# Single-Call Stochastic Extragradient Methods for Structured Non-monotone Variational Inequalities: Improved Analysis under Weaker Conditions

## TL;DR

This paper advances the analysis of single-call stochastic extragradient methods for non-monotone variational inequalities, providing convergence guarantees under weaker conditions and arbitrary sampling strategies, applicable to large-scale machine learning problems.

## Contribution

It introduces the expected residual condition, enabling convergence analysis under weaker assumptions and broad sampling strategies for structured non-monotone VIPs.

## Key findings

- Convergence guarantees under weaker conditions than bounded variance.
- Applicability to importance sampling and mini-batching.
- Theoretical analysis for various step-size rules.

## Abstract

Single-call stochastic extragradient methods, like stochastic past extragradient (SPEG) and stochastic optimistic gradient (SOG), have gained a lot of interest in recent years and are one of the most efficient algorithms for solving large-scale min-max optimization and variational inequalities problems (VIP) appearing in various machine learning tasks. However, despite their undoubted popularity, current convergence analyses of SPEG and SOG require a bounded variance assumption. In addition, several important questions regarding the convergence properties of these methods are still open, including mini-batching, efficient step-size selection, and convergence guarantees under different sampling strategies. In this work, we address these questions and provide convergence guarantees for two large classes of structured non-monotone VIPs: (i) quasi-strongly monotone problems (a generalization of strongly monotone problems) and (ii) weak Minty variational inequalities (a generalization of monotone and Minty VIPs). We introduce the expected residual condition, explain its benefits, and show how it can be used to obtain a strictly weaker bound than previously used growth conditions, expected co-coercivity, or bounded variance assumptions. Equipped with this condition, we provide theoretical guarantees for the convergence of single-call extragradient methods for different step-size selections, including constant, decreasing, and step-size-switching rules. Furthermore, our convergence analysis holds under the arbitrary sampling paradigm, which includes importance sampling and various mini-batching strategies as special cases.