Learning Variational Inequalities from Data: Fast Generalization Rates under Strong Monotonicity

Abstract

Variational inequalities (VIs) are a broad class of optimization problems encompassing machine learning problems ranging from standard convex minimization to more complex scenarios like min-max optimization and computing the equilibria of multi-player games. In convex optimization, strong convexity allows for fast statistical learning rates requiring only $\hat{I}˜(1/\hat{I}\mu )$ stochastic first-order oracle calls to find an $\hat{I}\mu$-optimal solution, rather than the standard $\hat{I}˜(1/\hat{I}{\mu }^2)$ calls. This note provides a simple overview of how one can similarly obtain fast $\hat{I}˜(1/\hat{I}\mu )$ rates for learning VIs that satisfy strong monotonicity, a generalization of strong convexity. Specifically, we demonstrate that standard stability-based generalization arguments for convex minimization extend directly to VIs when the domain admits a small covering, or when the operator is integrable and suboptimality is measured by potential functions; such as when finding equilibria in multi-player games.