A regularized adaptive steplength stochastic approximation scheme for monotone stochastic variational inequalities

We consider the solution of monotone stochastic variational inequalities and present an adaptive steplength stochastic approximation framework with possibly multivalued mappings. Traditional implementations of SA have been characterized by two challenges. First, convergence of standard SA schemes requires a strongly or strictly monotone single-valued mapping, a requirement that is rarely met. Second, while convergence requires that the steplength sequences need to satisfy Σ_kγ_k = ∞ and Σ_kγ_k² <; ∞, little guidance is provided on a choice of sequences. In fact, standard choices such as γ_k = 1/k may often perform poorly in practice. Motivated by the minimization of a suitable error bound, a recursive rule for prescribing steplengths is proposed for strongly monotone problems. By introducing a regularization sequence, extensions to merely monotone regimes are proposed. Finally, an iterative smoothing extension is suggested for accommodating multivalued mappings. Preliminary numerical results suggest that the schemes prove effective.