A Sharp Uniform-in-Time Error Estimate for Stochastic Gradient Langevin Dynamics

Abstract

We establish a sharp uniform-in-time error estimate for the stochastic gradient Langevin dynamics (SGLD), which is a widely-used sampling algorithm. Under mild assumptions, we obtain a uniform-in-time $\mathcal{O}\left({\eta }^{2}\right)$ bound for the Kullback-Leibler divergence between the SGLD iteration and the Langevin diffusion, where $\eta $ is the step size (or learning rate). Our analysis is also valid for varying step sizes. Consequently, we are able to derive an $\mathcal{O}\left(\eta \right)$ bound for the distance between the invariant measures of the SGLD iteration and the Langevin diffusion, in terms of Wasserstein or total variation distances. Our result can be viewed as a significant improvement compared with existing analysis for SGLD in related literature.

Keywords

Random batch / Euler-Maruyama scheme / Fokker-Planck equation / log-Sobolev inequality