On smoothing, regularization, and averaging in stochastic approximation methods for stochastic variational inequality problems

Traditionally, most stochastic approximation (SA) schemes for stochastic variational inequality (SVI) problems have required the underlying mapping to be either strongly monotone or monotone and Lipschitz continuous. In contrast, we consider SVIs with merely monotone and non-Lipschitzian maps. We develop a regularized smoothed SA (RSSA) scheme wherein the stepsize, smoothing, and regularization parameters are reduced after every iteration at a prescribed rate. Under suitable assumptions on the sequences, we show that the algorithm generates iterates that converge to the least norm solution in an almost sure sense, extending the results in Koshal et al. (IEEE Trans Autom Control 58(3):594–609, 2013) to the non-Lipschitzian regime. Additionally, we provide rate estimates that relate iterates to their counterparts derived from a smoothed Tikhonov trajectory associated with a deterministic problem. To derive non-asymptotic rate statements, we develop a variant of the RSSA scheme, denoted by aRSSA$$_r$$r, in which we employ a weighted iterate-averaging, parameterized by a scalar r where $$r = 1$$r=1 provides us with the standard averaging scheme. The main contributions are threefold: (i) when $$r<1$$r<1 and the parameter sequences are chosen appropriately, we show that the averaged sequence converges to the least norm solution almost surely and a suitably defined gap function diminishes at an approximate rate $$\mathcal{O}({1}\slash {\root 6 \of {k}})$$O(1/k6) after k steps; (ii) when $$r<1$$r<1, and smoothing and regularization are suppressed, the gap function admits the rate $$\mathcal{O}({1}\slash {\sqrt{k}})$$O(1/k), thus improving the rate $$\mathcal{O}(\ln (k)/\sqrt{k})$$O(ln(k)/k) under standard averaging; and (iii) we develop a window-based variant of this scheme that also displays the optimal rate for $$r < 1$$r<1. Notably, we prove the superiority of the scheme with $$r < 1$$r<1 with its counterpart with $$r=1$$r=1 in terms of the constant factor of the error bound when the size of the averaging window is sufficiently large. We present the performance of the developed schemes on a stochastic Nash–Cournot game with merely monotone and non-Lipschitzian maps.