Variance reduction techniques for stochastic proximal point algorithms

TL;DR

This paper introduces a unified framework for variance reduction in stochastic proximal point algorithms, improving convergence rates and stability over traditional methods, especially for difficult optimization problems.

Contribution

It presents the first unified study of variance reduction techniques for stochastic proximal point algorithms, including new convergence results and practical algorithms.

Findings

Faster convergence rates than vanilla stochastic proximal point algorithms.

Proximal variance reduction methods show greater stability with respect to step size.

Numerical experiments confirm advantages over gradient-based methods.

Abstract

In the context of finite sums minimization, variance reduction techniques are widely used to improve the performance of state-of-the-art stochastic gradient methods. Their practical impact is clear, as well as their theoretical properties. Stochastic proximal point algorithms have been studied as an alternative to stochastic gradient algorithms since they are more stable with respect to the choice of the step size. However, their variance-reduced versions are not as well studied as the gradient ones. In this work, we propose the first unified study of variance reduction techniques for stochastic proximal point algorithms. We introduce a generic stochastic proximal-based algorithm that can be specified to give the proximal version of SVRG, SAGA, and some of their variants. For this algorithm, in the smooth setting, we provide several convergence rates for the iterates and the objective function values, which are faster than those of the vanilla stochastic proximal point algorithm. More specifically, for convex functions, we prove a sublinear convergence rate of $O(1/k)$. In addition, under the Polyak-{\L}ojasiewicz (PL) condition, we obtain linear convergence rates. Finally, our numerical experiments demonstrate the advantages of the proximal variance reduction methods over their gradient counterparts in terms of the stability with respect to the choice of the step size in most cases, especially for difficult problems.