Last Iterate Convergence of Popov Method for Non-monotone Stochastic Variational Inequalities

Abstract: This paper focuses on non-monotone stochastic variational inequalities (SVIs) that may not have a unique solution. A commonly used efficient algorithm to solve VIs is the Popov method, which is known to have the optimal convergence rate for VIs with Lipschitz continuous and strongly monotone operators. We introduce a broader class of structured non-monotone operators, namely $p$-quasi sharp operators ($p> 0$), which allows tractably analyzing convergence behavior of algorithms. We show that the stochastic Popov method converges almost surely to a solution for all operators from this class under a linear growth. In addition, we obtain the last iterate convergence rate (in expectation) for the method under a linear growth condition for $2$-quasi sharp operators. Based on our analysis, we refine the results for smooth $2$-quasi sharp and $p$-quasi sharp operators (on a compact set), and obtain the optimal convergence rates. We further provide numerical experiments that demonstrate advantages of stochastic Popov method over stochastic projection method for solving SVIs.