Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models

Gen Li, Yuting Wei, Yuxin Chen, Yuejie Chi

TL;DR

The paper delivers a non-asymptotic, discrete-time analysis of diffusion-based generative models, focusing on both deterministic (probability-flow ODE) and stochastic (DDPM-type) samplers. It shows that with access to $ ilde ext{l}_2$-accurate score estimates, the ODE-based sampler achieves a $1/ ext{ε}$ convergence rate (in total variation) and the DDPM-type sampler achieves a $1/ ext{ε}^2$ rate, aligning with state-of-the-art results in the stochastic case. Importantly, the framework only requires minimal assumptions on the target data distribution and provides explicit bounds that separate discretization error from score-estimation error, enabling two accelerated variants that reach $1/ ext{ε}^2$ and $1/ ext{ε}$-type improvements for the ODE- and DDPM-based samplers, respectively. The analysis is deliberately elementary and non-asymptotic, avoiding heavy SDE/ODE machinery, and the results offer practical guidance for accelerating diffusion-based data generation while clarifying how score estimation quality influences generation fidelity.

Abstract

Diffusion models, which convert noise into new data instances by learning to reverse a Markov diffusion process, have become a cornerstone in contemporary generative modeling. While their practical power has now been widely recognized, the theoretical underpinnings remain far from mature. In this work, we develop a suite of non-asymptotic theory towards understanding the data generation process of diffusion models in discrete time, assuming access to $\ell_2$-accurate estimates of the (Stein) score functions. For a popular deterministic sampler (based on the probability flow ODE), we establish a convergence rate proportional to $1/T$ (with $T$ the total number of steps), improving upon past results; for another mainstream stochastic sampler (i.e., a type of the denoising diffusion probabilistic model), we derive a convergence rate proportional to $1/\sqrt{T}$, matching the state-of-the-art theory. Imposing only minimal assumptions on the target data distribution (e.g., no smoothness assumption is imposed), our results characterize how $\ell_2$ score estimation errors affect the quality of the data generation processes. In contrast to prior works, our theory is developed based on an elementary yet versatile non-asymptotic approach without resorting to toolboxes for SDEs and ODEs. Further, we design two accelerated variants, improving the convergence to $1/T^2$ for the ODE-based sampler and $1/T$ for the DDPM-type sampler, which might be of independent theoretical and empirical interest.

Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models

TL;DR

The paper delivers a non-asymptotic, discrete-time analysis of diffusion-based generative models, focusing on both deterministic (probability-flow ODE) and stochastic (DDPM-type) samplers. It shows that with access to -accurate score estimates, the ODE-based sampler achieves a convergence rate (in total variation) and the DDPM-type sampler achieves a rate, aligning with state-of-the-art results in the stochastic case. Importantly, the framework only requires minimal assumptions on the target data distribution and provides explicit bounds that separate discretization error from score-estimation error, enabling two accelerated variants that reach and -type improvements for the ODE- and DDPM-based samplers, respectively. The analysis is deliberately elementary and non-asymptotic, avoiding heavy SDE/ODE machinery, and the results offer practical guidance for accelerating diffusion-based data generation while clarifying how score estimation quality influences generation fidelity.

Abstract

Diffusion models, which convert noise into new data instances by learning to reverse a Markov diffusion process, have become a cornerstone in contemporary generative modeling. While their practical power has now been widely recognized, the theoretical underpinnings remain far from mature. In this work, we develop a suite of non-asymptotic theory towards understanding the data generation process of diffusion models in discrete time, assuming access to -accurate estimates of the (Stein) score functions. For a popular deterministic sampler (based on the probability flow ODE), we establish a convergence rate proportional to (with the total number of steps), improving upon past results; for another mainstream stochastic sampler (i.e., a type of the denoising diffusion probabilistic model), we derive a convergence rate proportional to , matching the state-of-the-art theory. Imposing only minimal assumptions on the target data distribution (e.g., no smoothness assumption is imposed), our results characterize how score estimation errors affect the quality of the data generation processes. In contrast to prior works, our theory is developed based on an elementary yet versatile non-asymptotic approach without resorting to toolboxes for SDEs and ODEs. Further, we design two accelerated variants, improving the convergence to for the ODE-based sampler and for the DDPM-type sampler, which might be of independent theoretical and empirical interest.
Paper Structure (104 sections, 17 theorems, 330 equations)

This paper contains 104 sections, 17 theorems, 330 equations.

Table of Contents

  1. Introduction
  2. This paper.
  3. Notation.
  4. Preliminaries
  5. Diffusion generative models
  6. The forward process.
  7. The reverse process.
  8. Deterministic vs. stochastic samplers: a continuous-time interpretation
  9. Algorithms and main results
  10. Assumptions and learning rates
  11. Score estimates.
  12. Target data distributions.
  13. Learning rate schedule.
  14. Deterministic samplers
  15. An ODE-based deterministic sampler
  16. ...and 89 more sections

Key Result

Theorem 1

Suppose that eq:assumption-data-bounded holds true. Assume that the score estimates $s_t(\cdot)$$(1 \le t \le T)$ satisfy Assumptions assumption:score-estimate and assumption:score-estimate-Jacobi. Then the sampling process eqn:ode-sampling with the learning rate schedule eqn:alpha-t satisfies for some universal constants $C_1>0$, where we recall that $p_1$ (resp. $q_1$) represents the distributi

Theorems & Definitions (26)

  • Definition 1: Score function
  • Remark 1
  • Theorem 1
  • Remark 2
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • ...and 16 more