Solving Stochastic Variational Inequalities without the Bounded Variance Assumption

2026-02-05

AI Summary

This work addresses the challenge of solving stochastic variational inequalities (VIs) under unbounded variance and unbounded domains, a setting particularly relevant to nonconvex-nonconcave min-max optimization. Focusing on monotone VIs and structured non-monotone VIs satisfying a weak Minty condition, the paper introduces a novel stochastic optimization algorithm that dispenses with the commonly assumed bounded-variance condition on gradient noise. Under a milder assumption—where the noise variance grows at most quadratically with the norm of the variable—the proposed method achieves an oracle complexity of $\tilde{O}(\varepsilon^{-4})$, matching the current best-known theoretical guarantee for constrained stochastic VIs. This result significantly relaxes traditional restrictions on noise distributions and substantially broadens the class of solvable stochastic VI problems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We analyze algorithms for solving stochastic variational inequalities (VI) without the bounded variance or bounded domain assumptions, where our main focus is min-max optimization with possibly unbounded constraint sets. We focus on two classes of problems: monotone VIs; and structured nonmonotone VIs that admit a solution to the weak Minty VI. The latter assumption allows us to solve structured nonconvex-nonconcave min-max problems. For both classes of VIs, to make the expected residual norm less than $\varepsilon$, we show an oracle complexity of $\widetilde{O}(\varepsilon^{-4})$, which is the best-known for constrained VIs. In our setting, this complexity had been obtained with the bounded variance assumption in the literature, which is not even satisfied for bilinear min-max problems with an unbounded domain. We obtain this complexity for stochastic oracles whose variance can grow as fast as the squared norm of the optimization variable.

Problem

• stochastic variational inequalities
• min-max optimization
• unbounded domain
• bounded variance assumption
• nonconvex-nonconcave

Innovation

• stochastic variational inequalities
• unbounded variance
• min-max optimization
• weak Minty condition
• oracle complexity