Learning Variational Inequalities from Data: Fast Generalization Rates under Strong Monotonicity
Year: 2024
Type: preprint
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 $Θ(1/ε)$ stochastic first-order oracle calls to find an $ε$-optimal solution, rather than the standard $Θ(1/ε^2)$ calls. This note provides a simple overview of how one can similarly obtain fast $Θ(1/ε)$ 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.
Source: arXiv (Cornell University)
Authors: Zhao, Eric, Chavdarova, Tatjana, Jordan, Michael
Institution: University of California, Berkeley
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4404314268