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

• URL: http://arxiv.org/abs/2207.09304v1
• Date: Tue, 19 Jul 2022 14:38:52 GMT
• Title: A sharp uniform-in-time error estimate for Stochastic Gradient Langevin Dynamics
• Authors: Lei Li and Yuliang Wang
• Abstract summary: We establish a sharp uniform-in-time error estimate for the Gradient Langevin Dynamics (SGLD) We obtain a uniform-in-time $O(eta2)$ bound for the KL-divergence between the SGLD iteration and the Langevin diffusion.
• Score: 11.437892757715877
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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