Solving Stochastic Variational Inequalities without the Bounded Variance Assumption

TL;DR

This paper develops algorithms for solving stochastic variational inequalities without assuming bounded variance or bounded domains, achieving optimal complexity for challenging unbounded min-max problems.

Contribution

It introduces methods that handle unbounded variance in stochastic VIs, extending the scope beyond previous bounded variance assumptions, especially for unbounded domains.

Findings

Achieves oracle complexity of (\u03b5) = \u007eO(\u03b5^{-4}) for constrained VIs.

Extends complexity results to problems with unbounded variance growing with the variable norm.

Provides solutions for structured nonmonotone VIs with weak Minty solutions.

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.