Solving Stochastic Variational Inequalities without the Bounded Variance Assumption

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.

💡 Research Summary

The paper tackles stochastic variational inequalities (SVI) without relying on the standard bounded‑variance (BV) or bounded‑domain (BD) assumptions, which are often violated in practical min‑max problems with unbounded constraints. The authors focus on two problem families: (i) monotone SVIs, which correspond to convex‑concave min‑max formulations where the operator (G) is monotone, and (ii) structured non‑monotone SVIs that admit a solution to a weak Minty variational inequality (wMVI). The wMVI condition introduces a non‑monotonicity parameter (\rho\ge 0); (\rho=0) recovers the classic Minty VI, while (\rho>0) allows a controlled amount of non‑monotonicity.

A key technical contribution is the replacement of the classical BV condition (\mathbb{E}|eG(z)-G(z)|^{2}\le\sigma^{2}) with a much weaker growth condition \