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For min-max optimization and variational inequalities problems (VIP)encountered in diverse machine learning tasks, Stochastic Extragradient (SEG)and Stochastic Gradient Descent Ascent (SGDA) have emerged as preeminentalgorithms. Constant step-size variants of SEG/SGDA have gained popularity,with appealing benefits such as easy tuning and rapid forgiveness of initialconditions, but their convergence behaviors are more complicated even inrudimentary bilinear models. Our work endeavors to elucidate and quantify theprobabilistic structures intrinsic to these algorithms. By recasting theconstant step-size SEG/SGDA as time-homogeneous Markov Chains, we establish afirst-of-its-kind Law of Large Numbers and a Central Limit Theorem,demonstrating that the average iterate is asymptotically normal with a uniqueinvariant distribution for an extensive range of monotone and non-monotoneVIPs. Specializing to convex-concave min-max optimization, we characterize therelationship between the step-size and the induced bias with respect to theVon-Neumann's value. Finally, we establish that Richardson-Rombergextrapolation can improve proximity of the average iterate to the globalsolution for VIPs. Our probabilistic analysis, underpinned by experimentscorroborating our theoretical discoveries, harnesses techniques fromoptimization, Markov chains, and operator theory.

Stochastic Methods in Variational Inequalities: Ergodicity, Bias and Refinements