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

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 Theta(1/ε) stochastic first-order oracle calls to find an ε-optimal solution, rather than the standard Theta(1/ε2) calls. This note provides a simple overview of how one can similarly obtain fast Theta(1/ε) 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.