Solving Stochastic Variational Inequalities without the Bounded Variance Assumption
This work addresses a fundamental limitation in solving stochastic VIs for optimization problems with unbounded domains, such as bilinear min-max, by removing the bounded variance assumption, which is incremental but practically important for broader applicability.
The paper tackles stochastic variational inequalities (VI) without requiring bounded variance or bounded domain assumptions, focusing on monotone and structured nonmonotone VIs, including nonconvex-nonconcave min-max problems. It achieves an oracle complexity of O~(ε^{-4}) to reduce the expected residual norm below ε, matching the best-known rate previously only attainable under stricter assumptions.
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.