@Article{CSIAM-AM-6-711,
author = {Li , Lei and Wang , Yuliang},
title = {A Sharp Uniform-in-Time Error Estimate for Stochastic Gradient Langevin Dynamics},
journal = {CSIAM Transactions on Applied Mathematics},
year = {2025},
volume = {6},
number = {4},
pages = {711--759},
abstract = {<p style="text-align: justify;">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 $\mathcal{O}(η^2)$ 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 $\mathcal{O}(\eta)$&nbsp;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.</p>},
issn = {2708-0579},
doi = {https://doi.org/10.4208/csiam-am.SO-2024-0039},
url = {http://global-sci.org/intro/article_detail/csiam-am/24501.html}
}