Optimal Convergence Analysis of DDPM for General Distributions

• URL: http://arxiv.org/abs/2510.27562v1
• Date: Fri, 31 Oct 2025 15:44:50 GMT
• Title: Optimal Convergence Analysis of DDPM for General Distributions
• Authors: Yuchen Jiao, Yuchen Zhou, Gen Li,
• Abstract summary: Denoising Diffusion Probabilistic Model (DDPM) is one of the most widely used samplers.<n>We provide a refined convergence analysis of the DDPM sampler.<n>We show that our convergence analysis is tight for a wide array of target distributions.
• Score: 11.155024379105788
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Score-based diffusion models have achieved remarkable empirical success in generating high-quality samples from target data distributions. Among them, the Denoising Diffusion Probabilistic Model (DDPM) is one of the most widely used samplers, generating samples via estimated score functions. Despite its empirical success, a tight theoretical understanding of DDPM -- especially its convergence properties -- remains limited. In this paper, we provide a refined convergence analysis of the DDPM sampler and establish near-optimal convergence rates under general distributional assumptions. Specifically, we introduce a relaxed smoothness condition parameterized by a constant $L$, which is small for many practical distributions (e.g., Gaussian mixture models). We prove that the DDPM sampler with accurate score estimates achieves a convergence rate of $$\widetilde{O}\left(\frac{d\min\{d,L^2\}}{T^2}\right)~\text{in Kullback-Leibler divergence},$$ where $d$ is the data dimension, $T$ is the number of iterations, and $\widetilde{O}$ hides polylogarithmic factors in $T$. This result substantially improves upon the best-known $d^2/T^2$ rate when $L < \sqrt{d}$. By establishing a matching lower bound, we show that our convergence analysis is tight for a wide array of target distributions. Moreover, it reveals that DDPM and DDIM share the same dependence on $d$, raising an interesting question of why DDIM often appears empirically faster.

Related papers

• Generalization Properties of Score-matching Diffusion Models for Intrinsically Low-dimensional Data [32.72306410557258]
  We study the statistical convergence of score-based diffusion models for learning an unknown distribution $$ from finitely many samples.<n>Our results demonstrate that diffusion models naturally adapt to the intrinsic geometry of data.<n>Our theory conceptually bridges the analysis of diffusion models with that of GANs and the sharp minimax rates established in optimal transport.
  arXiv  Detail & Related papers  (2026-03-04T03:59:02Z)
• O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions [6.76974373198208]
  We establish a fast convergence theory for the denoising diffusion probabilistic model (DDPM) under minimal assumptions.<n>We show that the convergence rate improves to $O(k/T)$, where $k$ is the intrinsic dimension of the target data distribution.<n>This highlights the ability of DDPM to automatically adapt to unknown low-dimensional structures.
  arXiv  Detail & Related papers  (2024-09-27T17:59:10Z)
• Non-asymptotic bounds for forward processes in denoising diffusions: Ornstein-Uhlenbeck is hard to beat [49.1574468325115]
  This paper presents explicit non-asymptotic bounds on the forward diffusion error in total variation (TV)<n>We parametrise multi-modal data distributions in terms of the distance $R$ to their furthest modes and consider forward diffusions with additive and multiplicative noise.
  arXiv  Detail & Related papers  (2024-08-25T10:28:31Z)
• Convergence Analysis of Probability Flow ODE for Score-based Generative Models [5.939858158928473]
  We study the convergence properties of deterministic samplers based on probability flow ODEs from both theoretical and numerical perspectives.<n>We prove the total variation between the target and the generated data distributions can be bounded above by $mathcalO(d3/4delta1/2)$ in the continuous time level.
  arXiv  Detail & Related papers  (2024-04-15T12:29:28Z)
• Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions [11.222970035173372]
  kernel-based score estimator achieves an optimal mean square error of $widetildeOleft(n-1 t-fracd+22(tfracd2 vee 1)right) We show that a kernel-based score estimator achieves an optimal mean square error of $widetildeOleft(n-1/2 t-fracd4right)$ upper bound for the total variation error of the distribution of the sample generated by the diffusion model under a mere sub-Gaussian
  arXiv  Detail & Related papers  (2024-02-23T20:51:31Z)
• Broadening Target Distributions for Accelerated Diffusion Models via a Novel Analysis Approach [49.97755400231656]
  We show that a new accelerated DDPM sampler achieves accelerated performance for three broad distribution classes not considered before.<n>Our results show an improved dependency on the data dimension $d$ among accelerated DDPM type samplers.
  arXiv  Detail & Related papers  (2024-02-21T16:11:47Z)
• Nearly $d$-Linear Convergence Bounds for Diffusion Models via Stochastic Localization [40.808942894229325]
  We provide the first convergence bounds which are linear in the data dimension. We show that diffusion models require at most $tilde O(fracd log2(1/delta)varepsilon2)$ steps to approximate an arbitrary distribution.
  arXiv  Detail & Related papers  (2023-08-07T16:01:14Z)
• Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models [49.81937966106691]
  We develop a suite of non-asymptotic theory towards understanding the data generation process of diffusion models. In contrast to prior works, our theory is developed based on an elementary yet versatile non-asymptotic approach.
  arXiv  Detail & Related papers  (2023-06-15T16:30:08Z)
• Stochastic Approximation Approaches to Group Distributionally Robust Optimization and Beyond [89.72693227960274]
  This paper investigates group distributionally robust optimization (GDRO) with the goal of learning a model that performs well over $m$ different distributions. To reduce the number of samples in each round from $m$ to 1, we cast GDRO as a two-player game, where one player conducts and the other executes an online algorithm for non-oblivious multi-armed bandits. In the second scenario, we propose to optimize the average top-$k$ risk instead of the maximum risk, thereby mitigating the impact of distributions.
  arXiv  Detail & Related papers  (2023-02-18T09:24:15Z)
• A Unified Framework for Multi-distribution Density Ratio Estimation [101.67420298343512]
  Binary density ratio estimation (DRE) provides the foundation for many state-of-the-art machine learning algorithms. We develop a general framework from the perspective of Bregman minimization divergence. We show that our framework leads to methods that strictly generalize their counterparts in binary DRE.
  arXiv  Detail & Related papers  (2021-12-07T01:23:20Z)
• Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction [63.41789556777387]
  Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP) We show that the number of samples needed to yield an entrywise $varepsilon$-accurate estimate of the Q-function is at most on the order of $frac1mu_min (1-gamma)5varepsilon2+ fract_mixmu_min (1-gamma)$ up to some logarithmic factor.
  arXiv  Detail & Related papers  (2020-06-04T17:51:00Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.