A Proximal-Point Algorithm with Variable Sample-sizes (PPAWSS) for Monotone Stochastic Variational Inequality Problems

TL;DR

This paper introduces PPAWSS, a novel proximal-point algorithm with variable sample sizes for monotone stochastic variational inequality problems, achieving optimal rates and improved complexity.

Contribution

It develops the first proximal-point algorithms with variable sample-sizes for monotone SVIs, providing both linear and sublinear convergence guarantees.

Findings

Achieves linear convergence in strongly monotone regimes.

Attains sublinear convergence for merely monotone maps.

Preliminary results show competitive performance against existing methods.

Abstract

We consider a stochastic variational inequality (SVI) problem with a continuous and monotone mapping over a closed and convex set. In strongly monotone regimes, we present a variable sample-size averaging scheme (VS-Ave) that achieves a linear rate with an optimal oracle complexity. In addition, the iteration complexity is shown to display a muted dependence on the condition number compared with standard variance-reduced projection schemes. To contend with merely monotone maps, we develop amongst the first proximal-point algorithms with variable sample-sizes (PPAWSS), where increasingly accurate solutions of strongly monotone SVIs are obtained via (VS-Ave) at every step. This allows for achieving a sublinear convergence rate that matches that obtained for deterministic monotone VIs. Preliminary numerical evidence suggests that the schemes compares well with competing schemes.