First-Order Stationarity of Reverse Diffusions

Zhifeng Chen; Chenyang Jiang; Yazhen Wang

stat.MLSep 25, 2026

First-Order Stationarity of Reverse Diffusions

Authors: Zhifeng Chen, Chenyang Jiang, Yazhen Wang

Organizations: Department of Statistics, UW-Madison

Abstract

Recent literature has shown a strong connection between optimization and sampling. We develop the corresponding first-order theory for diffusion models. First, the SDE-based reverse-time flows of overdamped and underdamped Langevin diffusions contract relative Fisher divergences at explicit exponential rates whenever the stationary potential of the forward process is strongly convex---a condition on the noising process one chooses, not on the data. This is a unique advantage of SDE-based reverse diffusion, absent in the reverse process based on ODEs. Second, we incorporate discretization and establish averaged first-order stationarity bounds---the sampling analog of averaged gradient-norm guarantees in nonconvex optimization---for samplers of both overdamped and underdamped diffusion models. As in nonconvex optimization, the convexity-free certificate is local: it guarantees score consistency, not global mode weights.

Figures & tables

Smooth optimization	Reverse diffusion sampling
objective $f(x)$	relative entropy $\operatorname{KL}(p\,\\|\,q)$
gradient $\nabla f$	relative score $\nabla\log(p/q)$ in $L^{2}(p)$
squared gradient norm $\\|\nabla f\\|^{2}$	Fisher divergence $\mathcal{F}(p\,\\|\,q)$
learning rate $\eta$	step size $h$
descent identity $\frac{\,\mathrm{d}}{\,\mathrm{d}t}f=-\\|\nabla f\\|^{2}$	KL dissipation (Corollary 2.2 )
strong convexity of $f$	strong log-concavity of the forward potential $U$

Table 1: The optimization dictionary used throughout the paper.

Figure 1: One-dimensional Gaussian mixture with exact scores, wrong initialization

p_{0}=N(10,0.5^{2})

. (a)

\mathcal{F}/\mathcal{F}_{0}

against the bound

e^{-2t}

of Theorem 3.2 ,

\operatorname{KL}/H_{0}

, and

\mathrm{TV}

along the exact reverse SDE. (b) Terminal laws of the reverse SDE and of the probability-flow ODE from the same initialization. (c) Gap between the time-averaged

\mathcal{F}(\hat{p}_{t}\,\|\,q_{t})

of the frozen-score integrator and its exact-flow counterpart

\mathcal{F}(p_{t}\,\|\,q_{t})

versus step size (log–log).

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 2: The critically damped experiment (

\gamma=2

\xi=1

), presented as in Figure 1 . (a) Along the exact reverse flow, the modified Fisher divergence

\mathcal{F}_{P}

contracts faster than the guaranteed

e^{-t}

\operatorname{KL}

decreases monotonically, and

\mathrm{TV}

plateaus. (b) Terminal position marginal. (c) The frozen-score exponential integrator ( 5.4 ): the time-averaged

\mathcal{F}(\hat{p}\,\|\,\hat{q})

of Theorem 5.5 (circles) and the time-averaged

\mathcal{F}(\hat{p}_{t}\,\|\,q_{t})

against the exact marginals along the frozen-score SDE interpolation (squares) versus step size, with the exact-flow value

\frac{1}{T}\int_{0}^{T}\mathcal{F}(p_{t}\,\|\,q_{t})\,\mathrm{d}t

(gray). (d) Gap between the time-averaged

\mathcal{F}(\hat{p}_{t}\,\|\,q_{t})

of (c) and its exact-flow counterpart

\mathcal{F}(p_{t}\,\|\,q_{t})

versus step size (log–log; dashed: slope one), as in Figure 1 (c).

Panel of Figure 3 — Figure 3: CelebA-HQ with the exact empirical score: $64$ images generated by the sampler of Section 4 from the standard initialization and the two mean shifts, $T=5.06$ , $n=28000$ training images. Every image is a training image; the initialization decides which ones are produced.

Panel of Figure 4 — Figure 4: Pretrained CelebA-HQ DDPM ( Ho et al., 2020 ) , $64$ samples of the $250$ -step ancestral sampler from the same three initializations of the initial noise as in Figure 3 . The mean shifts along the all-ones direction nevertheless bias them towards dark ( $\beta<0$ ) or bright ( $\beta>0$ ) photographs, exactly as with the exact score.

Explore similar work

Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Models
Jun 16, 2025Daniel Zhengyu Huang, Jiaoyang Huang, Zhengjiang LinDiffusion Model Sampling Diffusion Sampling
Beijing International Center for Mathematical Research, Center for Machine Learning Research, Peking University, Beijing, China · Department of Statistics and Data Science, University of Pennsylvania, Philadelphia, PA, USA · Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, 02139 Cambridge MA, USA
Diffusion Removes Langevin's Conditioning Dependence: A Sharp Gaussian Analysis
Oct 8, 2026Adam Perbost, Francis Bach, Pierre MarionDiffusion Sampling Score-Based Generative Modeling
Inria, École normale supérieure – PSL Research University Paris, France
Probability flow ODEs in score-based and reflected diffusion models
Oct 2, 2026Rama CONTDiffusion Sampling Diffusion Models
Mathematical Institute, University of Oxford

Jun 16, 2025J/K move · Enter open · S save

Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Models

Daniel Zhengyu Huang, Jiaoyang Huang, Zhengjiang Lin

Beijing International Center for Mathematical Research, Center for Machine Learning Research, Peking University, Beijing, China · Department of Statistics and Data Science, University of Pennsylvania, Philadelphia, PA, USA · Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, 02139 Cambridge MA, USA

Diffusion Model Sampling Diffusion Sampling

Open paper AskPDF arXiv

Diffusion probabilistic models generate samples by learning to reverse a noise-injection process that transforms data into noise. A key development is the reformulation of the reverse sampling process as a deterministic probability flow ordinary differential equation (ODE), which allows for efficient sampling using high-order numerical solvers. Unlike traditional time integrator analysis, the accuracy of this sampling procedure depends not only on numerical integration errors but also on the approximation quality and regularity of the learned score function, as well as their interaction. In this work, we present a rigorous convergence analysis of deterministic samplers derived from probability flow ODEs for general forward processes with arbitrary variance schedules. Specifically, we develop and analyze

p

-th order (exponential) Runge-Kutta schemes, under the practical assumption that the first and second derivatives of the learned score function are bounded. We prove that the total variation distance between the generated and target distributions can be bounded as \begin{align*} O\bigl(d^{\frac{7}{4}}\varepsilon_{\text{score}}^{\frac{1}{2}} +d(dH_{\max})^p\bigr), \end{align*} where

\varepsilon^2_{\text{score}}

denotes the

L^2

error in the score function approximation,

d

is the data dimension, and

H_{\max}

represents the maximum solver step size. Numerical experiments on benchmark datasets further confirm that the derivatives of the learned score function are bounded in practice.

Abstract

Explore similar work

Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Models

Diffusion Removes Langevin's Conditioning Dependence: A Sharp Gaussian Analysis

Probability flow ODEs in score-based and reflected diffusion models