  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.
