Stochastic Approximation Based Confidence Regions for Stochastic Variational Inequalities

TL;DR

This paper develops methods to construct asymptotic confidence regions for solutions to stochastic variational inequalities using stochastic approximation, focusing on the stochastic dual average method and providing practical procedures with simulations.

Contribution

It introduces a framework for creating confidence regions for SVIP solutions using SA, including establishing asymptotic normality and covariance estimation techniques.

Findings

Asymptotic normality of SA solutions in ergodic and non-ergodic senses.

Methods for online covariance matrix estimation.

Numerical simulations demonstrating confidence region construction.

Abstract

The sample average approximation (SAA) and the stochastic approximation (SA) are two popular schemes for solving the stochastic variational inequalities problem (SVIP). In the past decades, theories on the consistency of the SAA solutions and SA solutions have been well studied. More recently, the asymptotic confidence regions of the true solution to SVIP have been constructed when the SAA scheme is implemented. It is of fundamental interest to develop confidence regions of the true solution to the SVIP when the SA scheme is employed. In this paper, we discuss the framework of constructing asymptotic confidence regions for the true solution of SVIP with a focus on stochastic dual average method. We first establish the asymptotic normality of the SA solutions both in ergodic sense and non-ergodic sense. Then the online methods of estimating the covariance matrices in the normal distributions are studied. Finally, practical procedures of building the asymptotic confidence regions of solutions to SVIP with numerical simulations are presented.