Expected Variational Inequalities

TL;DR

This paper introduces expected variational inequalities (EVIs), a relaxation of traditional VIs, which can be solved efficiently and generalize many existing models across various fields.

Contribution

The paper proposes EVIs, a new relaxation of VIs, demonstrating polynomial-time solvability and broad applicability beyond game theory.

Findings

EVIs can be solved in polynomial time under general operators.

EVIs generalize correlated equilibria and other models.

Framework captures diverse settings like smooth games and constrained utilities.

Abstract

Variational inequalities (VIs) encompass many fundamental problems in diverse areas ranging from engineering to economics and machine learning. However, their considerable expressivity comes at the cost of computational intractability. In this paper, we introduce and analyze a natural relaxation -- which we refer to as expected variational inequalities (EVIs) -- where the goal is to find a distribution that satisfies the VI constraint in expectation. By adapting recent techniques from game theory, we show that, unlike VIs, EVIs can be solved in polynomial time under general (nonmonotone) operators. EVIs capture the seminal notion of correlated equilibria, but enjoy a greater reach beyond games. We also employ our framework to capture and generalize several existing disparate results, including from settings such as smooth games, and games with coupled constraints or nonconcave utilities.