Expected Variational Inequalities

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.

Many real-world problems—such as predicting how consumers and companies make decisions in markets or training deep learning models—can be described using a mathematical framework called variational inequalities (VIs). The catch is that VIs are often too complex to solve, especially when the problem is large.Our main research question is whether there’s a way to combine the flexibility of VIs with the ability to find solutions efficiently. We came up with a new formulation called expected variational inequalities (EVIs). Rather than finding a point that solves the problem exactly, EVIs aim for a solution that performs well on average across many points. EVIs are inspired by ideas from economics, specifically game theory. There, moving from VIs to EVIs involves introducing a correlation device that enables the players to coordinate their actions—much like a traffic light helps two drivers at a crossroad coordinate who goes first.