On the Theory of Variance Reduction for Stochastic Gradient Monte Carlo

TL;DR

This paper establishes convergence guarantees in Wasserstein distance for various variance-reduction stochastic gradient Langevin methods under strong convexity and smoothness assumptions, providing theoretical bounds and empirical validation.

Contribution

It introduces a novel proof technique combining optimization and sampling analysis to derive convergence bounds for variance-reduction Langevin methods.

Findings

Variance-reduction methods achieve improved convergence rates.

Theoretical bounds identify regimes where each method excels.

Experimental results confirm the theoretical predictions.

Abstract

We provide convergence guarantees in Wasserstein distance for a variety of variance-reduction methods: SAGA Langevin diffusion, SVRG Langevin diffusion and control-variate underdamped Langevin diffusion. We analyze these methods under a uniform set of assumptions on the log-posterior distribution, assuming it to be smooth, strongly convex and Hessian Lipschitz. This is achieved by a new proof technique combining ideas from finite-sum optimization and the analysis of sampling methods. Our sharp theoretical bounds allow us to identify regimes of interest where each method performs better than the others. Our theory is verified with experiments on real-world and synthetic datasets.