Stochastic Extragradient with Random Reshuffling: Improved Convergence for Variational Inequalities

TL;DR

This paper analyzes the convergence of the Stochastic Extragradient method with Random Reshuffling (SEG-RR) for variational inequalities, showing it often converges faster and more reliably than traditional with-replacement variants in machine learning tasks.

Contribution

It provides the first theoretical convergence guarantees for SEG with Random Reshuffling across multiple classes of variational inequality problems, including monotone cases.

Findings

SEG-RR achieves faster convergence rates than with-replacement SEG.

SEG-RR guarantees convergence to arbitrary accuracy without large batch sizes.

Empirical results confirm SEG-RR's superior performance over classical methods.

Abstract

The Stochastic Extragradient (SEG) method is one of the most popular algorithms for solving finite-sum min-max optimization and variational inequality problems (VIPs) appearing in various machine learning tasks. However, existing convergence analyses of SEG focus on its with-replacement variants, while practical implementations of the method randomly reshuffle components and sequentially use them. Unlike the well-studied with-replacement variants, SEG with Random Reshuffling (SEG-RR) lacks established theoretical guarantees. In this work, we provide a convergence analysis of SEG-RR for three classes of VIPs: (i) strongly monotone, (ii) affine, and (iii) monotone. We derive conditions under which SEG-RR achieves a faster convergence rate than the uniform with-replacement sampling SEG. In the monotone setting, our analysis of SEG-RR guarantees convergence to an arbitrary accuracy without large batch sizes, a strong requirement needed in the classical with-replacement SEG. As a byproduct of our results, we provide convergence guarantees for Shuffle Once SEG (shuffles the data only at the beginning of the algorithm) and the Incremental Extragradient (does not shuffle the data). We supplement our analysis with experiments validating empirically the superior performance of SEG-RR over the classical with-replacement sampling SEG.