On Forgetting and Stability of Score-based Generative models

TL;DR

This paper analyzes the stability and error propagation in score-based generative models using Markov chain properties, providing theoretical bounds and insights into their long-term behavior.

Contribution

It introduces a framework with Lyapunov and Doeblin conditions to quantify stability and error propagation in score-based models.

Findings

Quantitative bounds on sampling error derived

Stability ensured by contraction in reverse diffusion dynamics

Framework clarifies the role of stochasticity in generative models

Abstract

Understanding the stability and long-time behavior of generative models is a fundamental problem in modern machine learning. This paper provides quantitative bounds on the sampling error of score-based generative models by leveraging stability and forgetting properties of the Markov chain associated with the reverse-time dynamics. Under weak assumptions, we provide the two structural properties to ensure the propagation of initialization and discretization errors of the backward process: a Lyapunov drift condition and a Doeblin-type minorization condition. A practical consequence is quantitative stability of the sampling procedure, as the reverse diffusion dynamics induces a contraction mechanism along the sampling trajectory. Our results clarify the role of stochastic dynamics in score-based models and provide a principled framework for analyzing propagation of errors in such approaches.