A Unified Kullback--Leibler Divergence Analysis of Generative Diffusion Models via Entropy Production Rate

2026-08-03

AI Summary

This work proposes a unified framework to precisely quantify the impact of various error sources on the terminal KL divergence in generative diffusion models. By analyzing the entropy production rate of forward–backward diffusion process pairs at the marginal distribution level, the terminal KL divergence is decomposed into three components: initialization error, score approximation error, and time discretization error. The approach uniquely circumvents path-space analysis by leveraging the continuity equation and modulation of diffusion coefficients, enabling a theoretical proof that the Euler–Maruyama sampler achieves an $O(h^2)$ convergence rate—substantially improving upon the conventional $O(h)$ result. This framework cohesively encompasses score-based SDEs, probability flow ODEs, and stochastic interpolation methods. Numerical experiments corroborate the predicted scaling relationships between step size, terminal time, and sampling accuracy.

Abstract

We introduce a unified framework for the error analysis of generative models based on the entropy production rate of the forward-reverse diffusion process pair. For a pair of continuity equation flows, the rate admits a closed velocity form identity whose time integral decomposes the terminal Kullback--Leibler (KL) divergence into the sum of an initialization error, a score approximation error, and a time-discretization error. By analyzing the entropy production at the level of marginal distributions, rather than in path space, our framework yields a sharp convergence rate of $\mathcal{O}(h^2)$ for the Euler-Maruyama sampler, where $h$ is the step size. This improves upon the $\mathcal{O}(h)$ rates typically obtained from Girsanov's path-space analyses. Furthermore, our framework unifies the analysis of score-based SDEs, probability-flow ODEs, and stochastic interpolants by varying diffusion coefficients within a single inequality, revealing the trade-off between deterministic and stochastic sampling. Numerical experiments confirm the predicted scaling with step size and terminal time.

Problem

• Kullback-Leibler divergence
• generative diffusion models
• entropy production rate
• error analysis
• convergence rate

Innovation

• entropy production rate
• Kullback-Leibler divergence
• diffusion models
• convergence rate
• unified analysis