  We consider the stochastic variational inequality problem in which the map is
expectation-valued in a component-wise sense. Much of the available convergence
theory and rate statements for stochastic approximation schemes are limited to
monotone maps. However, non-monotone stochastic variational inequality problems
are not uncommon and are seen to arise from product pricing, fractional
optimization problems, and subclasses of economic equilibrium problems.
Motivated by the need to address a broader class of maps, we make the following
contributions: (i) We present an extragradient-based stochastic approximation
scheme and prove that the iterates converge to a solution of the original
problem under either pseudomonotonicity requirements or a suitably defined
acute angle condition. Such statements are shown to be generalizable to the
stochastic mirror-prox framework; (ii) Under strong pseudomonotonicity, we show
that the mean-squared error in the solution iterates produced by the
extragradient SA scheme converges at the optimal rate of O(1/k) statements that
were hitherto unavailable K in this regime. Notably, we optimize the initial
steplength by obtaining an {\epsilon}-infimum of a discontinuous nonconvex
function. Similar statements are derived for mirror-prox generalizations and
can accommodate monotone SVIs under a weak-sharpness requirement. Finally, both
the asymptotics and the empirical rates of the schemes are studied on a set of
variational problems where it is seen that the theoretically specified initial
steplength leads to significant performance benefits.
