(untitled) (CSIAM Transactions on Applied Mathematics, 2025)
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(η^2) bound for the Kullback-Leibler divergence between the SGLD iteration and the Langevin diffusion, where η is the step size (or learning rate). Our analysis is also valid for varying step sizes. Consequently, we are able to derive an O(η) 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.