Rényi divergence improves understanding of diffusion model convergence
Convergence of Denoising Diffusion Probabilistic Models: A Rényi Divergence Perspective
Information Theory
Summary
Denoising diffusion probabilistic models (DDPMs) generate data by gradually turning noise into realistic outputs, but measuring how well they work can miss subtle errors in rare cases. The authors use a stronger method called Rényi divergence to better capture these differences. They break down the error into parts related to mixing, discretization, and score estimation, and show how these errors combine when the output is close to the true distribution. Their analysis also applies to a variant called critically damped DDPMs, showing it likely offers no big theoretical improvements.
What this means in practice
- •For machine learning engineers: Design diffusion-based generative models with improved theoretical guarantees on convergence in rare-event distributions.
- •For statistical physicists: Analyze diffusion processes using Rényi divergence to better characterize convergence behavior in kinetic models.
A theory result. No direct application yet.
Authors
Wenliang Liu, Santiago Mazuelas, Guanding Yu, Renjie Liao, Lele Wang, Yuan Shen
Abstract
Denoising diffusion probabilistic models (DDPMs) have emerged as a powerful framework for generative modeling by transforming standard Gaussian inputs into outputs that approximate a target distribution. While DDPMs have been proved to converge in total variation, KL divergence, and Wasserstein distance, these discrepancy measures share a fundamental limitation of being insensitive to distributional mismatch in low-probability regions. To address this, we adopt the stronger Rényi divergence as the discrepancy measure. Leveraging a Girsanov-type change-of-measure argument, we establish the first Rényi characterization of DDPMs to the best of our knowledge, showing that the total error decomposes into a mixing error, a discretization error, and a score estimation error. We assume the target distribution is sub-Gaussian and characterize the score error via the cumulant generating function, relaxing both the log-Sobolev-inequality and $L^\infty$-score-error conditions used in prior Rényi divergence analyses of inexact Langevin Monte Carlo, a score-based sampling algorithm closely related to DDPM. Our results also establish an iteration complexity for achieving $ε$-accuracy in $α$-Rényi divergence that is asymptotically proportional to $α^3/ε$. Finally, by extending our analysis, we establish the first Rényi divergence convergence guarantee for critically damped DDPMs, a kinetic variant governed by critically damped Langevin dynamics. Similar to existing characterizations of critically damped DDPMs, this result shares the structural scaling of standard DDPMs, differing by at most a logarithmic factor in the mixing error exponent, and supports the conjecture that critically damped DDPMs do not yield a theoretical improvement.