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cs.LG updates on arXiv.org

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Learning Variational Inequalities from Data: Fast General...
Eric Zhao, Tatjana Chavdarova, Michael Jordan · 2024-10-28 · via cs.LG updates on arXiv.org

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.