Convergence of the Inexact Langevin Algorithm in KL Divergence with Application to Score-based Generative Models

Abstract

Motivated by the increasingly popular Score-based Generative Modeling (SGM), we study the Inexact Langevin Dynamics (ILD) and Inexact Langevin Algorithm (ILA) where a score function estimate is used in place of the exact score. We establish {\em stable} biased convergence guarantees in terms of the Kullback-Leibler (KL) divergence. To achieve these guarantees, we impose two key assumptions: 1) the target distribution satisfies the log-Sobolev inequality, and 2) the error of score estimator exhibits a sub-Gaussian tail, referred to as Moment Generating Function (MGF) error assumption. Under the stronger $L^\infty$ score error assumption, we obtain a stable convergence bound in R\'enyi divergence. We also generalize the proof technique to SGM, and derive a stable convergence bound in KL divergence. In addition, we explore the question of how to obtain a provably accurate score estimator. We demonstrate that a simple estimator based on kernel density estimation fulfills the MGF error assumption for sub-Gaussian target distributions, at the population level.