A sharp uniform-in-time error estimate for Stochastic Gradient Langevin Dynamics

TL;DR

This paper provides a precise, uniform-in-time error estimate for SGLD, showing that the divergence from the ideal Langevin diffusion remains controlled over time, with bounds depending on the step size.

Contribution

The authors derive a sharp uniform-in-time $O( ext{step size}^2)$ KL-divergence bound for SGLD, improving upon previous analyses and valid for varying step sizes.

Findings

KL-divergence between SGLD and Langevin diffusion is bounded by $O( ext{step size}^2)$ uniformly over time.

The invariant measures of SGLD and Langevin diffusion are close within $O( ext{step size})$ in Wasserstein or total variation distance.

Analysis applies under mild assumptions and accommodates varying step sizes.

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 $O(\eta^2)$ bound for the KL-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 $O(\eta)$ 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.