On Same-Sample and Independent-Sample Stochastic Extragradient for Monotone Variational Inequalities

2026-08-06

AI Summary

This work addresses the convergence gap and lack of theoretical guarantees between single-sample (S-SEG) and independent-sample (I-SEG) stochastic extragradient methods for monotone variational inequalities over unbounded domains. It systematically analyzes the behavior of both algorithms and establishes that mere averaged Lipschitz continuity and bounded variance are insufficient to ensure S-SEG convergence, revealing its sensitivity to per-sample Lipschitz constants. The study further demonstrates that the asymmetric two-stepsize strategy effective for I-SEG may fail for S-SEG. Under weaker assumptions, it derives sharp high-probability bounds on the restricted gap function for both methods, showing these bounds are unimprovable. Through carefully constructed counterexamples and high-probability analysis, the work clarifies the fundamental distinctions between S-SEG and I-SEG, thereby filling a critical theoretical void in the convergence analysis of S-SEG over unbounded domains.

Abstract

We study stochastic extragradient (SEG) methods for solving monotone variational inequality problems (VIPs) over a feasible set. Although extragradient is a foundational algorithm for VIPs and its deterministic convergence theory is well developed, its stochastic counterpart remains less understood. Most existing analyses focus on independent-sample SEG (I-SEG) and assume either that the domain is compact or that the variance of the stochastic operator is uniformly bounded. The behavior of same-sample SEG (S-SEG), a natural variant with materially different properties, has received far less attention. In this work, we address these gaps in the literature. We first show that S-SEG is sensitive to samplewise Lipschitz parameters: mean Lipschitzness and bounded variance alone do not ensure convergence, even on a compact set. Then, for possibly unbounded domains, we establish a high-probability restricted-gap convergence for each SEG variant under a relaxed set of assumptions, and show that certain fundamental improvements to these results are impossible in general. Finally, we show that a known asymmetric double step-size selection that guarantees almost sure last-iterate convergence for I-SEG can fail for S-SEG: there exists a stochastic monotone VIP for which S-SEG diverges almost surely even under the modified step-sizes.

Problem

• stochastic extragradient
• monotone variational inequalities
• same-sample SEG
• independent-sample SEG
• convergence analysis

Innovation

• stochastic extragradient
• monotone variational inequalities
• same-sample SEG
• convergence analysis
• step-size selection