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We consider the stochastic variational inequality problem in which the map isexpectation-valued in a component-wise sense. Much of the available convergencetheory and rate statements for stochastic approximation schemes are limited tomonotone maps. However, non-monotone stochastic variational inequality problemsare not uncommon and are seen to arise from product pricing, fractionaloptimization problems, and subclasses of economic equilibrium problems.Motivated by the need to address a broader class of maps, we make the followingcontributions: (i) We present an extragradient-based stochastic approximationscheme and prove that the iterates converge to a solution of the originalproblem under either pseudomonotonicity requirements or a suitably definedacute angle condition. Such statements are shown to be generalizable to thestochastic mirror-prox framework; (ii) Under strong pseudomonotonicity, we showthat the mean-squared error in the solution iterates produced by theextragradient SA scheme converges at the optimal rate of O(1/k) statements thatwere hitherto unavailable K in this regime. Notably, we optimize the initialsteplength by obtaining an {\epsilon}-infimum of a discontinuous nonconvexfunction. Similar statements are derived for mirror-prox generalizations andcan accommodate monotone SVIs under a weak-sharpness requirement. Finally, boththe asymptotics and the empirical rates of the schemes are studied on a set ofvariational problems where it is seen that the theoretically specified initialsteplength leads to significant performance benefits.

Optimal stochastic extragradient schemes for pseudomonotone stochastic variational inequality problems and their variants