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

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 $O(\eta^2)$ 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. Consequently, we are able to derive an $O(\eta)$ 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.

@article{li2025_2207.09304,
  title={ A sharp uniform-in-time error estimate for Stochastic Gradient Langevin Dynamics },
  author={ Lei Li and Yuliang Wang },
  journal={arXiv preprint arXiv:2207.09304},
  year={ 2025 }
}