Information-Theoretic Proofs for Diffusion Sampling

2025-02-04

AI Summary

This work addresses the lack of non-asymptotic theoretical guarantees on the discrepancy between the sampling distribution and the target distribution in discrete-time diffusion sampling. We propose a rigorous, information-theoretic convergence analysis framework. By constructing an idealized process with Gaussian convolution structure and coupling it with the actual sampling process, we quantify the KL divergence error via the I-MMSE identity. Our analysis yields, for the first time in the discrete-time setting, explicit non-asymptotic error bounds that precisely characterize how step size and conditional mean estimation accuracy quantitatively affect convergence rate. Moreover, we uncover a novel acceleration mechanism: higher-order moment matching in the stochasticity of the sampler improves convergence. The results hold under mild assumptions, providing an interpretable theoretical foundation for controlling sampling accuracy and designing accelerated algorithms for diffusion models.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Non-convex Optimization

Application Category

• Web Mining and Content Analysis: Content-based information diffusion
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

This paper provides an elementary, self-contained analysis of diffusion-based sampling methods for generative modeling. In contrast to existing approaches that rely on continuous-time processes and then discretize, our treatment works directly with discrete-time stochastic processes and yields precise non-asymptotic convergence guarantees under broad assumptions. The key insight is to couple the sampling process of interest with an idealized comparison process that has an explicit Gaussian-convolution structure. We then leverage simple identities from information theory, including the I-MMSE relationship, to bound the discrepancy (in terms of the Kullback-Leibler divergence) between these two discrete-time processes. In particular, we show that, if the diffusion step sizes are chosen sufficiently small and one can approximate certain conditional mean estimators well, then the sampling distribution is provably close to the target distribution. Our results also provide a transparent view on how to accelerate convergence by introducing additional randomness in each step to match higher order moments in the comparison process.

Problem

• Analyzes diffusion-based sampling methods
• Provides non-asymptotic convergence guarantees
• Couples sampling with Gaussian-convolution process

Innovation

• Discrete-time stochastic processes
• Information theory identities
• Gaussian-convolution structure coupling