Improving the Last-Iterate Guarantees of Anytime Algorithms for Stochastic Monotone Variational Inequalities

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.