On Smoothing, Regularization and Averaging in Stochastic Approximation Methods for Stochastic Variational Inequalities

Abstract

Traditionally, stochastic approximation schemes for SVIs have relied on strong monotonicity and Lipschitzian properties of the underlying map. In contrast, we consider monotone stochastic variational inequality (SVI) problems where the strong monotonicity and Lipschitzian assumptions on the mappings are weakened. In the first part of the paper, to address such shortcomings, a regularized smoothed SA (RSSA) scheme is developed wherein the stepsize, smoothing, and regularization parameters are diminishing sequences updated after every iteration. Under suitable assumptions on the sequences, we show that the algorithm generates iterates that converge to a solution in an almost sure sense, extending the results in [16] to the non-Lipschitzian regime. Motivated by the need to develop non-asymptotic rate statements, in the second part of the paper, we develop a variant of the RSSA scheme, denoted by aRSSA$_r$, in which we employ a weighted iterate-averaging, parametrized by a scalar $r$ where $r = 1$ provides us with the standard averaging scheme. We make several contributions in this context: First, we show that the gap function associated with the sequences by the aRSSA$_r$ scheme tends to zero when the parameter sequences are chosen appropriately. Second, we show that the gap function associated with the averaged sequence diminishes to zero at the optimal rate $\cal{O}(1/\sqrt{K})$ after $K$ steps when smoothing and regularization are suppressed and $r < 1$, thus improving the rate statement for the standard averaging which admits a rate of $\cal{O}(\ln(K)/\sqrt{K})$. Third, we develop a window-based variant of this scheme that also displays the optimal rate for $r < 1$. Notably, we prove the superiority of the scheme with $r < 1$ with its counterpart with $r=1$ in terms of the constant factor of the error bound when the size of the averaging window is sufficiently large.