Stochastic Extragradient Methods Beyond Lipschitz Continuity

Stochastic variational inequality problems have received significant attention due to their ability to model large-scale machine learning tasks. Typically, the operators involved are assumed to be globally Lipschitz, with only a handful of works extending beyond the standard setting to the so-called $(L_0,L_1)$-Lipschitz setting. In this work, we propose a novel stochastic past extragradient algorithm for variational inequality problems under a broad regularity assumption that is even less restrictive than $(L_0,L_1)$-Lipschitz continuity and provide convergence guarantees under bounded variance of the stochastic oracle. More precisely, we show a.s. convergence with suitable stepsize schedules and obtain convergence rates in expectation and in probability for fixed horizons. Finally, in numerical simulations we compare our proposed algorithm against standard methods.