Stochastic approximation of CVaR-based variational inequalities

TL;DR

This paper develops stochastic approximation algorithms for solving variational inequalities defined by CVaR, providing convergence guarantees under various conditions and demonstrating their effectiveness through simulations.

Contribution

It introduces novel stochastic approximation schemes for CVaR-based VIs, analyzing their convergence properties and establishing relationships between sample size, error, and solution accuracy.

Findings

Algorithms converge to exact or neighborhood solutions depending on CVaR estimation accuracy.

Explicit bounds relate sample size and estimation error to convergence neighborhood.

Simulation confirms theoretical convergence results.

Abstract

In this paper we study variational inequalities (VI) defined by the conditional value-at-risk (CVaR) of uncertain functions. We introduce stochastic approximation schemes that employ an empirical estimate of the CVaR at each iteration to solve these VIs. We investigate convergence of these algorithms under various assumptions on the monotonicity of the VI and accuracy of the CVaR estimate. Our first algorithm is shown to converge to the exact solution of the VI when the estimation error of the CVaR becomes progressively smaller along any execution of the algorithm. When the estimation error is nonvanishing, we provide two algorithms that provably converge to a neighborhood of the solution of the VI. For these schemes, under strong monotonicity, we provide an explicit relationship between sample size, estimation error, and the size of the neighborhood to which convergence is achieved. A simulation example illustrates our theoretical findings.