A Sharp Uniform-in-Time Error Estimate for Stochastic Gradient Langevin Dynamics (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. Undermild assumptions, we obtain a uniform-in-time $\mathcal{O}(η^2)$ bound for the Kullback-Leiblerdivergence between the SGLD iteration and the Langevin diffusion, where $η$ is the stepsize (or learning rate). Our analysis is also valid for varying step sizes. Consequently,we are able to derive an $\mathcal{O}(η)$ bound for the distance between the invariant measuresof 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 withexisting analysis for SGLD in related literature.
Keywords: Random batch; Euler-Maruyama scheme; Fokker-Planck equation; log-Sobolev inequality.