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Motivated by multi-user optimization problems and non-cooperative Nash gamesin uncertain regimes, we consider stochastic Cartesian variational inequalities(SCVI) where the set is given as the Cartesian product of a collection ofcomponent sets. First, we consider the case where the number of the componentsets is large. For solving this type of problems, the classical stochasticapproximation methods and their prox generalizations are computationallyinefficient as each iteration becomes very costly. To address this challenge,we develop a randomized block stochastic mirror-prox (B-SMP) algorithm, whereat each iteration only a randomly selected block coordinate of the solution isupdated through implementing two consecutive projection steps. Under standardassumptions on the problem and settings of the algorithm, we show that when themapping is strictly pseudo-monotone, the algorithm generates a sequence ofiterates that converges to the solution of the problem almost surely. To deriverate statements, we assume that the maps are strongly pseudo-monotone andobtain {a non-asymptotic mean squared error$\mathcal{O}\left(\frac{d}{k}\right)$, where $k$ is the iteration number and$d$ is the number of component sets. Second, we consider large-scale stochasticoptimization problems with convex objectives. For this class of problems, wedevelop a new averaging scheme for the B-SMP algorithm. Unlike the classicalaveraging stochastic mirror-prox (SMP) method where a decreasing set of weightsfor the averaging sequence is used, here we consider a different set of weightsthat are characterized in terms of the stepsizes and a {parameter}. We showthat using such weights, the objective values of the averaged sequenceconverges to the optimal value in the mean sense with the rate$\mathcal{O}\left(\frac{\sqrt{d}}{\sqrt{k}}\right)$.

On stochastic mirror-prox algorithms for stochastic Cartesian variational inequalities: randomized block coordinate and optimal averaging schemes