Paper ID: 2207.09304

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

Lei Li, Yuliang Wang

We establish a sharp uniform-in-time error estimate for the Stochastic Gradient Langevin Dynamics (SGLD), which is a popular 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. Based on this, we are able to obtain an $O(\eta)$ bound for the distance between the SGLD iteration and the invariant distribution of the Langevin diffusion, in terms of Wasserstein or total variation distances.

Submitted: Jul 19, 2022