Regularized Iterative Stochastic Approximation Methods for Variational Inequality Problems

We consider a Cartesian stochastic variational inequality problem with a monotone map. For this problem, we develop and analyze distributed iterative stochastic approximation algorithms. Such a problem arises, for example, as an equilibrium problem in monotone stochastic Nash games over continuous strategy sets. We introduce two classes of stochastic approximation methods, each of which requires exactly one projection step at every iteration, and we provide convergence analysis for them. Of these, the first is the stochastic iterative Tikhonov regularization method which necessitates the update of regularization parameter after every iteration. The second method is a stochastic iterative proximal-point method, where the centering term is updated after every iteration. Notably, we present a generalization of this method where the weighting in the proximal-point method can also be updated after every iteration. Conditions are provided for recovering global convergence in limited coordination extensions of such schemes where players are allowed to choose their steplength sequences independently, while meeting a suitable coordination requirement. We apply the proposed class of techniques and their limited coordination versions to stochastic networked rate allocation problem.