Min-Max Optimization Is Strictly Easier Than Variational Inequalities

TL;DR

This paper demonstrates that solving convex-concave min-max problems directly can be faster than solving their associated variational inequalities, especially in unconstrained quadratic cases, due to the asymmetry of min and max variables.

Contribution

It shows that min-max optimization can be strictly easier than variational inequalities in certain settings, providing new insights into algorithmic convergence rates.

Findings

Min-max algorithms outperform variational inequality methods in unconstrained quadratic problems.

Asymmetry of min and max variables enables faster convergence in min-max problems.

Sharp convergence rate characterizations are derived using extremal polynomials, Green's functions, and conformal mappings.

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.