On stochastic gradient Langevin dynamics with dependent data streams: the fully non-convex case

TL;DR

This paper provides non-asymptotic convergence analysis of Stochastic Gradient Langevin Dynamics (SGLD) algorithms for sampling from complex, non-log-concave distributions, even with dependent data streams, improving existing bounds.

Contribution

It offers sharper, uniform convergence estimates for SGLD in non-convex settings with dependent data, extending prior work to more general data dependencies.

Findings

Non-asymptotic $L^1$-Wasserstein convergence bounds established.

Analysis accommodates dependent data streams in gradient estimation.

Results are sharper and uniform across iterations.

Abstract

We consider the problem of sampling from a target distribution, which is \emph {not necessarily logconcave}, in the context of empirical risk minimization and stochastic optimization as presented in Raginsky et al. (2017). Non-asymptotic analysis results are established in the $L^1$-Wasserstein distance for the behaviour of Stochastic Gradient Langevin Dynamics (SGLD) algorithms. We allow the estimation of gradients to be performed even in the presence of \emph{dependent} data streams. Our convergence estimates are sharper and \emph{uniform} in the number of iterations, in contrast to those in previous studies.