ODE solvers to accelerate the sampling. Song et al. [3] used the RK45 ODE solver [28] for the diffusion ODEs, which generates samples in $\\sim 6 0$ function evaluations to reach comparable quality with a 1000-step SDE solver for Eq. (2.5) on the CIFAR-10 dataset [29]. However, existing general-purpose ODE solvers still cannot generate satisfactory samples in the few-step $\\sim 1 0$ steps) sampling regime. To the best of our knowledge, there is still a lack of training-free samplers for DPMs in the few-step sampling regime, and the sampling speed of DPMs is still a critical issue. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 Customized Fast Solvers for Diffusion ODEs ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            285,
+            576,
+            303
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As highlighted in Sec. 2.2, discretizing SDEs is generally difficult in high dimensions [27, Chap. 11] and it is hard to converge within few steps. In contrast, ODEs are easier to solve, yielding a potential for fast samplers. However, as mentioned in Sec. 2.2, the general black-box ODE solver used in previous work [3] empirically fails to converge in few steps. This motivates us to design a dedicated solver for diffusion ODEs to enable fast and high-quality few-step sampling. We start with a detailed investigation of the specific structure of diffusion ODEs. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 Simplified Formulation of Exact Solutions of Diffusion ODEs ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            412,
+            637,
+            429
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The key insight of this work is that given an initial value $\\mathbf { \\delta } _ { \\mathbf { \\mathcal { X } } _ { s } }$ at time $s > 0$ , the solution $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ at each time $t < s$ of diffusion ODEs in Eq. (2.7) can be simplified into a very special exact formulation which can be efficiently approximated. ",
+        "bbox": [
+            176,
+            439,
+            823,
+            482
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our first key observation is that a part of the solution $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ can be exactly computed by considering the particular structure of diffusion ODEs. The r.h.s. of diffusion ODEs in Eq. (2.7) consists of two parts: the part $f ( t ) x _ { t }$ is a linear function of $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ , and the other part $\\frac { g ^ { 2 } ( t ) } { 2 \\sigma _ { t } } \\epsilon _ { \\theta } ( \\pmb { x } _ { t } , t )$ is generally a nonlinear function of $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ because of the neural network $\\epsilon _ { \\theta } ( x _ { t } , t )$ . This type of ODE is referred to as semi-linear ODE. The black-box ODE solvers adopted by previous work [3] are ignorant of this semi-linear structure as they take the whole $h _ { \\theta } ( x _ { t } , \\bar  t ) $ in Eq. (2.7) as the input, which causes discretization errors of both the linear and nonlinear term. We note that for semi-linear ODEs, the solution at time $t$ can be exactly formulated by the “variation of constants” formula [30]: ",
+        "bbox": [
+            173,
+            487,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6f1153f8f759866e204632c40d87e1d736b7589259f6710fccd3282a51e399e5.jpg",
+        "text": "$$\n\\pmb { x } _ { t } = e ^ { \\int _ { s } ^ { t } f ( \\tau ) \\mathrm { d } \\tau } \\pmb { x } _ { s } + \\int _ { s } ^ { t } \\left( e ^ { \\int _ { \\tau } ^ { t } f ( r ) \\mathrm { d } r } \\frac { g ^ { 2 } \\big ( \\tau \\big ) } { 2 \\sigma _ { \\tau } } \\epsilon _ { \\theta } ( \\pmb { x } _ { \\tau } , \\tau ) \\right) \\mathrm { d } \\tau .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            307,
+            607,
+            691,
+            643
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This formulation decouples the linear part and the nonlinear part. In contrast to black-box ODE solvers, the linear part is now exactly computed, which eliminates the approximation error of the linear term. However, the integral of the nonlinear part is still complicated because it couples the coefficients about the noise schedule (i.e., $f ( \\tau ) , g ( \\tau ) \\bar { , } \\sigma _ { \\tau } )$ and the complex neural network $\\epsilon _ { \\theta }$ , which is still hard to approximate. ",
+        "bbox": [
+            173,
+            643,
+            825,
+            713
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our second key observation is that the integral of the nonlinear part can be greatly simplified by introducing a special variable. Let $\\lambda _ { t } : = \\log \\bar { ( \\alpha _ { t } / \\sigma _ { t } ) }$ (one half of the log-SNR), then $\\lambda _ { t }$ is a strictly decreasing function of $t$ (due to the definition of DPMs as discussed in Sec. 2.1). We can rewrite $g ( t )$ in Eq. (2.3) as ",
+        "bbox": [
+            173,
+            719,
+            825,
+            775
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/28428e28211e5384d76d1e8330c965590495d098f46b2f50a448a40982e1a8f0.jpg",
+        "text": "$$\ng ^ { 2 } ( t ) = \\frac { \\mathrm { d } \\sigma _ { t } ^ { 2 } } { \\mathrm { d } t } - 2 \\frac { \\mathrm { d } \\log { \\alpha _ { t } } } { \\mathrm { d } t } \\sigma _ { t } ^ { 2 } = 2 \\sigma _ { t } ^ { 2 } \\left( \\frac { \\mathrm { d } \\log { \\sigma _ { t } } } { \\mathrm { d } t } - \\frac { \\mathrm { d } \\log { \\alpha _ { t } } } { \\mathrm { d } t } \\right) = - 2 \\sigma _ { t } ^ { 2 } \\frac { \\mathrm { d } \\lambda _ { t } } { \\mathrm { d } t } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            254,
+            776,
+            743,
+            813
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Combining with $f ( t ) = \\mathrm { d } \\log \\alpha _ { t } / \\mathrm { d } t$ in Eq. (2.3), we can rewrite Eq. (3.1) as ",
+        "bbox": [
+            171,
+            813,
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+            829
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d804fb43f91c9fdbe50275bbfdd61568c9d23d13e7c4cea3f8e0751dee5a2bb6.jpg",
+        "text": "$$\n\\pmb { x } _ { t } = \\frac { \\alpha _ { t } } { \\alpha _ { s } } \\pmb { x } _ { s } - \\alpha _ { t } \\int _ { s } ^ { t } \\left( \\frac { \\mathrm { d } \\lambda _ { \\tau } } { \\mathrm { d } \\tau } \\right) \\frac { \\sigma _ { \\tau } } { \\alpha _ { \\tau } } \\pmb { \\epsilon } _ { \\theta } ( \\pmb { x } _ { \\tau } , \\tau ) \\mathrm { d } \\tau .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            344,
+            830,
+            651,
+            866
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As $\\lambda ( t ) = \\lambda _ { t }$ is a strictly decreasing function of $t$ , it has an inverse function $t _ { \\lambda } ( \\cdot )$ satisfying $t = t _ { \\lambda } ( \\lambda ( t ) )$ . We further change the subscripts of $_ { \\textbf { \\em x } }$ and $\\epsilon _ { \\theta }$ from $t$ to $\\lambda$ and denote $\\hat { \\pmb x } _ { \\lambda } : = \\pmb x _ { t _ { \\lambda } ( \\lambda ) }$ $\\hat { \\epsilon } _ { \\boldsymbol { \\theta } } ( \\hat { x } _ { \\lambda } , \\lambda ) : = \\epsilon _ { \\boldsymbol { \\theta } } ( x _ { t _ { \\lambda } ( \\lambda ) } , t _ { \\lambda } ( \\lambda ) )$ . Rewrite Eq. (3.3) by “change-of-variable” for $\\lambda$ , then we have: ",
+        "bbox": [
+            173,
+            867,
+            825,
+            912
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3.1 (Exact solution of diffusion ODEs). Given an initial value $\\mathbf { \\delta } _ { \\mathbf { \\mathcal { X } } _ { s } }$ at time $s > 0$ , the solution $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ at time $t \\in [ 0 , s ]$ of diffusion ODEs in Eq. (2.7) is: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9abcfd7effacf68fb53c21394f34e41f5f607171cf80a13282a4bf3971c34831.jpg",
+        "text": "$$\n\\pmb { x } _ { t } = \\frac { \\alpha _ { t } } { \\alpha _ { s } } \\pmb { x } _ { s } - \\alpha _ { t } \\int _ { \\lambda _ { s } } ^ { \\lambda _ { t } } e ^ { - \\lambda } \\hat { \\pmb { \\epsilon } } _ { \\theta } ( \\hat { \\pmb { x } } _ { \\lambda } , \\lambda ) \\mathrm { d } \\lambda .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            125,
+            632,
+            162
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We call the integral $\\begin{array} { r } { \\int e ^ { - \\lambda } \\hat { \\epsilon } _ { \\theta } ( \\hat { \\pmb x } _ { \\lambda } , \\lambda ) \\mathrm { d } \\lambda } \\end{array}$ the exponentially weighted integral of $\\scriptstyle { \\hat { \\epsilon } } _ { \\theta }$ , which is very special and highly related to the exponential integrators in the literature of ODE solvers [25]. To the best of our knowledge, such formulation has not been revealed in prior work of diffusion models. ",
+        "bbox": [
+            173,
+            174,
+            825,
+            217
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Eq. (3.4) provides a new perspective for approximating the solutions of diffusion ODEs. Specifically, given $\\mathbf { \\delta } _ { \\mathbf { \\mathcal { X } } _ { s } }$ at time $s$ , According to Eq. (3.4), approximating the solution at time $t$ is equivalent to directly approximating the exponentially weighted integral of $\\hat { \\epsilon } _ { \\theta }$ from $\\lambda _ { s }$ to $\\lambda _ { t }$ , which avoids the error of the linear terms and is well-studied in the literature of exponential integrators [25, 31]. Based on this insight, we propose fast solvers for diffusion ODEs, as detailed in the following sections. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 High-Order Solvers for Diffusion ODEs ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            308,
+            490,
+            324
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we propose high-order solvers for diffusion ODEs with convergence order guarantee by leveraging our proposed solution formulation Eq. (3.4). The proposed solvers and analysis are highly motivated by the methods of exponential integrators [25, 31] in the ODE literature. ",
+        "bbox": [
+            173,
+            333,
+            825,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Specifically, given an initial value $\\mathbf { \\nabla } _ { \\mathbf { x } _ { T } }$ at time $T$ and $M + 1$ time steps $\\{ t _ { i } \\} _ { i = 0 } ^ { M }$ decreasing from $t _ { 0 } = T$ to $t _ { M } = 0$ . Let $\\tilde { \\mathbf { x } } _ { t _ { 0 } } = \\mathbf { x } _ { T }$ be the initial value. The proposed solvers use $M$ steps to iteratively compute a sequence $\\{ \\tilde { { \\pmb { x } } } _ { t _ { i } } \\} _ { i = 0 } ^ { M }$ to approximate the true solutions at time steps $\\{ t _ { i } \\} _ { i = 0 } ^ { M }$ . In particular, the last iterate $\\tilde { \\boldsymbol { x } } _ { t _ { M } }$ approximates the true solution at time 0. ",
+        "bbox": [
+            173,
+            381,
+            825,
+            439
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In order to reduce the approximation error between $\\tilde { \\pmb { x } } _ { t _ { M } }$ and the true solution at time 0, we need to reduce the approximation error for each $\\tilde { \\mathbf { x } } _ { t _ { i } }$ at every step [30]. Starting with the previous value $\\tilde { \\pmb { x } } _ { t _ { i - 1 } }$ at time $t _ { i - 1 }$ , according to Eq. (3.4), the exact solution $\\pmb { x } _ { t _ { i - 1 }  t _ { i } }$ at time $t _ { i }$ is given by ",
+        "bbox": [
+            174,
+            444,
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+            487
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e941fd95cbf3e1c496583b82dd623ff65158438a8e8529e43b6a17ad2855df95.jpg",
+        "text": "$$\n\\pmb { x } _ { t _ { i - 1 }  t _ { i } } = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\pmb { x } } _ { t _ { i - 1 } } - \\alpha _ { t _ { i } } \\int _ { \\lambda _ { t _ { i - 1 } } } ^ { \\lambda _ { t _ { i } } } e ^ { - \\lambda } \\hat { \\pmb { \\epsilon } } _ { \\theta } ( \\hat { \\pmb { x } } _ { \\lambda } , \\lambda ) \\mathrm { d } \\lambda .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            492,
+            679,
+            532
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Therefore, to compute the value $\\tilde { \\boldsymbol { x } } _ { t _ { i } }$ for approximating $\\pmb { x } _ { t _ { i - 1 }  t _ { i } }$ , we need to approximate the exponentially weighted integral of $\\hat { \\epsilon } _ { \\theta }$ from $\\lambda _ { t _ { i - 1 } }$ to $\\lambda _ { t _ { i } }$ . Denote $h _ { i } : = \\lambda _ { t _ { i } } - \\lambda _ { t _ { i - 1 } }$ , and $\\hat { \\epsilon } _ { \\theta } ^ { ( n ) } ( \\hat { { \\mathbf x } } _ { \\lambda } , \\lambda ) \\mathrel { \\mathop : } =$ dnϵˆθ(xˆλ,λ)dλn as the n-th order total derivative of ϵˆθ(xˆλ, λ) w.r.t. λ. For k ≥ 1, the (k − 1)-th order Taylor expansion of $\\hat { \\epsilon } _ { \\theta } ( \\hat { \\pmb x } _ { \\lambda } , \\lambda )$ w.r.t. $\\lambda$ at $\\lambda _ { t _ { i - 1 } }$ is ",
+        "bbox": [
+            173,
+            536,
+            825,
+            603
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/976243e2861066edc99111ffc150a8141dda0494f026576eb3e22e90ae405d03.jpg",
+        "text": "$$\n\\hat { \\epsilon } _ { \\theta } ( \\hat { x } _ { \\lambda } , \\lambda ) = \\sum _ { n = 0 } ^ { k - 1 } \\frac { ( \\lambda - \\lambda _ { t _ { i - 1 } } ) ^ { n } } { n ! } \\hat { \\epsilon } _ { \\theta } ^ { ( n ) } ( \\hat { x } _ { \\lambda _ { t _ { i - 1 } } } , \\lambda _ { t _ { i - 1 } } ) + \\mathcal { O } ( ( \\lambda - \\lambda _ { t _ { i - 1 } } ) ^ { k } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            608,
+            732,
+            651
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Substituting the above Taylor expansion into Eq. (3.5) yields ",
+        "bbox": [
+            173,
+            655,
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+            670
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aa02a3d9e9670da70428ff51d8dccbfd33121e4f699e839306505e69a37a124c.jpg",
+        "text": "$$\n\\pmb { x } _ { t _ { i - 1 }  t _ { i } } = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\pmb { x } } _ { t _ { i - 1 } } - \\alpha _ { t _ { i } } \\sum _ { n = 0 } ^ { k - 1 } \\hat { \\pmb { \\epsilon } } _ { \\theta } ^ { ( n ) } ( \\hat { \\pmb { x } } _ { { \\pmb { \\lambda } } _ { t _ { i - 1 } } } , \\lambda _ { t _ { i - 1 } } ) \\int _ { \\lambda _ { t _ { i - 1 } } } ^ { \\lambda _ { t _ { i } } } e ^ { - \\lambda } \\frac { ( \\lambda - \\lambda _ { t _ { i - 1 } } ) ^ { n } } { n ! } \\mathrm { d } \\lambda + \\mathcal { O } ( h _ { i } ^ { k + 1 } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            674,
+            787,
+            718
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where the integral $\\begin{array} { r } { \\int e ^ { - \\lambda } \\frac { ( \\lambda - \\lambda _ { t _ { i - 1 } } ) ^ { n } } { n ! } \\mathrm { d } \\lambda } \\end{array}$ can be analytically computed by repeatedly applying $n$ times of integration-by-parts (see Appendix B.2). Therefore, to approximate $\\pmb { x } _ { t _ { i - 1 }  t _ { i } }$ , we only need to approximate the $n$ -th order total derivatives $\\hat { \\epsilon } _ { \\theta } ^ { ( n ) } ( \\hat { \\pmb { x } } _ { \\lambda } , \\lambda )$ for $n \\leq k - 1$ , which is a well-studied problem in the ODE literature [31, 32]. By dropping the $\\mathcal { O } ( h _ { i } ^ { k + 1 } )$ error term and approximating the first $( k - 1 )$ -th total derivatives with the “stiff order conditions” [31, 32], we can derive $k$ -th-order ODE solvers for diffusion ODEs. We name such solvers as DPM-Solver overall, and DPM-Solver- $k$ for a specific order $k$ . Here we take $k = 1$ for demonstration. In this case, Eq. (3.6) becomes ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e1a7a77f08fcf2344ebba70e31c391e6092d3781368a0af49fae27dbeb5b255e.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle { \\boldsymbol { x } } _ { t _ { i - 1 } \\to t _ { i } } = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } - \\alpha _ { t _ { i } } \\epsilon _ { \\theta } ( \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } , t _ { i - 1 } ) \\int _ { \\lambda _ { t _ { i - 1 } } } ^ { \\lambda _ { t _ { i } } } e ^ { - \\lambda } \\mathrm { d } \\lambda + \\mathcal { O } ( h _ { i } ^ { 2 } ) } \\\\ { \\displaystyle = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } - \\sigma _ { t _ { i } } ( e ^ { h _ { i } } - 1 ) \\epsilon _ { \\theta } ( \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } , t _ { i - 1 } ) + \\mathcal { O } ( h _ { i } ^ { 2 } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            837,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "By dropping the high-order error term $\\mathcal { O } ( h _ { i } ^ { 2 } )$ , we can obtain an approximation for $\\pmb { x } _ { t _ { i - 1 }  t _ { i } }$ . As $k = 1$ here, we call this solver DPM-Solver- $^ { l }$ , and the detailed algorithm is as following. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "DPM-Solver-1. Given an initial value $\\mathbf { \\nabla } _ { \\mathbf { x } _ { T } }$ and $M + 1$ time steps $\\{ t _ { i } \\} _ { i = 0 } ^ { M }$ decreasing from $t _ { 0 } = T$ to $t _ { M } = 0$ . Starting with $\\tilde { \\mathbf { x } } _ { t _ { 0 } } = \\mathbf { x } _ { T }$ , the sequence $\\{ \\tilde { { x } } _ { t _ { i } } \\} _ { i = 1 } ^ { M }$ is computed iteratively as follows: ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8deaa6e7603de31da2fd31418b2d224dd8ec375a261d169ca354b6f5b2802791.jpg",
+        "text": "$$\n\\tilde { \\boldsymbol { x } } _ { t _ { i } } = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } - \\sigma _ { t _ { i } } ( e ^ { h _ { i } } - 1 ) \\boldsymbol { \\epsilon } _ { \\boldsymbol { \\theta } } ( \\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } } , t _ { i - 1 } ) , \\quad \\mathrm { w h e r e ~ } h _ { i } = \\lambda _ { t _ { i } } - \\lambda _ { t _ { i - 1 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            243,
+            162,
+            753,
+            194
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For $k \\geq 2$ , approximating the first $k$ terms of the Taylor expansion needs additional intermediate points between $t$ and $s$ [31]. The derivation is more technical so we defer it to Appendix B. Below we propose algorithms for $k = 2 , 3$ and name them as DPM-Solver-2 and DPM-Solver-3, respectively. ",
+        "bbox": [
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+            207,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 DPM-Solver-2. ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            267,
+            362,
+            281
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Require: initial value $\\mathbf { \\nabla } _ { \\mathbf { \\mathcal { X } } T }$ , time steps $\\{ t _ { i } \\} _ { i = 0 } ^ { M }$ , model $\\epsilon _ { \\theta }$ ",
+        "bbox": [
+            174,
+            286,
+            544,
+            303
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 DPM-Solver-3. ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            462,
+            364,
+            476
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Require: initial value $\\mathbf { \\nabla } _ { \\mathbf { \\mathcal { X } } T }$ , time steps $\\{ t _ { i } \\} _ { i = 0 } ^ { M }$ , model $\\epsilon _ { \\theta }$ ",
+        "bbox": [
+            174,
+            482,
+            544,
+            498
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ad4609832a6419c85fb2132d9d91d2c95beec92e10bd4907c45b4ed7d6c517e6.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\quad _ { s 2 i - 1 }  _ { t _ { \\lambda } } ( \\overline { { \\lambda } } _ { t _ { i - 1 } } + r _ { 1 } h _ { i } ) , \\quad s _ { 2 i }  t _ { \\lambda } ( \\lambda _ { t _ { i - 1 } } + r _ { 2 } h _ { i } ) } \\\\ & { u _ { 2 i - 1 }  \\frac { \\alpha _ { s _ { 2 i - 1 } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { x } _ { t _ { i - 1 } } - \\sigma _ { s _ { 2 i - 1 } } ( e ^ { r _ { 1 } h _ { i } } - 1 ) \\epsilon _ { \\theta } ( \\tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) } \\\\ & { D _ { 2 i - 1 }  \\epsilon _ { \\theta } ( u _ { 2 i - 1 } , s _ { 2 i - 1 } ) - \\epsilon _ { \\theta } ( \\tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) } \\\\ & { u _ { 2 i }  \\frac { \\alpha _ { s _ { 2 i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { x } _ { t _ { i - 1 } } - \\sigma _ { s _ { 2 i } } ( e ^ { r _ { 2 } h _ { i } } - 1 ) \\epsilon _ { \\theta } ( \\tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) - \\frac { \\sigma _ { s _ { 2 i } } r _ { 2 } } { r _ { 1 } } ( \\frac { e ^ { r _ { 2 } h _ { i } } - 1 } { r _ { 2 } h _ { i } } - 1 ) D _ { 2 i - 1 } } \\\\ & { D _ { 2 i }  \\epsilon _ { \\theta } ( u _ { 2 i } , s _ { 2 i } ) - \\epsilon _ { \\theta } ( \\tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) } \\\\ & { \\tilde { x } _ { t _ { i } }  \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { x } _ { t _ { i - 1 } } - \\sigma _ { t _ { i } } ( e ^ { h _ { i } } - 1 ) \\epsilon _ { \\theta } ( \\tilde { x } _ { t _ { i - 1 } } , t _ { i - 1 } ) - \\frac { \\sigma _ { t _ { i } } } { r _ { 2 } } ( \\frac { e ^ { h _ { i } } - 1 } { h } - 1 ) D _ { 2 i } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            222,
+            522,
+            789,
+            645
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "10: return $\\tilde { \\pmb { x } } _ { t _ { M } }$ ",
+        "bbox": [
+            176,
+            655,
+            279,
+            670
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Here, $t _ { \\lambda } ( \\cdot )$ is the inverse function of $\\lambda ( t )$ , which has an analytical formulation for the practical noise schedule used in [2, 16], as shown in Appendix D. The chosen intermediate points are $( s _ { i } , \\pmb { u } _ { i } )$ for DPM-Solver-2 and $\\left( s _ { 2 i - 1 } , { \\pmb u } _ { 2 i - 1 } \\right)$ and $( s _ { 2 i } , { \\pmb u } _ { 2 i } )$ for DPM-Solver-3. As shown in the algorithm, DPM-Solver- $k$ requires $k$ function evaluations per step for $k = 1 , 2 , 3$ . Despite the more expensive steps, higher-order solvers $( k = 2 , 3$ ) are usually more efficient since they require much fewer steps to converge, due to their higher convergence order. We show that DPM-Solver- $k$ is $k$ -th-order solver, as stated in the following theorem. The proof is in Appendix B. ",
+        "bbox": [
+            173,
+            698,
+            826,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.2 (DPM-Solver- $k$ as a $k$ -th-order solver). Assume $\\epsilon _ { \\theta } ( x _ { t } , t )$ follows the regularity conditions detailed in Appendix B.1, then for $k = 1 , 2 , 3$ , DPM-Solver- $k$ is a $k$ -th order solver for diffusion ODEs, i.e., for the sequence $\\{ \\tilde { \\pmb { x } } _ { t _ { i } } \\} _ { i = 1 } ^ { M }$ computed by DPM-Solver- $k$ , the approximation error at time 0 satisfies $\\tilde { \\pmb { x } } _ { t _ { M } } - \\pmb { x } _ { 0 } = \\mathcal { O } ( h _ { \\operatorname* { m a x } } ^ { k } )$ , where $h _ { m a x } = \\mathrm { m a x } _ { 1 \\leq i \\leq M } ( \\lambda _ { t _ { i } } - \\lambda _ { t _ { i - 1 } } )$ . ",
+        "bbox": [
+            173,
+            800,
+            825,
+            861
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Finally, solvers with $k \\geq 4$ need much more intermediate points as shown by previous work [31, 32] for exponential integrators. Therefore, we only consider $k$ from 1 to 3 in this work, while leaving the solvers with higher $k$ for future study. ",
+        "bbox": [
+            174,
+            868,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3 Step Size Schedule ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            343,
+            106
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The proposed solvers in Sec. 3.2 need to specify the time steps $\\{ t _ { i } \\} _ { i = 0 } ^ { M }$ in advance. We propose two choices of the time step schedule. One choice is handcrafted, which is to uniformly split the interval $[ \\lambda _ { T } , \\lambda _ { 0 } ]$ , i.e. $\\begin{array} { r } { \\lambda _ { t _ { i } } = \\dot { \\lambda _ { T } } + \\frac { i } { M } ( \\lambda _ { 0 } - \\lambda _ { T } ) } \\end{array}$ , $i = 0 , \\ldots , M$ . Note that this is different from previous work [2, 3] which chooses uniform steps for $t _ { i }$ . Empirically, DPM-Solver with uniform time steps $\\lambda _ { t _ { i } }$ can already generate quite good samples in few steps, where results are listed in Appendix E. As the other choice, we propose an adaptive step size algorithm, which dynamically adjusts the step size by combining different orders of DPM-Solver. The adaptive algorithm is inspired by [20] and we defer its implementation details to Appendix C. ",
+        "bbox": [
+            173,
+            116,
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+            228
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For few-step sampling, we need to use up all the number of function evaluations (NFE). When the NFE is not divisible by 3, we firstly apply DPM-Solver-3 as much as possible, and then add a single step of DPM-Solver-1 or DPM-Solver-2 (dependent on the reminder of $K$ divided by 3), as detailed in Appendix D. In the subsequent experiments, we use such combination of solvers with the uniform step size schedule for $\\mathrm { N F E } \\leq 2 0$ , and otherwise the adaptive step size schedule. ",
+        "bbox": [
+            174,
+            233,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.4 Sampling from Discrete-Time DPMs ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            320,
+            467,
+            335
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Discrete-time DPMs [2] train the noise prediction model at $N$ fixed time steps $\\{ t _ { n } \\} _ { n = 1 } ^ { N }$ , and the noise prediction model is parameterized by $\\tilde { \\epsilon } _ { \\theta } ( { \\boldsymbol x } _ { n } , n )$ for $n = 0 , \\ldots , N - 1$ , where each ${ \\pmb x } _ { n }$ is corresponding to the value at time $t _ { n + 1 }$ . We can transform the discrete-time noise prediction model to the continuous version by letting $\\begin{array} { r } { \\epsilon _ { \\theta } ( x , t ) : = \\tilde { \\epsilon } _ { \\theta } ( x , \\frac { ( N - 1 ) t } { T } ) } \\end{array}$ , for all $\\pmb { x } \\in \\mathbb { R } ^ { d } , t \\in [ 0 , T ]$ . Note that the input time of $\\tilde { \\epsilon } _ { \\theta }$ may not be integers, but we find that the noise prediction model can still work well, and we hypothesize that it is because of the smooth time embeddings (e.g., position embeddings [2]). By such reparameterization, the noise prediction model can adopt the continuous-time steps as input, and thus we can also use DPM-Solver for fast sampling. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Comparison with Existing Fast Sampling Methods ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Here, we discuss the relationship and highlight the difference between DPM-Solver and existing ODE-based fast sampling methods for DPMs. We further briefly discuss the advantage of training-free samplers over those training-based ones. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 DDIM as DPM-Solver-1 ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Denoising Diffusion Implicit Models (DDIM) [19] design a deterministic method for fast sampling from DPMs. For two adjacent time steps $t _ { i - 1 }$ and $t _ { i }$ , assume that we have a solution $\\tilde { \\boldsymbol { x } } _ { t _ { i - 1 } }$ at time $t _ { i - 1 }$ , then a single step of DDIM from time $t _ { i - 1 }$ to time $t _ { i }$ is ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3ad93903d54c31c8256e52d597943286bbf9fba385901a6d278538f4eb730eed.jpg",
+        "text": "$$\n\\tilde { \\pmb { x } } _ { t _ { i } } = \\frac { \\alpha _ { t _ { i } } } { \\alpha _ { t _ { i - 1 } } } \\tilde { \\pmb { x } } _ { t _ { i - 1 } } - \\alpha _ { t _ { i } } \\left( \\frac { \\sigma _ { t _ { i - 1 } } } { \\alpha _ { t _ { i - 1 } } } - \\frac { \\sigma _ { t _ { i } } } { \\alpha _ { t _ { i } } } \\right) \\epsilon _ { \\theta } ( \\tilde { \\pmb { x } } _ { t _ { i - 1 } } , t _ { i - 1 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            307,
+            646,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Although motivated by entirely different perspectives, we show that the updates of DPM-Solver-1 and Denoising Diffusion Implicit Models (DDIM) [19] are identical. By the definition of $\\lambda$ , we have $\\frac { \\sigma _ { t _ { i - 1 } } } { \\alpha _ { t _ { i - 1 } } } = e ^ { - \\lambda _ { t _ { i - 1 } } ^ { - } }$ and $\\begin{array} { r } { \\frac { \\sigma _ { t _ { i } } } { \\alpha _ { t _ { i } } } = e ^ { - \\lambda _ { t _ { i } } } } \\end{array}$ . Plugging these and $h _ { i } = \\lambda _ { t _ { i } } - \\lambda _ { t _ { i - 1 } }$ to Eq. (4.1) results in exactly a step of DPM-Solver-1 in Eq. (3.7). However, the semi-linear ODE formulation of DPM-Solver allows for principled generalization to higher-order solvers and convergence order analysis. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Recent work [13] also show that DDIM is a first-order discretization of diffusion ODEs by differentiating both sides of Eq. (4.1). However, they cannot explain the difference between DDIM and the first-order Euler discretization of diffusion ODEs. In contrast, by showing that DDIM is a special case of DPM-Solver, we reveal that DDIM makes full use of the semi-linearity of diffusion ODEs, which explains its superiority over traditional Euler methods. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 Comparison with Traditional Runge-Kutta Methods ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "One can obtain a high-order solver by directly applying traditional explicit Runge-Kutta (RK) methods to the diffusion ODE in Eq. (2.7). Specifically, RK methods write the solution of Eq. (2.7) in the ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/114b9edc4cc71bed7f9d053944dde80589018ddf790160907d7334f50b5c35fb.jpg",
+        "table_caption": [
+            "Table 1: FID ↓ on CIFAR-10 for different orders of Runge-Kutta (RK) methods and DPM-Solvers, varying the number of function evaluations (NFE). For RK methods, we evaluate diffusion ODEs w.r.t. both $t$ (Eq. (2.7)) and $\\lambda$ (Eq. (E.1)). We use uniform step size in $t$ for RK (t), and uniform step size in $\\lambda$ for RK $( \\lambda )$ and DPM-Solvers. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Sampling method\\NFE</td><td>12</td><td>18</td><td>24</td><td>30</td><td>36</td><td>42</td><td>48</td></tr><tr><td>RK2 (t)</td><td>16.40</td><td>7.25</td><td>3.90</td><td>3.63</td><td>3.58</td><td>3.59</td><td>3.54</td></tr><tr><td>RK2(入)</td><td>107.81</td><td>42.04</td><td>17.71</td><td>7.65</td><td>4.62</td><td>3.58</td><td>3.17</td></tr><tr><td>DPM-Solver-2</td><td>5.28</td><td>3.43</td><td>3.02</td><td>2.85</td><td>2.78</td><td>2.72</td><td>2.69</td></tr><tr><td>RK3 (t)</td><td>48.75</td><td>21.86</td><td>10.90</td><td>6.96</td><td>5.22</td><td>4.56</td><td>4.12</td></tr><tr><td>RK3 (入)</td><td>34.29</td><td>4.90</td><td>3.50</td><td>3.03</td><td>2.85</td><td>2.74</td><td>2.69</td></tr><tr><td>DPM-Solver-3</td><td>6.03</td><td>2.90</td><td>2.75</td><td>2.70</td><td>2.67</td><td>2.65</td><td>2.65</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "following integral form: ",
+        "bbox": [
+            174,
+            275,
+            333,
+            290
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dbc1e1d3bec4cc742d44654340bd9da788aefebc8fc4b4e2a9e38d739f018ceb.jpg",
+        "text": "$$\n{ \\bf { x } } _ { t } = { \\bf { x } } _ { s } + \\int _ { s } ^ { t } h _ { \\theta } ( { \\bf { x } } _ { \\tau } , \\tau ) \\mathrm { { d } } \\tau = { \\bf { x } } _ { s } + \\int _ { s } ^ { t } \\left( f ( \\tau ) { \\bf { x } } _ { \\tau } + \\frac { g ^ { 2 } ( \\tau ) } { 2 \\sigma _ { \\tau } } \\epsilon _ { \\theta } ( { \\bf { x } } _ { \\tau } , \\tau ) \\right) \\mathrm { { d } } \\tau ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "and use some intermediate time steps between $[ t , s ]$ and combine the evaluations of $h _ { \\theta }$ at these time steps to approximate the whole integral. The approximation error of explicit RK methods depends on $h _ { \\theta }$ , which consists of the error corresponding to both the linear term $f ( \\tau ) x _ { \\tau }$ and the nonlinear noise prediction model $\\epsilon _ { \\theta }$ . However, the error of the linear term may increase exponentially because the exact solution of the linear term has an exponential coefficient (as shown in Eq. (3.1)). There are many empirical evidence [25, 31] showing that directly using explicit RK methods for semi-linear ODEs may suffer from unstable numerical issues for large step size. We also demonstrate the empirical difference of the proposed DPM-Solver and the traditional explicit RK methods in Sec. 5.1, which shows that DPM-Solver have smaller discretization errors than the RK methods with the same order. ",
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+        "type": "text",
+        "text": "4.3 Training-based Fast Sampling Methods for DPMs ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Samplers that need extra training or optimization include knowledge distillation [13, 14], learning the noise level or variance [15, 16, 33], and learning the noise schedule or sample trajectory [17, 18]. Although the progressive distillation method [13] can obtain a fast sampler within 4 steps, it needs further training costs and loses part of the information in the original DPM (e.g., after distillation, the noise prediction model cannot predict the noise (score function) at every time step between $[ 0 , T ] )$ . In contrast, training-free samplers can keep all the information of the original model, and thereby can be directly extended to the conditional sampling by combining the original model and an external classifier [4] (e.g. see Appendix D for the conditional sampling with classifier guidance). ",
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+        "type": "text",
+        "text": "Beyond directly designing fast samplers for DPMs, several works also propose novel types of DPMs which supports faster sampling. For instance, defining a low-dimensional latent variable for DPMs [34]; designing special diffusion processes with bounded score functions [35]; combining GANs with the reverse process of DPMs [36]. The proposed DPM-Solver may also be suitable for accelerating the sampling of these DPMs, and we leave them for future work. ",
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+        "text": "5 Experiments ",
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+        "text": "In this section, we show that as a training-free sampler, DPM-Solver can greatly speedup the sampling of existing pre-trained DPMs, including both continuous-time and discrete-time ones, with both linear noise schedule [2, 19] and cosine noise schedule [16]. We vary different number of function evaluations (NFE) which is the number of calls to the noise prediction model $\\epsilon _ { \\theta } ( x _ { t } , t )$ , and compare the sample quality between DPM-Solver and other methods. For each experiment, We draw 50K samples and use the widely adopted FID score [37] to evaluate the sample quality, where lower FID usually implies better sample quality. ",
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+        "text": "Unless explicitly mentioned, we always use the solver combination with the uniform step size schedule in Sec. 3.3 if the NFE budget is less than 20, and otherwise the DPM-Solver-3 with the adaptive step size schedule in Sec. 3.3. We refer to Appendix D for other implementation details of DPM-Solver and Appendix E for detailed settings. ",
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+        "image_caption": [
+            "Figure 2: Sample quality measured by FID $\\downarrow$ of different sampling methods for DPMs on CIFAR-10 with both continuous-time and discrete-time models, CelebA 64x64, ImageNet 64x64, ImageNet $1 2 8 \\mathrm { x } 1 2 8$ and LSUN bedroom $2 5 6 \\times 2 5 6$ with discrete-time models, varying the number of function evaluations (NFE). The method $^ { \\dag } { \\bf G } { \\bf G } { \\bf D } { \\bf M }$ [18] needs extra training to optimize the sample trajectory, while other methods are training-free. To get the strongest baseline, we use the quadratic step size for DDIM on CelebA, which has a better FID than that of the uniform step size in the original paper [19]. "
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+        "text": "5.1 Comparison with Continuous-Time Sampling Methods ",
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+        "text": "We firstly compare DPM-Solver with other continuous-time sampling methods for DPMs. The compared methods include the Euler-Maruyama discretization for diffusion SDEs [3], the adaptive step size solver for diffusion SDEs [20] and the RK methods for diffusion ODEs [3, 28] in Eq. (2.7). We compare these methods for sampling from a pre-trained continuous-time “VP deep” model [3] on the CIFAR-10 dataset [29] with the linear noise schedule. ",
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+        "text": "Fig. 2a shows the efficiency of compared solvers. We use uniform time steps with 50, 200, 1000 NFE for the diffusion SDE with Euler discretization, and vary the tolerance hyperparameter [3, 20] for the adaptive step size SDE solver [20] and RK45 ODE solver [28] to control the NFE. DPM-Solver can generate good sample quality within around 10 NFE, while other solvers have large discretization error even in 50 NFE, which shows that DPM-Solver can achieve ${ \\sim } 5 $ speedup of the previous best solver. In particular, we achieve 4.70 FID with 10 NFE, 3.75 FID with 12 NFE, 3.24 FID with 15 NFE, and 2.87 FID with 20 NFE, which is the fastest sampler on CIFAR-10. ",
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+        "text": "As an ablation study, we also compare the second-order and third-order DPM-Solver and RK methods, as shown in Table 1. We compare RK methods for diffusion ODEs w.r.t. both time $t$ in Eq. (2.7) and half-log-SNR $\\lambda$ by applying change-of-variable (see detailed formulations in Appendix E.1). The results show that given the same NFE, the sample quality of DPM-Solver is consistently better than RK methods with the same order. The superior efficiency of DPM-Solver is particularly evident in the few-step regime under 15 NFE, where RK methods have rather large discretization errors. This is mainly because DPM-Solver analytically computes the linear term, avoiding the corresponding discretization error. Besides, the higher order DPM-Solver-3 converges faster than DPM-Solver-2, which matches the order analysis in Theorem 3.2. ",
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+        "text": "5.2 Comparison with Discrete-Time Sampling Methods ",
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+    {
+        "type": "text",
+        "text": "We use the method in Sec. 3.4 for using DPM-Solver in discrete-time DPMs, and then compare DPM-Solver with other discrete-time training-free samplers, including DDPM [2], DDIM [19], Analytic-DDPM [21], Analytic-DDIM [21], PNDM [22], FastDPM [38] and Itô-Taylor [24]. We also compare with GGDM [18], which uses the same pre-trained model but needs further training for the sampling trajectory. We compare the sample quality by varying NFE from 10 to 1000. ",
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+    {
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+        "text": "Specifically, we use the discrete-time model trained by $L _ { \\mathrm { s i m p l e } }$ in [2] on the CIFAR-10 dataset with linear noise schedule; the discrete-time model in [19] on CelebA 64x64 [39] with linear noise schedule; the discrete-time model trained by $L _ { \\mathrm { h y b r i d } }$ in [16] on ImageNet 64x64 [26] with cosine noise schedule; the discrete-time model with classifier guidance in [4] on ImageNet 128x128 [26] with linear noise schedule; the discrete-time model in [4] on LSUN bedroom $2 5 6 \\times 2 5 6$ [40] with linear noise schedule. For the models trained on ImageNet, we only use their “mean” model and omit the “variance” model. As shown in Fig. 2, on all datasets, DPM-Solver can obtain reasonable samples within 12 steps (FID 4.65 on CIFAR-10, FID 3.71 on CelebA 64x64 and FID 19.97 on ImageNet 64x64, FID 4.08 on ImageNet $1 2 8 \\mathbf { x } 1 2 8 _ { \\rho }$ ), which is $4 \\sim 1 6 \\times$ faster than the previous fastest training-free sampler. DPM-Solver even outperforms GGDM, which requires additional training. ",
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+    {
+        "type": "text",
+        "text": "6 Conclusions ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We tackle the problem of fast and training-free sampling from DPMs. We propose DPM-Solver, a fast dedicated training-free solver of diffusion ODEs for fast sampling of DPMs in around 10 steps of function evaluations. DPM-Solver leverages the semi-linearity of diffusion ODEs and it directly approximates a simplified formulation of exact solutions of diffusion ODEs, which consists of an exponentially weighted integral of the noise prediction model. Inspired by numerical methods for exponential integrators, we propose first-order, second-order and third-order DPMSolver to approximate the exponentially weighted integral of noise prediction models with theoretical convergence guarantee. We propose both handcrafted and adaptive step size schedule, and apply DPM-Solver for both continuous-time and discrete-time DPMs. Our experimental results show that DPM-Solver can generate high-quality samples in around 10 function evaluations on various datasets, and it can achieve $4 \\sim 1 6 \\times$ speedup compared with previous state-of-the-art training-free samplers. ",
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+    {
+        "type": "text",
+        "text": "Limitations and broader impact Despite the promising speedup performance, DPM-Solver is designed for fast sampling, which may be not suitable for accelerating the likelihood evaluations of DPMs. Besides, compared to the commonly-used GANs, diffusion models with DPM-Solver are still not fast enough for real-time applications. In addition, like other deep generative models, DPMs may be used to generate adverse fake contents, and the proposed solver may further amplify the potential undesirable influence of deep generative models for malicious applications. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This work was supported by National Key Research and Development Project of China (No. 2021ZD0110502); NSF of China Projects (Nos. 62061136001, 61620106010, 62076145, U19B2034, U1811461, U19A2081, 6197222, 62106120); Beijing NSF Project (No. JQ19016); Beijing Outstanding Young Scientist Program NO. BJJWZYJH012019100020098; a grant from Tsinghua Institute for Guo Qiang; the NVIDIA NVAIL Program with GPU/DGX Acceleration; the High Performance Computing Center, Tsinghua University; the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (22XNKJ13). J.Z is also supported by the XPlorer Prize. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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Ermon, “Maximum likelihood training of score-based diffusion models,” in Advances in Neural Information Processing Systems, vol. 34, 2021, pp. 1415–1428.   \n[42] K. Yang, J. Yau, L. Fei-Fei, J. Deng, and O. Russakovsky, “A study of face obfuscation in ImageNet,” arXiv preprint arXiv:2103.06191, 2021. ",
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+        "text": "1. For all authors... ",
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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] See section 6.   \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] See section 6.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+        "text": "2. If you are including theoretical results... ",
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+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Appendix B. (b) Did you include complete proofs of all theoretical results? [Yes] See Appendix B. ",
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+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Code is attached in the supplemental materials, with the appendix.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Our method is training-free. But we also report the hyperparameters for evaluations used in our proposed solver.   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We observe that the standard deviation of the FID evaluations of DPM-Solver are rather small (mainly less than 0.01) because the FID is already averaged over 50K samples, following existing work [18, 20, 21]. The small standard deviation does not change the conclusion.   \n(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] The GPU type and amount is detailed in Appendix E. ",
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+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes]   \n(b) Did you mention the license of the assets? [Yes] See Appendix E   \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We include our code in the supplemental materials.   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No] All of the datasets used in the experiments are publicly available.   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] We mentioned the human privacy issues of the ImageNet dataset in Appendix E. ",
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+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
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+# LOCAL AUGMENTATION FOR GRAPH NEURAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Data augmentation has been widely used in image data and linguistic data but remains under-explored for Graph Neural Networks (GNNs). Existing methods focus on augmenting the graph data from a global perspective and largely fall into two genres: structural manipulation and adversarial training with feature noise injection. However, recent graph data augmentation methods ignore the importance of local information for the GNNs’ message passing mechanism. In this work, we introduce the local augmentation, which enhances the locality of node representations by their subgraph structures. Specifically, we model the data augmentation as a feature generation process. Given a node’s features, our local augmentation approach learns the conditional distribution of its neighbors’ features and generates more neighbors’ features to boost the performance of downstream tasks. Based on the local augmentation, we further design a novel framework: LA-GNN, which can apply to any GNN models in a plug-and-play manner. Extensive experiments and analyses show that local augmentation consistently yields performance improvement for various GNN architectures across a diverse set of benchmarks.
+
+# 1 INTRODUCTION
+
+Graph Neural Networks (GNNs) and their variants (Abu-El-Haija et al., 2019; Kipf & Welling, 2017; Velickovi ˇ c et al., 2018) have achieved state-of-the-art performance for many tasks on graphs such as ´ recommendation system (Ying et al., 2018) and traffic prediction (Guo et al., 2019). However, most of the GNN models, such as GCN (Kipf & Welling, 2017) and GAT (Velickovi ˇ c et al., 2018), learn ´ the node representations by aggregating information over only the 2-hop neighborhood. Such shallow architectures limit their ability to extract information from higher-layer neighborhoods (Wang & Derr, 2021). But deep GNNs are prone to over-smoothing (Li et al., 2018), which suggests the node representations tend to converge to a certain vector and thus become indistinguishable. One solution to address this problem is to preserve the locality of node representations when increasing the number of layers. For example, JKNet (Xu et al., 2018) densely connects (Huang et al., 2017) each hidden layer to the final layer. GCNII (Chen et al., 2020) employs an initial residual to construct a skip connection from the input layer. Besides, Zeng et al. (2021) pointed out that the key for GNN is to smooth the local neighborhood into informative representation, no matter how deep it is. And they decouple the depth and scope of GNNs to help capture local graph structure. Prior works have emphasized the importance of local information, but one property of the graph is that the number of nodes in the local neighborhood is far fewer than higher-order neighbors. And this property limits the expressive power of GNNs due to the limited neighbors in the local structure. A very intuitive idea is to use data augmentation to increase the number of nodes in the local substructure.
+
+However, existing graph data augmentation methods ignore the importance of local information and only perturb at the topology-level and feature-level from a global perspective, which can be divided into two categories: topology-level augmentation (Rong et al., 2020; Wang et al., 2020b; Zhao et al., 2021) and feature-level augmentation (Deng et al., 2019; Feng et al., 2019; Kong et al., 2020). Topology-level augmentation perturbs the adjacency matrix, yielding different graph structures. On the other hand, existing feature-level augmentation mainly exploits perturbation of node attributes guided by adversarial training (Deng et al., 2019; Feng et al., 2019; Kong et al., 2020). These augmentation techniques have two drawbacks. 1) Some of they employ full-batch training for augmentation, which is computationally expensive, and introduce some additional side effects such as over-smoothing. 2) The type of feature-level augmentation is coarse-grained, which focuses on global augmentation and overlooks the local information of the neighborhood. Moreover, to our best knowledge, none of the existing approaches combines both the feature representations and the graph topology, especially the local subgraph structures, for graph-level data augmentation.
+
+In this work, we propose a framework: Local Augmentation for Graph Neural Networks (LA-GNNs), to further enhance the locality of node representations based on both the topology-level and featurelevel information in the substructure. The term "local augmentation" refers to the generation of neighborhood features via a generative model conditioned on local structures and node features. Specifically, our proposed framework learns the conditional distribution of the connected neighbors’ representations given the representation of the central node, bearing some similarities with the Skipgram (Mikolov et al., 2013) and Deepwalk Perozzi et al. (2014), with the difference that our method does not base on word or graph embedding.
+
+The motivation behind this work concludes three-fold. 1) Existing feature-level augmentation works primarily pay attention to global augmentation without considering the informative neighborhood. 2) The distributions of the representations of the neighbors are closely connected to the central node, making ample room for feature augmentation. 3) Preserving the locality of node representations is key to avoiding over-smoothing $\mathrm { { X u } }$ et al., 2018; Klicpera et al., 2019; Chen et al., 2020). And there are several benefits in applying local augmentation for the GNN training. First, local augmentation is essentially a data augmentation technique that can improve the generalization of the GNN models and prevent over-fitting. Second, we can recover some missing contextual information of the local neighborhood in an attributed graph via the generative model (Jia & Benson, 2020). Third, our proposed framework is flexible and can be applied to various popular backbone networks such as GCN (Kipf & Welling, 2017), GAT (Velickovi ˇ c et al., 2018), GCNII (Chen et al., 2020), and ´ GRAND (Feng et al., 2020) to enhance their performance. Extensive experimental results demonstrate that our proposed framework could improve the performance of GNN variants on 7 benchmark datasets.
+
+# 2 BACKGROUND
+
+Notations. Let $G = ( V , E )$ represent the graph, where $V$ is the set of vertices $\{ v _ { 1 } , \cdots , v _ { N } \}$ with $| V | = N$ and $E$ is the set of edges. The adjacency matrix is defined as $\mathbf { A } \in \{ 0 , 1 \} ^ { N \times N }$ , and nod $\mathbf { A } _ { i j } = 1$ f and only if denote the d $( v _ { i } , v _ { j } ) \in E$ . Let ee ma $\mathcal { N } _ { i } \overset { \cdot } { = } \{ v _ { j } \vert  \mathbf { A } _ { i j } = 1 \}$ the neighborhood of. The feature matrix $v _ { i }$ $\mathbf { D }$ $\begin{array} { r } { \dot { \bf D } _ { i i } = \dot { \sum } _ { j = 1 } ^ { n } { \bf A } _ { i j } } \end{array}$ is denoted as $\mathbf { X } \in \mathbb { R } ^ { N \times F }$ where each node $v$ is associated with a $F$ -dimensional feature vector $\mathbf { X } _ { v }$ . $\mathbf { Y } \in \{ 0 , 1 \} ^ { N \times C }$ denote the one-hot label matrix, where $\mathbf { Y } _ { i } \in \{ 0 , 1 \} ^ { C }$ is a one-hot vector and $\begin{array} { r } { \sum _ { j = 1 } ^ { C } \mathbf { Y } _ { i j } = 1 } \end{array}$ for any $v _ { i } \in V$ .
+
+GNN. Graph Neural Network (GNN) is a type of neural network that directly operates on the graph structure, such as GCN and GAT (Kipf & Welling, 2017; Velickovi ˇ c et al., 2018), that capture the ´ dependence of graphs via message passing between the nodes of a graph as
+
+$$
+\mathbf { H } ^ { ( \ell ) } = f ( \mathbf { A } , \mathbf { H } ^ { ( \ell - 1 ) } ) ,
+$$
+
+where $f$ denotes the specific GNN layer for different models, $\mathbf { H } ^ { ( \ell ) }$ are the hidden vectors of the $\ell$ -th layer and $\mathbf { H } ^ { ( 0 ) } = \mathbf { X }$ . For example, $\dot { f ( \mathbf { A } , \mathbf { H } ) } = \sigma ( \hat { \mathbf { A } } \mathbf { H } \mathbf { W } )$ for GCN, where $\hat { \mathbf { A } } = \tilde { \mathbf { D } } ^ { - \frac { 1 } { 2 } } \tilde { \mathbf { A } } \tilde { \mathbf { D } } ^ { - \frac { 1 } { 2 } } ,$ $\tilde { \bf D }$ is the degree matrix of $\tilde { \mathbf { A } }$ , i.e., $\begin{array} { r } { \tilde { \bf D } _ { i i } = \sum _ { j } \tilde { \bf A } _ { i j } } \end{array}$ , and $\tilde { \mathbf { A } } = \mathbf { A } + \mathbf { I }$ .
+
+Topology-level Augmentation. Topology-level augmentation usually perturbs $\mathbf { A }$ to generate different graph structures, which can be formulated as $\mathbf { A } ^ { \prime } = \mathcal { F } ( \mathbf { A } , \mathbf { X } )$ , where $\mathcal F ( \cdot )$ is a structure perturbation function. For example, DropEdge (Rong et al., 2020) considers $\mathcal { F } ( \mathbf { A } , \mathbf { X } ) = \mathbf { A } - \mathbf { A _ { s } }$ which is independent of $\mathbf { X }$ , where $\mathbf { A _ { s } }$ is a sparse matrix consists of a subset of the original edges $E$ . GAUG-O (Zhao et al., 2021) leverages their proposed neural edge predictors to produce a different structure $\mathbf { A } ^ { \prime }$ where $\begin{array} { r } { \mathbf { A } _ { i j } ^ { \prime } = \left\lfloor \frac { 1 } { 1 + e ^ { - \left( \log \mathbf { P } _ { i j } + G \right) / \tau } } + \frac { 1 } { 2 } \right\rfloor } \end{array}$ , $\mathbf { P } _ { i j } = \alpha \mathbf { M } _ { i j } + ( 1 - \alpha ) \mathbf { A } _ { i j }$ , $\mathbf { M } = { \boldsymbol { \sigma } } \left( \mathbf { Z } \mathbf { Z } ^ { T } \right)$ , $\mathbf { Z } = f \left( \mathbf { A } , f ( \mathbf { A } , \mathbf { X } ) \right)$ , $\tau$ is the temperature of Gumbel-Softmax distribution, $G \sim { \mathrm { G u m b e l } } ( 0 , 1 )$ is a Gumbel random variate, and $\alpha$ is a hyperparameter mediating the influence of edge predictor on the original graph.
+
+# Feature-level Augmentation.
+
+Besides, feature-level augmentation function can be defines as $\mathbf { X } ^ { \prime } \ = \ \mathcal { H } ( \mathbf { A } , \mathbf { X } )$ , where $\mathcal { H } ( \cdot )$ is a feature perturbation function. FLAG (Kong et al., 2020) defines the perturbation function as $\begin{array} { r } { \mathcal { H } ( \mathbf { A } , \mathbf { X } ) = \textbf { X } + \boldsymbol { \delta } } \end{array}$ where
+
+Table 1: Comparison of existing graph data augmentation.   
+
+<table><tr><td colspan="4">GraphData Augmentation</td></tr><tr><td>Method</td><td>ConsideredPart</td><td>Type</td><td>Perturbed Part</td></tr><tr><td>DropEdge</td><td>A</td><td>Sampling</td><td>A</td></tr><tr><td>GAUG-O</td><td>A&amp;X</td><td>Reconstruction</td><td></td></tr><tr><td>FLAG</td><td>X</td><td>Noise Injection</td><td></td></tr><tr><td>G-GCN</td><td>A&amp;X</td><td>Reconstruction</td><td>AXX</td></tr><tr><td>Local Augmentation</td><td>A&amp;X</td><td>Generation</td><td>X</td></tr></table>
+
+perturbation $\pmb { \delta }$ is updated iteratively during the adversarial training phase. G-GCN (plain) (Zhu et al., 2020) obtains the global attribute feature matrix $\mathbf { X } ^ { ( a ) } \in \bar { \mathbb { R } ^ { N \times d _ { a } } }$ through minimizing the objective Qv∈V Qa∈CA(v) v a Pk∈U expX(a)v ·Vk where $U$ is the set of all attributes, $C A ( v )$ is the sampled context attributes of $v$ , and $\mathbf { V } \in \mathbb { R } ^ { d _ { a } \times F }$ denotes the parameters. Obviously, the perturbation function of G-GCN has no close-form solution. In this work, we propose a novel feature-level augmentation method, named local augmentation. And the comparison of the details of various graph data augmentation techniques can be found in Table 1.
+
+# 3 LOCAL AUGMENTATION
+
+In this section, we describe details of the proposed method. The local augmentation framework consists of three modules: learning the conditional distribution via a generative model, the active learning trick, and the downstream GNN models, as illustrated in Figure 1. Note that the proposed algorithm enhances the locality of node representations through augmenting 1-hop neighbors in a generative way. Specifically, we exploit a generative model to learn the conditional distribution of the connected neighbors’ representations given the representation of a node. We describe the details of learning the conditional distribution and the motivation for why local augmentation is able to improve the performance in a probabilistic view in Sec. 3.1, detail the architecture of downstream GNN models in Sec. 3.2. We finally elaborate the training procedure of both the generative model and the downstream GNN models with the active learning trick in Sec. 3.3.
+
+![](images/a25e0a452d09cb122f2cc05b960b1e984bdab4b6769823194b1ac823b6b03d23.jpg)  
+Figure 1: A schematic depiction of our local augmentation. The purple and yellow circles on the graph correspond to the central node and its augmented neighbors respectively. After augmenting the neighborhood, we exploit the initial and the generated feature matrix as input for downstream GNNs.
+
+# 3.1 LEARNING THE CONDITIONAL DISTRIBUTION
+
+We start by reviewing the semi-supervised learning of GNNs in a probabilistic view. Most existing GNN models (Kipf & Welling, 2017; Velickovi ˇ c et al., 2018) are viewed as a classification function ´ to predict the class labels of the graph nodes. In this work, we use a GNN classification estimator $P _ { \theta } ( \mathbf { Y } | \mathbf { A } , \mathbf { X } )$ $\theta$ is the parameter) to model the conditional distribution of label $\mathbf { Y }$ with respect to the graph structure A and feature matrix X. Given training samples $\{ \mathbf { A } , \mathbf { X } , \mathbf { Y } \}$ , the parameter $\theta$ can be estimated using Maximum Likelihood Estimation (MLE), by optimizing the following likelihood function:
+
+$$
+\operatorname* { m a x } \prod _ { k \in \mathbf { K } } P _ { \theta } \left( \mathbf { Y } _ { k } | \mathbf { A } , \mathbf { X } \right) ,
+$$
+
+where $\mathbf { K }$ is the set of node indices of the training dataset whose labels are visible during the semi-supervised training. To further boost the performance of GNN, we introduce a new model $P _ { \theta } ( { \bf Y } , \overline { { { \bf X } } } | { \bf A } , { \bf X } )$ , where $\overline { { \mathbf { X } } }$ is generated features by feature-level augmentation. For this model, the MLE method needs to optimize a marginalized probability $P _ { \theta }$ over the generated feature matrix $\overline { { \mathbf { X } } }$ :
+
+$$
+\operatorname* { m a x } \prod _ { k \in \mathbf { K } } \int _ { \overline { { \mathbf { X } } } } P _ { \theta } \left( \mathbf { Y } _ { k } , \overline { { \mathbf { X } } } | \mathbf { A } , \mathbf { X } \right) .
+$$
+
+For Bayesian tractability, we decompose $P _ { \theta }$ in Eq.(3) as a product of two posterior probabilities:
+
+$$
+\begin{array} { r } { P _ { \theta , \phi } ( \mathbf { Y } _ { k } , \overline { { \mathbf { X } } } | \mathbf { A } , \mathbf { X } ) : = P _ { \theta } ( \mathbf { Y } _ { k } | \mathbf { A } , \mathbf { X } , \overline { { \mathbf { X } } } ) Q _ { \phi } ( \overline { { \mathbf { X } } } | \mathbf { A } , \mathbf { X } ) , } \end{array}
+$$
+
+where $P _ { \theta } ( \mathbf { Y } _ { k } | \mathbf { A } , \mathbf { X } , \mathbf { \overline { { X } } } )$ and $Q _ { \phi } ( { \overline { { \mathbf { X } } } } | \mathbf { A } , \mathbf { X } )$ denote the probabilistic distributions approximated by the downstream GNN and the (feature-level augmentation) generator respectively, parameterized by $\theta$ and $\phi$ . There are two benefits in the decomposition above. First, it allows us to decouple the training of the downstream predictor $P _ { \theta }$ and the generator $Q _ { \phi }$ , enabling the generator to easily generalize to other downstream tasks. Moreover, inspired by the successes of data augmentation via deep-learning-based generative modeling (Antoniou et al., 2017), the representation power of Eq.(4) is superior than that of a single predictor $P _ { \theta } \left( \mathbf { Y } _ { k } | \mathbf { A } , \mathbf { X } \right)$ without data augmentation.
+
+Consequently, once a generator $Q _ { \phi }$ is trained very well, our training procedure can optimize $P _ { \theta } ( \mathbf { Y } _ { k } | \mathbf { A } , \mathbf { X } , \mathbf { \overline { { X } } } )$ with samples $\overline { { \mathbf { X } } }$ drawn from the fixed conditional distribution $Q _ { \phi }$ . Now, we show how to train the generator as follows.
+
+Generator To learn a feature augmentation generator, a naive solution is to learn one single distribution for all the neighbors using the MLE method, i.e., solving the following optimization problem
+
+$$
+\operatorname* { m a x } _ { \psi } \sum _ { j \in \mathcal { N } _ { i } } \log p _ { \psi } \left( \mathbf { X } _ { j } | \mathbf { X } _ { i } \right) = \operatorname* { m a x } _ { \psi } \log \prod _ { j \in \mathcal { N } _ { i } } p _ { \psi } \left( \mathbf { X } _ { j } | \mathbf { X } _ { i } \right) ,
+$$
+
+where $\{ \mathbf { X } _ { j \mid j \in \mathcal { N } _ { i } } , \mathbf { X } _ { i } \}$ . Then $p _ { \psi }$ can be used to augment features for all the neighbors. However, this method ignores the differences between all the neighbors, which may induce severe noise.
+
+To overcome the limitation, we assume that each neighbor satisfies a different conditional distribution. Specifically, there exists a conditional distribution $p ( \cdot | \mathbf { X } _ { i } , \mathbf { z } _ { j } )$ with latent random variable $\mathbf { z } _ { j }$ , such that we have $\mathbf { X } _ { j } \sim p ( \mathbf { X } | \mathbf { X } _ { i } , \mathbf { z } _ { j } )$ for $\mathbf { X } _ { j \mid j \in \mathcal { N } _ { i } }$ . Once we obtain $p ( \cdot | \mathbf { X } _ { i } , \mathbf { z } _ { j } )$ in some way, we can generate augmented features $\overline { { \mathbf { X } } }$ , and then we can train $P _ { \theta } ( \mathbf { Y } _ { k } | \mathbf { A } , \mathbf { X } , \mathbf { \overline { { X } } } )$ instead of $P _ { \theta } ( \mathbf { Y } _ { k } | \mathbf { A } , \mathbf { X } )$ to improve the final performance of $P _ { \theta }$ . Below, we will present how to find $p ( \cdot | \mathbf { X } _ { i } , \mathbf { z } _ { j } )$ , which will produce the generator $Q _ { \phi }$ .
+
+To achieve our purpose, a suitable method is the conditional variational auto-encoder (CVAE) (Kingma & Welling, 2013; Sohn et al., 2015), which can help learn the distribution of the latent variable $\mathbf { z } _ { j }$ , and the conditional distribution $p ( \cdot | \mathbf { X } _ { i } , \mathbf { z } _ { j } )$ . So, a CVAE model $Q _ { \phi } \left( \overline { { \mathbf { X } } } | \mathbf { A } , \mathbf { X } \right)$ is adopted as our generator, where $\phi = \{ \varphi , \psi \}$ , $\varphi$ denotes the variational parameters and $\psi$ represents the generative parameters. To derive the optimization problem for CVAE, $\log p _ { \psi } \left( \mathbf { X } _ { j } | \mathbf { X } _ { i } \right)$ can be written with latent variables $\mathbf { z }$ as follows, following previous work (Pandey & Dukkipati, 2017; Sohn et al., 2015):
+
+$$
+\begin{array} { l } { \log p _ { \psi } ( \mathbf { X } _ { j } | \mathbf { X } _ { i } ) = \displaystyle \int q _ { \varphi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) \log \frac { p _ { \psi } ( \mathbf { X } _ { j } , \mathbf { z } | \mathbf { X } _ { i } ) } { q _ { \varphi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) } \mathrm { d } \mathbf { z } + K L ( q _ { \varphi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) \| p _ { \psi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) ) } \\ { \displaystyle \qquad \geq \int q _ { \varphi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) \log \frac { p _ { \psi } ( \mathbf { X } _ { j } , \mathbf { z } | \mathbf { X } _ { i } ) } { q _ { \varphi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) } \mathrm { d } \mathbf { z } , } \end{array}
+$$
+
+and the evidence lower bound (ELBO) can be written as:
+
+$$
+\mathcal { L } ( \mathbf { X } _ { j } , \mathbf { X } _ { i } ; \psi , \varphi ) = - K L ( q _ { \varphi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) | | p _ { \psi } ( \mathbf { z } | \mathbf { X } _ { i } ) ) + \int q _ { \varphi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) \log p _ { \psi } ( \mathbf { X } _ { j } | \mathbf { X } _ { i } , \mathbf { z } ) \mathrm { d } \mathbf { z } ,
+$$
+
+where the encoder $q _ { \varphi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) \ = \ { \mathcal { N } } ( f ( \mathbf { X } _ { j } , \mathbf { X } _ { i } ) , g ( \mathbf { X } _ { j } , \mathbf { X } _ { i } ) )$ and decoder $p _ { \psi } ( { \bf X } _ { j } | { \bf X } _ { i } , { \bf z } ) =$ $\mathcal { N } ( h ( \mathbf { X } _ { i } , \mathbf { z } ) , c I )$ . The encoder is a two-layer MLP. $f$ and $g$ share the first layer, and their second layers employ different parameters. The decoder $h$ is two-layer MLP. For simplicity and tractability, the implemented generator $Q \left( \overline { { \mathbf { X } } } | \mathbf { A } , \mathbf { X } \right)$ uses the same parameters across all nodes $v _ { i } \in V$ .
+
+Optimization of the MLE Now, we present how to optimize the MLE Eq.(4) using the feature matrix produced from the generator. Once the augmented feature matrix can be sampled from the generator, we can optimize the parameters of Eq.(4) in the following way. Firstly, the parameter $\bar { \phi } = \{ \psi , \varphi \}$ can be optimized by maximizing the ELBO of the generator (6), i.e., we train the generator. Secondly, the parameter $\theta$ is optimized by maximizing the MLE Eq.(4) with $\phi$ fixed, which is the conditional distribution of ${ \bf Y } _ { k }$ given A, $\mathbf { X }$ , and $\overline { { \mathbf { X } } }$ , i.e., we train the downstream GNN model.
+
+In this paper, the MLE is formulated by a downstream GNN model as follows:
+
+$$
+P _ { \theta } \left( \mathbf { Y } _ { k } \mid \mathbf { A } , \mathbf { X } , { \overline { { \mathbf { X } } } } \right) \propto - { \overline { { \mathcal { L } } } } ( \theta | \mathbf { A } , \mathbf { X } , { \overline { { \mathbf { X } } } } , \phi ) ,
+$$
+
+$$
+\begin{array} { r } { \overline { { \mathcal { L } } } ( \theta | \mathbf { A } , \mathbf { X } , \overline { { \mathbf { X } } } , \phi ) = - \sum _ { k \in \mathbf { T } } \sum _ { f = 1 } ^ { C } \mathbf { Y } _ { k f } \ln \Big ( \mathrm { s o f t m a x } \big ( \mathrm { G N N } ( \mathbf { A } , \mathbf { X } , \overline { { \mathbf { X } } } ) \big ) _ { k f } \Big ) . } \end{array}
+$$
+
+# 3.2 THE ARCHITECTURE OF LA-GNN
+
+We discuss the details of downstream GNN models. And we use GCN, GAT, GCNII, and GRAND as the backbones and test them on semi-supervised node classification tasks. We name the modified GNN architecture as LA-GNN, where LA means local augmentation.
+
+LA-GCN A 2-layer LA-GCN is defined as follows:
+
+$$
+\mathbf { H } ^ { ( 2 ) } = \sigma \left( \hat { \mathbf { A } } \left( \sigma \left( \hat { \mathbf { A } } \mathbf { X } \mathbf { W } _ { 1 } ^ { ( 1 ) } \right) \bigg | \bigg | \sigma \left( \hat { \mathbf { A } } \overline { { \mathbf { X } } } _ { 1 } \mathbf { W } _ { 2 } ^ { ( 1 ) } \right) \bigg | \bigg | \cdots \bigg | \bigg | \sigma \left( \hat { \mathbf { A } } \overline { { \mathbf { X } } } _ { n } \mathbf { W } _ { n + 1 } ^ { ( 1 ) } \right) \right) \mathbf { W } ^ { ( 2 ) } \right) ,
+$$
+
+where $\overline { { \mathbf { X } } } _ { i }$ $\bar { \mathsf { \bar { c } } } _ { i } ( i = 1 , 2 , \cdots , n )$ is the augmented feature matrix produced by the generator, $\parallel$ denotes an operator of column-wise concatenation, $\mathbf { W } _ { i } ^ { ( 1 ) } \left( i = 1 , 2 , \cdots , n \right)$ denotes the parameters of the first LA-GCN layer, and $\mathbf { W } ^ { ( 2 ) }$ denotes the parameters of the second LA-GCN layer.
+
+LA-GCNII Since GCNII (Chen et al., 2020) applies a fully-connected neural network on $\mathbf { X }$ to obtain a lower-dimensional initial representation $\mathbf { H } ^ { ( 0 ) }$ before the forward propagation, we apply a fully-connected neural network on $\mathbf { X }$ and $\overline { { \mathbf { X } } }$ to obtain $\mathbf { H } ^ { ( 0 ) }$ for LA-GCNII as follows:
+
+$$
+\mathbf { H } ^ { ( 0 ) } = \sigma ( \mathbf { X } \mathbf { W } _ { 1 } ^ { ( 0 ) } ) \| \sigma ( \overline { { \mathbf { X } } } _ { 1 } \mathbf { W } _ { 2 } ^ { ( 0 ) } ) \| \cdots \| \sigma ( \overline { { \mathbf { X } } } _ { n } \mathbf { W } _ { n + 1 } ^ { ( 0 ) } ) .
+$$
+
+$\mathbf { H } ^ { ( 0 ) }$ is fed into the next forward propagation layer. Besides, we do not modify the architecture of GAT and GRAND, and just add our generated feature matrix to the input.
+
+# 3.3 ACTIVE LEARNING
+
+In this section, we introduce a trick for the overall training framework. After the training of the generator finishes, it contains an issue of using $Q _ { \phi } ( { \overline { { \mathbf { X } } } } | \mathbf { A } , \mathbf { X } )$ of Eq.(4) for inference because $Q$ may generate some samples from the side part of the distribution. This critical question makes the inferences inefficient. Inspired by Nielsen & Okoniewski (2019), we introduce active learning to capture the suitable generated feature matrix and the corresponding generator, which improves the inference efficiency and helps the optimization of the MLE. During active learning, the probability of each feature is proportional to its uncertainty evaluated by an acquisition function. We adopt the Bayesian Active Learning by Disagreement (BALD) acquisition function (Houlsby et al., 2011) to sample the most important inferences with the approximation from the Monte Carlo (MC) dropout samples as
+
+$$
+{ \cal U } ( \overline { { \mathbf { X } } } ) \approx H \left[ \frac { 1 } { N } \sum _ { n = 1 } ^ { N } P \left( \mathbf { Y } _ { k } | \overline { { \mathbf { X } } } , \omega _ { n } \right) \right] - \frac { 1 } { N } \sum _ { n = 1 } ^ { N } H \left[ P \left( \mathbf { Y } _ { k } | \overline { { \mathbf { X } } } , \omega _ { n } \right) \right] ,
+$$
+
+where $N$ is the number of MC samples and $\omega _ { n }$ are the parameters of the network sampled for the $n$ -th MC dropout sample. A high BLAD score indicates a network with high uncertainty about the generated feature matrix. So it tends to be selected to improve the GNN model. Finally, the overall algorithm framework is summarized in Algorithm 1, which shows the optimization of Eq.(4).
+
+Algorithm 1 The framework to train the Generator $Q _ { \phi }$ and the downstream GNN $P _ { \theta }$ using the initial feature matrix $\mathbf { X }$ and the generated feature matrix $\overline { { \mathbf { X } } }$ selected by the acquisition function
+
+1: Initialize $U { = }$ -inf, $\overline { { \mathbf { X } } }$ , $Q _ { \phi }$ , $\overline { { \mathbf { X } } } ^ { \prime }$ , and $Q _ { \phi } ^ { \prime }$   
+2: for $i = 1$ to the number of generator iterations do   
+3: Train the generator $Q _ { \phi }$ using $\mathbf { A }$ and $\mathbf { X }$   
+4: Generate feature matrix $\overline { { \mathbf { X } } }$ using $Q _ { \phi }$   
+5: Compute $U ( { \overline { { \mathbf { X } } } } )$ using Eq.(10).   
+6: if $U ( { \overline { { \mathbf { X } } } } ) > U$ then   
+7: $U = U ( { \overline { { \mathbf { X } } } } )$   
+8: if $i > N _ { w a r m u p }$ then   
+9: Train GNN $P _ { \theta }$ using $\mathbf { A }$ and $\overline { { \mathbf { X } } }$ for the number of continued GNN training iterations   
+10: X 0 = X , Q 0φ = Q φ   
+11: $\overline { { \mathbf { X } } } = \overline { { \mathbf { X } } } ^ { \prime }$ , $Q _ { \phi } = Q _ { \phi } ^ { \prime }$   
+12: Train the downstream GNN $P _ { \theta }$ with the generated feature matrix $\overline { { \mathbf { X } } }$ , and generator $Q _ { \phi }$
+
+# 4 DISCUSSION
+
+In this section, we discuss the motivation of this work and provide some analysis.
+
+Connection to EP-B and GraphSAGE We discuss how our proposed model distinguishes from the classical representation learning models on graphs. Previous methods such as EP-B (GarcíaDurán & Niepert, 2017) and GraphSAGE (Hamilton et al., 2017) rely on reconstruction loss function between the central node and its neighbors’ embeddings. EP-B aims to minimize the reconstruction error by optimizing the objective $\begin{array} { r } { \operatorname* { m i n } { \sum _ { u \in V \backslash \{ v \} } } \left[ \gamma + d ( \widetilde { \mathbf { X } } _ { v } , \mathbf { X } _ { v } ) - d ( \widetilde { \mathbf { X } } _ { v } , \mathbf { X } _ { u } ) \right] } \end{array}$ where $\mathbf { X } _ { v }$ represents the target node; $\mathbf { X } _ { u }$ denotes the neighbor nodes; $\widetilde { \mathbf { X } } _ { v } = \mathrm { A G G } ( \mathbf { X } _ { l } | l \in \mathcal { N } ( v ) )$ indicates the reconstruction from neighbors; and $\gamma$ refers to the bias. Besides, GraphSAGE exploits the negative sampling to differentiate the representations of remote node-pairs. GraphSAGE enforce nearby nodes to have similar representations and to enforce disparate nodes to be distinct by minimizing the objective $\operatorname* { m i n } - E _ { u \sim \mathcal { N } ( v ) } \overset { \cdot } { \log } \left( \left( \sigma ( \mathbf { X } _ { u } ^ { T } \mathbf { X } _ { v } ) \right) \right) - \lambda E _ { v _ { n } \sim P _ { n } ( v ) } \log \left( \left( \sigma ( - \mathbf { X } _ { v _ { n } } ^ { T } \mathbf { X } _ { v } ) \right) \right)$ where $\mathbf { X } _ { v }$ denotes target node; $\mathbf { X } _ { u }$ represents the neighbor node; ${ \bf X } _ { v _ { n } }$ is disparate node; and $P _ { n } ( v )$ is the negative sampling. These approaches build upon the assumption that adjacent nodes share similar attributes. In contrast, our model does not rely on such assumption and instead generates the neighboring node features from the conditional distribution of central node representations. Given the target node, $\mathbf { X } _ { v }$ , our aim is to learn the conditional distribution of the neighbor nodes, $\mathbf { X } _ { u }$ . A comparison between the reconstruction-based representation learning on graphs and our proposed framework is illustrated in Figure 2. And our local augmentation method is the third paradigm to exploit neighbors in a generative way.
+
+![](images/02db34de261ebadeed56e54daa7a15cd3eaa86f7795ea776ca63ea498b28e4a9.jpg)  
+Figure 2: (a) The original graph. (b) EP-B exploits the neighbors to reconstruct the central node’s embedding. (c) GraphSAGE encourages nearby nodes to have similar embeddings. (d) Given the representation of the central node, our aim is to infer the representations of the connected distribution of neighbors.
+
+Local Augmentation vs. General Augmentation General image augmentation algorithms include geometric transformations, feature space augmentation, adversarial training, and generative adversarial networks (Shorten & Khoshgoftaar, 2019). It is impossible to apply geometric transformations directly to graph data augmentation since graphs are sensitive to node permutation. General adversarial training, feature space augmentation, and generative adversarial networks don’t take the graph structure into account. Graphs consist of a set of identities with certain pairs of these identities connected by edges. We need to consider node features and the graph structure when designing the graph data augmentation framework. Our proposed method of local augmentation fully considers these two points. By extracting the neighbors’ feature vectors, we have enough data points to learn the distribution. There are two benefits to designing local augmentation. First, by taking the sub-graph structure and feature representation associated with this sub-graph structure as input for the generative model, we can learn the information of the sub-graph structure. Second, the number of data points to learn the distribution depends on the node degree. This assures that we have enough data points compared with the general feature augmentation and we can learn a better distribution.
+
+Complementing missing information Jia & Benson (2020) points out that some attribute information might be missing on a subset of vertices. By learning the distribution of node representations from the observed data, we can utilize the produced node representations from the generative model to complement the information missing in the nodes’ attributes, which boosts the robustness of downstream tasks. And we show that our model still works in the scenario that nodes lose a certain percentage of attributes. In other words, we can exploit the well-learned distribution to complement the contextual information of the local neighborhood to enhance the locality of the node representations.
+
+# 5 EXPERIMENTS
+
+In this section, we evaluate the performance of our proposed model on semisupervised node classification tasks on a variety of public graph datasets and compare our model with the state-of-the-art graph neural networks. We also carry out additional experiments to showcase the necessity of our design and its robustness to missing information.
+
+Table 2: Classification results on fixed split $( \% )$   
+
+<table><tr><td>Method</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td></tr><tr><td>Chebyshev (Defferrard et al.,2016)</td><td>81.2</td><td>69.8</td><td>74.4</td></tr><tr><td>APPNP (Klicpera et al.,2019)</td><td>83.8</td><td>71.6</td><td>79.7</td></tr><tr><td>MixHop (Abu-El-Haija et al.,2019)</td><td>81.9</td><td>71.4</td><td>80.8</td></tr><tr><td>Graph U-net (Gao&amp; Ji,2019)</td><td>84.4</td><td>73.2</td><td>79.6</td></tr><tr><td>GSNN-M (Wang et al.,2020a)</td><td>83.9</td><td>72.2</td><td>79.1</td></tr><tr><td>S²GC (Zhu &amp; Koniusz,2021)</td><td>83.5</td><td>73.6</td><td>80.2</td></tr><tr><td>GCN (Kipf &amp; Welling,2017)</td><td>81.6</td><td>70.3</td><td>78.9</td></tr><tr><td>G-GCN (Zhu et al.,2020)</td><td>83.7</td><td>71.3</td><td>80.9</td></tr><tr><td>DropEdge-GCN (Rong et al.,2020)</td><td>82.8</td><td>72.3</td><td>79.6</td></tr><tr><td>GAUG-O-GCN (Zhao et al.,2021)</td><td>83.6</td><td>73.3</td><td>79.3</td></tr><tr><td>LA-GCN</td><td>84.1</td><td>72.5</td><td>81.3</td></tr><tr><td>GAT (Velickovic et al., 2018)</td><td>83.0</td><td>70.4</td><td>0OM</td></tr><tr><td>LA-GAT</td><td>83.9</td><td>72.3</td><td>OOM</td></tr><tr><td>GCNII (Chen et al.,2020) LA-GCNII</td><td>85.2</td><td>73.1</td><td>80.0</td></tr><tr><td></td><td>85.2</td><td>73.7</td><td>81.6</td></tr><tr><td>GRAND (Feng et al.,2020)</td><td>85.4</td><td>75.4</td><td>82.7</td></tr><tr><td>LA-GRAND</td><td>85.8</td><td>75.8</td><td>83.3</td></tr></table>
+
+# 5.1 DATASETS
+
+We utilize seven public graph datasets (Cora, Citeseer, Pubmed, Squirrel, Actor, Chameleon, and Cornell) for semisupervised node classification tasks. The details of these datasets can be found in the appendix.
+
+# 5.2 SEMI-SUPERVISED NODE CLASSIFICATION
+
+Baselines and Experimental Setup. We apply the standard fixed splits (Yang et al., 2016) on three datasets Cora, Citeseer, and Pubmed, with 20 nodes per class for training, 500 nodes for validation, and 1,000 nodes for testing. And we consider four backbones: GCN (Kipf & Welling, 2017), GAT (Velickovi ˇ c et al., 2018), GCNII (Chen ´ et al., 2020), and GRAND (Feng et al., 2020) to evaluate our proposed framework and compare our model against state-of-the-art models including 1) backbone models: Chebyshev (Defferrard et al., 2016), GCN, GAT,
+
+Table 3: Classification results on random split $( \% )$   
+
+<table><tr><td>Method</td><td>Squirrel</td><td>Actor</td><td>Chameleon</td><td>Cornell</td></tr><tr><td>APPNP</td><td>21.6</td><td>32.1</td><td>33.0</td><td>58.7</td></tr><tr><td>S²GC</td><td>21.3</td><td>27.8</td><td>30.2</td><td>57.2</td></tr><tr><td>GCN</td><td>22.5</td><td>26.2</td><td>25.1</td><td>55.7</td></tr><tr><td>DropEdge-GCN</td><td>21.9</td><td>26.5</td><td>25.0</td><td>53.6</td></tr><tr><td>LA-GCN</td><td>23.2</td><td>27.0</td><td>28.9</td><td>56.1</td></tr><tr><td>GAT</td><td>24.2</td><td>27.2</td><td>34.8</td><td>55.8</td></tr><tr><td>LA-GAT</td><td>28.2</td><td>27.4</td><td>38.6</td><td>56.5</td></tr><tr><td>GCNII</td><td>25.3</td><td>31.9</td><td>30.2</td><td>57.3</td></tr><tr><td>LA-GCNII</td><td>28.6</td><td>32.7</td><td>32.5</td><td>56.6</td></tr></table>
+
+APPNP (Klicpera et al., 2019), Graph U-net (Gao & Ji, 2019), MixHop (Abu-El-Haija et al., 2019), GCNII, GSNN-M (Wang et al., 2020a), $\mathrm { { \cal S } ^ { 2 } { \cal G } { \cal C } }$ (Zhu & Koniusz, 2021), and GRAND and 2) featurelevel and topology-level augmentation models: G-GNNs (Zhu et al., 2020), DropEdge (Rong et al.,
+
+2020) and GAUG-O (Zhao et al., 2021). For four datasets Squirrel, Actor, Chameleon, and Cornell, we take 10 random splits (Shchur et al., 2018) where $10 \%$ , $30 \%$ , and $60 \%$ of the date for training, validation, testing; measure the performance of GCN, GAT, GCNII, and corresponding modified models.
+
+Results For three datasets Cora, Citeseer, and Pubmed, we report the mean classification accuracy on the test nodes of all our models after 100 runs and report the values after running the experiments of their models with our server under their setting hyperparameters in their original papers. The results of the evaluation experiments are summarized in Tables 2, 3, and in the appendix, which demonstrate that the backbone models equipped with our method achieve the best performance across all the datasets except the Cornell dataset. More specifically, we can improve upon GCN by a margin of $2 . 5 \%$ , $2 . 2 \%$ , and $2 . 4 \%$ on Cora, Citeseer, and Pubmed respectively. Moreover, LA-GNN outperforms other backbone models including GAT and GCNII as well as data augmentation models (Zhu et al., 2020; Rong et al., 2020; Zhao et al., 2021) on these citation network datasets. Besids, we also provide the analysis of the distribution of our generated feature matrix. And Figure 3 shows the distribution of the attributes of the original
+
+and inference neighbors, which can demonstrate our inference feature matrix follow the distribution of the initial feature matrix.
+
+![](images/410d67587b543d15cc65cba5cdc78b6ecc62b88dfffe4c1dfb7018aab80db729.jpg)  
+Figure 3: The distribution of the attribute bin of the inference neighbors vs. the distribution of the attribute bin of the original neighbors, with KL divergence $= 0 . 0 0 2 6$ . The value of each feature bin is the sum of the attribute values of multiple dimensions of the feature vector. We split the feature vector into multiple feature bins.
+
+# 5.3 ABLATION STUDY
+
+In this section, to demonstrate the effectiveness of our proposed generative framework, we conduct experiments that compare LA-GNN to several of its ablated variants without generative modeling. The results are shown in Table 4. ${ } " \mathrm { G C N } +$ width" only increases the first network layer width for GCN and GCNII to match LAGNN without giving generated samples as input. $" +$ concatenation" only replaces the generated feature matrix of LA-GNN with the original feature matrix of the central node. $" +$ plain neighborhood" replaces the generated feature matrix of LA-GNN with a neighborhood feature matrix where each row corresponds to the feature vector of the randomly sampled neighbor. The
+
+Table 4: Effects of different components of our framework evaluated on the standard split of the Cora, Citeseer and Pubmed dataset.   
+
+<table><tr><td>Method</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td></tr><tr><td>GCN</td><td>81.6</td><td>70.3</td><td>78.9</td></tr><tr><td>GCNII</td><td>85.2</td><td>73.1</td><td>80.0</td></tr><tr><td>GCN + width</td><td>82.0</td><td>71.4</td><td>79.5</td></tr><tr><td>GCN + concatenation</td><td>81.8</td><td>71.6</td><td>78.8</td></tr><tr><td>GCN + plain neighborhood</td><td>80.9</td><td>68.8</td><td>75.0</td></tr><tr><td>GCNII + width</td><td>85.1</td><td>73.1</td><td>80.2</td></tr><tr><td>GCNII + concatenation</td><td>85.2</td><td>73.3</td><td>80.2</td></tr><tr><td>GCNII + plain neighborhood</td><td>83.3</td><td>71.9</td><td>78.1</td></tr><tr><td>LA-GCN</td><td>84.1</td><td>72.5</td><td>81.3</td></tr><tr><td>LA-GCNII</td><td>85.2</td><td>73.7</td><td>81.6</td></tr></table>
+
+results show that the first two variants provide no notable improvement for the backbone models, and the third variant even results in degradation. By eliminating the possibility that these confounding factors irrelevant to our core approach may contribute to the final performance, it’s evident that the performance gain in Table 2 and 3 are due to our proposed generative local augmentation framework.
+
+# 5.4 ROBUSTNESS TO MISSING INFORMATION
+
+In this section, we conduct experiments to verify that our proposed framework can robustify downstream tasks against missing information in the feature attributes. Specifically, we mask a certain percentage of the attributes of each feature vector and use the same pipeline to do augmentation for the masked feature matrix. As shown in Table 5, we can see that as the mask ratio increases, the gap of the performance between the GCN and LA-GCN enlarges in most cases in Cora and Citeseer, which corroborates our insight discussed in Section 4. Since there exists large redundancy in the features of the Pubmed dataset, the performance of GCN and LA-GCN decreases little as the mask ratio increases and the gap of the performance does not enlarge. To conclude, our model can complement the contextual information of the local neighborhood to enhance the locality of the node representations.
+
+Table 5: Summary of results on recovering study in terms of classification accuracy $( \% )$ . $\downarrow$ means a decrease compared with the accuracy if features are not masked.   
+
+<table><tr><td>Dataset</td><td colspan="4">Cora</td><td colspan="4">Citeseer</td><td colspan="4">Pubmed</td></tr><tr><td>Mask Ratio</td><td>0.1</td><td>0.2</td><td>0.4</td><td>0.8</td><td>0.1</td><td>0.2</td><td>0.4</td><td>0.8</td><td>0.1</td><td>0.2</td><td>0.4</td><td>0.8</td></tr><tr><td>GCN</td><td>81.0(↓0.6)</td><td>80.6(↓1.0)</td><td>80.1(↓1.5)</td><td>76.0 (↓5.6)</td><td>70.1(↓0.2)</td><td>69.3 (↓1.0)</td><td>67.2 (↓3.1)</td><td>61.0(↓9.3)</td><td>78.5(↓0.4)</td><td>78.5(↓0.4)</td><td>77.5 (↓1.4)</td><td>76.9 (↓2.0)</td></tr><tr><td>LA-GCN</td><td>83.5 (↓0.6)</td><td>83.1(↓1.0)</td><td>81.6(↓2.5)</td><td>81.1 (↓3.0)</td><td>72.2(↓0.3)</td><td>71.7 (↓0.8)</td><td>69.3 (↓3.2)</td><td>65.9 (↓6.6)</td><td>81.4(↓0.1)</td><td>80.9 (↓0.6)</td><td>80.5 (↓1.0)</td><td>79.4 (↓2.1)</td></tr></table>
+
+# 6 RELATED WORK
+
+Graph Neural Networks In general, convolution in the graph domain involves non-spectral (spatial) and spectral approaches. Non-spectral methods generalize convolutions operating on spatially close neighbors to the graph domain, such as Duvenaud et al. (2015); Atwood & Towsley (2016); Niepert et al. (2016); Monti et al. (2017). Spectral approaches define the convolution operations based on the spectral formulation, such as Bruna et al. (2014); Defferrard et al. (2016); Kipf & Welling (2017). Recently, several methods (Abu-El-Haija et al., 2019; Liao et al., 2019) based on GCN have been proposed to obtain the higher-order filters. Besides, GAT (Velickovi ˇ c et al., 2018), Graph ´ U-Nets (Gao & Ji, 2019) combine attention networks and pooling operation with GNN separately, which achieve state-of-the-art performance on node and link classification tasks. In this work, local augmentation can be applied on various backbone models to improve performance.
+
+Graph Generative Models Generative models (Goodfellow et al., 2014; Kingma & Welling, 2013) are powerful tools of learning data distribution through unsupervised learning, and they have achieved tremendous success in various applications. Recently, researchers have proposed several interesting generative models for graph data generation. Variational graph auto-encoder (VGAE) (Kipf & Welling, 2016) makes use of latent variables and learns interpretable latent representations for undirected graphs. Salha et al. (2019) replace the GCN encoder in VGAE with a simple linear model and emphasize the effectiveness of a simple node encoding scheme. Xu et al. (2019) propose a generative model framework to learn node representations, by sampling graph generation sequences constructed from observed graph data. ConDgen (Yang et al., 2019) exploits the GCN encoder to handle the inherent challenges of flexible context-structure conditioning and permutation-invariant generation. Besides, some methods have been proposed to apply the graph generative models in various applications such as graph matching (Simonovsky & Komodakis, 2018), molecule design (Liu et al., 2018), retrosynthesis prediction (Shi et al., 2020) and chemical design (Samanta et al., 2018). Compared with these approaches mainly focusing on structure generation, our model takes full use of the power of the generative model for feature representation generation, which can serve as an enhanced technique for the downstream backbone models.
+
+# 7 CONCLUSION
+
+We propose local augmentation, a brand-new technique that exploits the generative model to learn the conditional distribution of the central node’s neighbors’ feature representations given its representation. We can augment more 1-hop neighbors from a well-trained generative model to enhance the performance of backbone GNN models. Experiments show that our model can improve performance across various GNN architectures and benchmark datasets by enriching local information. Besides, our model achieves new state-of-the-art results on various semi-supervised node classification tasks. One limitation of our proposed framework is that we do not exploit the 2-hop neighbors or use the random walk to find more related neighbors for the central node. And one future work is that we can extract more $^ { 2 / 3 }$ -hop neighbors if the central node’s degree is small and learn the conditional distribution for random sampling nodes if the graph is large.
+
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+A PROOF OF EQ.(6)
+
+We give more details of the derivation of the generator ELBO as follows:
+
+$$
+\begin{array} { r l } { \log _ { \rho } | \mathbf { X } _ { i } | \mathbb { X } _ { j } - j } & { \neq \langle z | \mathbf { z } | \mathbf { z } | \mathbf { X } _ { j } , ~ \mathbf { X } _ { i } \rangle \log _ { \rho } \langle ~ \mathbf { X } _ { i } | \mathbf { X } _ { j } \rangle \mathrm { d } \mathbf { z } } \\ & { = \int \langle z | \mathbf { z } | \mathbf { A } _ { \mathbf { x } } \mathbf { x } , ~ \mathbf { X } _ { i } | \log _ { \rho } | \mathbf { X } _ { i } \mathbf { X } _ { j } | \log _ { \rho } \langle ~ \mathbf { X } _ { i } | \mathbf { X } _ { j } , ~ \mathbf { X } _ { i } \rangle } \\ & { \quad - \int \langle z | \mathbf { z } | \mathbf { z } | \mathbf { X } _ { i } , ~ \mathbf { X } _ { i } \rangle \log _ { \rho } | \langle ~ \mathbf { X } _ { i } | \mathbf { X } _ { j } \rangle \mathrm { d } \mathbf { z } | } \\ & { = \int \langle \exp \{ \mathbf { X } _ { i } \mathbf { X } _ { j } \} | \exp \{ \exp \{ | \mathbf { X } _ { i } \mathbf { X } _ { j } | \} \} \exp \{ | \langle \mathbf { X } _ { i } \mathbf { X } _ { j } , ~ \mathbf { X } _ { i } , ~ \mathbf { X } _ { j } | \} \mathrm { d } \mathbf { z } } \\ & { \quad - \int \langle z | \mathbf { z } | \mathbf { X } _ { i } \rangle \exp \{ | \langle \mathbf { X } _ { j } | \mathbf { X } _ { i } , ~ \mathbf { X } _ { j } \rangle | \} } \\ & { \quad - \int \langle z | \mathbf { z } | \mathbf { z } | \mathrm { X } _ { i } \rangle \exp \{ | \langle \mathbf { X } _ { i } | \mathbf { X } _ { j } \rangle | \} } \\ & { = \int \langle z | \mathbf { z } | \mathbf { X } _ { i } \mathbf { X } _ { j } \rangle \log _ { \rho } \langle \mathbf { X } _ { i } | \mathbf { X } _ { j } \rangle \exp \{ | \langle \mathbf { X } _ { i } | \mathbf { X } _ { j } \rangle | } \\ & { \quad - \int \langle z | \mathbf { z } | \mathbf { Z } _ { j } \rangle \exp \{ | \langle \mathbf { X } _ { j } | \mathbf { X } _ { j } \rangle | } \\ &  \quad - \int \langle z | \mathbf { z } | \mathbf { Z } _ { j } \rangle \exp \{  \end{array}
+$$
+
+$$
+\begin{array} { r l } { L _ { E L B O } = \displaystyle { \int } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , \mathbf X _ { i } ) \log \frac { p _ { \phi } ( { \mathbf x } _ { j } , { \mathbf X _ { i } } ) \log } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , { \mathbf X _ { i } } ) } } \mathrm { d } { \mathbf z } } \\ { } & { ~ = \displaystyle { \int } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , \mathbf X _ { i } ) \log \frac { p _ { \psi } ( { \mathbf X _ { j } } , { \mathbf X _ { i } } , { \mathbf z } ) } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , { \mathbf X _ { i } } ) p _ { \psi } ( { \mathbf X _ { i } } ) } } \mathrm { d } { \mathbf z } } \\ { } & { ~ = \displaystyle { \int } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , \mathbf X _ { i } ) \log \frac { p _ { \phi } ( { \mathbf X _ { j } } | \mathbf X _ { i } , { \mathbf z } ) p _ { \psi } ( { \mathbf X _ { i } } , { \mathbf z } ) } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , { \mathbf X _ { i } } ) p _ { \psi } ( { \mathbf X _ { i } } ) } } \mathrm { d } { \mathbf z } } \\ { } & { ~ = \displaystyle { \int } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , \mathbf X _ { i } ) \log \frac { p _ { \phi } ( { \mathbf X _ { j } } | \mathbf X _ { i } , { \mathbf z } ) p _ { \psi } ( { \mathbf z } | \mathbf X _ { i } ) } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , { \mathbf X _ { i } } ) } } \mathrm { d } { \mathbf z } } \\ { } &  ~ = \displaystyle { \int } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , \mathbf X _ { i } ) \log \frac { p _ { \phi } ( { \mathbf z } | \mathbf X _ { j } , { \mathbf z } ) p _ { \phi } ( { \mathbf Z } | \mathbf X _ { i } ) } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , { \mathbf X _ { i } } ) } } \\ { } &  ~ = \displaystyle { \int }  q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , \mathbf X _ { i }  \end{array}
+$$
+
+# B REPRODUCIBILITY
+
+# B.1 DATASETS DETAILS
+
+Cora, Citeseer, and Pubmed are standard citation network benchmark datasets Sen et al. (2008). In these datasets, nodes represent documents, and edges denote citations; node feature corresponds to elements of a bag-of-words representation of a document, and node label corresponds to one of the academic topics. Besides, we utilize four datasets used in Pei et al. (2020) for evaluation. Chameleon and squirrel are two page-page networks on specific topics in Wikipedia Rozemberczki et al. (2021). In these datasets, nodes represent web pages, and edges denote mutual links between pages; node features correspond to several informative nouns in the Wikipedia pages and labels correspond to the number of the average monthly traffic of the web page. WebKB1 is a webpage dataset collected from various universities. We use the one subdataset of it, Cornell. In this dataset, nodes represent web pages, and edges are hyperlinks between them; node features correspond to the bag-of-words representation of web pages and labels correspond to five categories, student, project, course, staff, and faculty. Film dataset is the actor-only induced subgraph of the film-directoractor-writer network Tang et al. (2009). In this dataset, Nodes represent actors, and edges denote co-occurrence on the same Wikipedia page; node features correspond to some keywords in the Wikipedia pages and labels correspond to five categories in terms of words of actor’s Wikipedia. All the dataset statistics are summarized in Table 6.
+
+Table 6: Datasets statistics   
+
+<table><tr><td>Dataset</td><td>Cora</td><td>Cite.</td><td>Pubm.</td><td>Cham.</td><td>Squi.</td><td>Actor</td><td>Corn.</td></tr><tr><td>#Nodes</td><td>2708</td><td>3327</td><td>19717</td><td>2277</td><td>5201</td><td>7600</td><td>183</td></tr><tr><td>#Edges</td><td>5429</td><td>4732</td><td>44338</td><td>36101</td><td>217073</td><td>33544</td><td>295</td></tr><tr><td>#Features</td><td>1433</td><td>3703</td><td>500</td><td>2325</td><td>2089</td><td>931</td><td>1703</td></tr><tr><td># Classes</td><td>7</td><td>6</td><td>3</td><td>5</td><td>5</td><td>5</td><td>5</td></tr></table>
+
+# B.2 IMPLEMENTATION DETAILS
+
+We use Pytorch (Paszke et al., 2019) to implement LA-GNNs. The codes of $S ^ { 2 } G C$ (Zhu & Koniusz, 2021), LA-GCN, LA-GAT, LA-GCNII, LA-GRAND, and DropEdge-GCN are implemented referring to Pytorch implementation of $\mathrm { S } ^ { 2 } \mathrm { G } \mathrm { C } ^ { 2 }$ , $\mathrm { G C N } ^ { 3 }$ (Kipf & Welling, 2017), $\mathrm { G A T ^ { 4 } }$ (Velickovi ˇ c et al., 2018), ´ $\mathrm { G C N I I } ^ { 5 }$ (Chen et al., 2020) $\mathrm { G R A N D } ^ { 6 }$ (Feng et al., 2020), and DropEdge- $\mathbf { \Delta } G \mathbf { C N } ^ { 7 }$ (Rong et al., 2020). Besides, we implement APPNP (Klicpera et al., 2019) with DGL (Wang et al., 2019) version of APPNP8. The datasets Cora, Citeseer, Pubmed are downloaded from TensorFlow (Abadi et al., 2016) implementation of $\mathrm { G C N ^ { 9 } }$ , and the datasets Chameleon, Squirrel, Actor, and Cornell are downloaded from the implementation of Geom- $\mathrm { G C N ^ { 1 0 } }$ (Pei et al., 2020). All the experiments in this work are conducted on a single NVIDIA Tesla V100 with 32GB memory size. The operating system behind the Docker where the experiments are running is Red Hat 4.8.2-16. And the software that we use for experiments are Python 3.6.8, numpy 1.19.2, sklearn 0.0, scipy 1.5.4, networkx 2.5.1, torch 1.6.0, torchvision 0.7.0, CUDA 10.2.89, and CUDNN 8.0.2.
+
+# B.3 HYPERPARAMETER DETAILS
+
+LA-GNNs introduce an additional parameter, that is the hidden layer for generated feature matrix $\overline { { \mathbf { X } } }$ before concatenation. The difference of architectures between GCN and LA-GCN can be found in Figure 4, and the LA-GCNII architecture can be found in Figure 5.
+
+![](images/d7816fad620dd0825bfeea46796303662cc8614bc99cb04d713507ea99f5f1a5.jpg)  
+Figure 4: GCN and LA-GCN architectures. The difference between GCN and LA-GCN architectures is that the LA-GCN has an additional convolutional layer for $\overline { { \mathbf { X } } }$ and it uses a concatenation operation to mix the hidden representations.
+
+![](images/5443477c1a5cf6f340de4b0323cbbb8400ffee35c029505e3e5d44d84aa3e826.jpg)  
+Figure 5: LA-GCNII architecture. The difference between GCNII and LA-GCNII is that the LAGCNII has an additional MLP layer for $\overline { { \mathbf { X } } }$ and it uses a concatenation operation to mix the hidden representations.
+
+The difference of hyperparameters between the GCN and LA-GCN is only the hidden layer size before concatenation. For the LA-GCNII, LA-GAT, LA-GRAND, we tune the hyperparameters in the same way as described in their original papers with validation set.

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