Min-Max Optimization Is Strictly Easier Than Variational Inequalities

2025-11-04

AI Summary

This work investigates whether convex-concave minimax optimization can be solved directly—bypassing the standard variational inequality (VI) reformulation—to achieve faster convergence. Focusing on the canonical setting of unconstrained quadratic objectives, it establishes, for the first time, that the optimal first-order convergence rate for minimax problems strictly outperforms that of their VI counterparts. The key insight is the exploitation of inherent asymmetry between primal and dual variables—a structural property obscured in the VI formulation and responsible for its suboptimal rate. Using extremal polynomial techniques grounded in Green’s functions and conformal mapping, the authors precisely characterize the asymptotic optimal convergence rates for both formulations. Under the standard first-order oracle model, minimax optimization admits a strictly superior asymptotic convergence rate. This result provides new theoretical foundations and a structured pathway for algorithmic acceleration, challenging the conventional VI-based design paradigm.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Security and Privacy: Large-scale security measurements

Abstract

Classically, a mainstream approach for solving a convex-concave min-max problem is to instead solve the variational inequality problem arising from its first-order optimality conditions. Is it possible to solve min-max problems faster by bypassing this reduction? This paper initiates this investigation. We show that the answer is yes in the textbook setting of unconstrained quadratic objectives: the optimal convergence rate for first-order algorithms is strictly better for min-max problems than for the corresponding variational inequalities. The key reason that min-max algorithms can be faster is that they can exploit the asymmetry of the min and max variables--a property that is lost in the reduction to variational inequalities. Central to our analyses are sharp characterizations of optimal convergence rates in terms of extremal polynomials which we compute using Green's functions and conformal mappings.

Problem

• Min-max optimization bypasses variational inequalities for faster convergence
• Exploiting variable asymmetry improves min-max problem solving efficiency
• Optimal convergence rates characterized through extremal polynomial analysis

Innovation

• Min-max optimization bypasses variational inequality reduction
• Exploits asymmetry between min and max variables
• Uses extremal polynomials with Green's functions analysis