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

We study first-order methods for solving monotone variational inequalities arising in min-max optimization. Classical approaches such as the extragradient method rely on two gradient queries per iteration, which limits their analysis and applicability in the online and stochastic settings. We propose a family of Generalized Optimistic Methods with Anchoring (GOMA), which combine two-time-scale optimistic updates with an anchoring term inspired by Halpern iteration. In the deterministic setting, GOMA achieves the optimal accelerated last-iterate rate $O(1/k^2)$ on the squared gradient norm for monotone Lipschitz operators. In the stochastic setting with unbounded variance, a simplified single-call variant of GOMA achieves a last-iterate convergence rate of $O(1/\sqrt{k})$ on the squared gradient norm. To the best of our knowledge, this is the first such guarantee for stochastic monotone Lipschitz variational inequalities in the unconstrained setting without variance reduction or growing batches.

We study Polyak-type step-size selection for extragradient methods for solving deterministic and stochastic monotone root-finding problems. We show that the known projection-type correction for deterministic extragradient arises from minimizing an upper bound on the distance to a solution, paralleling the classical Polyak step-size construction. Using this viewpoint, we provide a unified deterministic analysis of the Polyak-type Extragradient Method (PolyakEG), based on a local critical condition controlling the variation of operator $F$ along the extrapolation direction. This analysis does not require global Lipschitz continuity, and covers sublinear convergence under broader conditions such as Hölder continuity or $(L_0, L_1)$-Lipschitzness and linear convergence under additional strong monotonicity, all through a single framework. We then study the stochastic extensions of this approach. We first prove convergence of a direct stochastic variant, PolyakSEG, when all stochastic component operators share a common solution. We also show that, without this condition, PolyakSEG with nonvanishing step-sizes may fail to converge to a zero of the mean operator. To address this limitation, we propose DecPolyakSEG, which combines decreasing step-sizes with Polyak-type updates, and establish a sublinear residual convergence result without requiring a common solution across the component operators. These results parallel recent developments in stochastic Polyak step-sizes from the convex minimization literature and establish an analogous research avenue in the broader root-finding regime.

We analyze a stochastic algorithm with Halpern anchoring for constrained convex-concave problems and monotone variational inequalities. This algorithm is single-loop and single-call since it uses one unbiased sample of the gradient operator at every iteration to be applicable to monotone games with noisy feedback. With $t$ denoting the iteration counter, we prove the anytime last-iterate convergence rate of $O(t^{-1/4})$ for both gradient-mapping norm and restricted gap, improving the best-known rate $O(t^{-1/5})$ that was obtained for the restricted gap function. Our rates cover constrained problems with a potentially unbounded feasible set as well as a structured class of stochastic oracles without a bounded variance.