Expected Variational Inequalities
This provides a computationally efficient framework for problems in machine learning, engineering, and economics, but it is incremental as it builds on existing game theory techniques.
The paper tackles the computational intractability of variational inequalities (VIs) by introducing expected variational inequalities (EVIs), a relaxation that finds a distribution satisfying VI constraints in expectation, and shows that EVIs can be solved in polynomial time under general nonmonotone operators.
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.