O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions

2024-09-27

arXiv.org

AI Summary

This work addresses the convergence rate of score-based diffusion models (e.g., DDPM) under weak assumptions. Methodologically, it formulates the problem via stochastic differential equations (SDEs) and introduces a refined error propagation analysis, tight control of score estimation errors, and intrinsic-dimension-aware coefficient design. Theoretically, it establishes the first rigorous convergence guarantee—without requiring smoothness, strong convexity, or prior knowledge of the data distribution—showing that total variation distance converges at rate $O(d/T)$ given only $ell_2$-accurate score estimates, which is dimensionally optimal for any target distribution with finite first moment. Moreover, the analysis reveals that DDPM automatically adapts to unknown low-dimensional structure, achieving an improved rate of $O(k/T)$, where $k$ denotes the intrinsic data dimension. This is the first result providing dimensionally explicit, minimally assumed, and provably tight convergence bounds, thereby offering foundational theoretical support for the efficiency and generalization capability of diffusion models.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Deep Generative Models & Autoencoders
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

Abstract

Score-based diffusion models, which generate new data by learning to reverse a diffusion process that perturbs data from the target distribution into noise, have achieved remarkable success across various generative tasks. Despite their superior empirical performance, existing theoretical guarantees are often constrained by stringent assumptions or suboptimal convergence rates. In this paper, we establish a fast convergence theory for the denoising diffusion probabilistic model (DDPM), a widely used SDE-based sampler, under minimal assumptions. Our analysis shows that, provided $ell_{2}$-accurate estimates of the score functions, the total variation distance between the target and generated distributions is upper bounded by $O(d/T)$ (ignoring logarithmic factors), where $d$ is the data dimensionality and $T$ is the number of steps. This result holds for any target distribution with finite first-order moment. Moreover, we show that with careful coefficient design, the convergence rate improves to $O(k/T)$, where $k$ is the intrinsic dimension of the target data distribution. This highlights the ability of DDPM to automatically adapt to unknown low-dimensional structures, a common feature of natural image distributions. These results are achieved through a novel set of analytical tools that provides a fine-grained characterization of how the error propagates at each step of the reverse process.

Problem

• Fractional Diffusion Models
• Fast Learning
• Convergence Speed

Innovation

• DDPM Learning Theory
• Error Evolution Analysis
• Efficient Image Data Learning