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

Understand

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.