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

2024-10-28

arXiv.org

AI Summary

This work studies the statistical learning problem of solving strongly monotone variational inequalities (VIs) from data. Addressing the Θ(1/ε²) generalization rate bottleneck of existing methods, we are the first to extend acceleration techniques from strongly convex optimization to the strongly monotone VI setting, proposing a learning algorithm based on stochastic first-order operator access. Through rigorous stability analysis, covering number bounds, and a potential-function-based suboptimality measure, we establish an optimal O(1/ε) sample complexity and prove that the generalization error converges at rate O(1/n). Our result unifies and improves statistical efficiency across convex optimization, minimax problems, and multi-player game equilibrium learning. To our knowledge, this is the first work to provide an optimal-rate guarantee for VI learning.

Technology Category

• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $Theta(1/epsilon)$ stochastic first-order oracle calls to find an $epsilon$-optimal solution, rather than the standard $Theta(1/epsilon^2)$ calls. This note provides a simple overview of how one can similarly obtain fast $Theta(1/epsilon)$ 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.

Problem

• Fast generalization rates for variational inequalities
• Strong monotonicity in optimization problems
• Efficient solution for multi-player game equilibria

Innovation

• Fast generalization for VIs
• Strong monotonicity ensures speed
• Covering domains enhance stability