On the Role of Constraints in the Complexity of Min-Max Optimization

TL;DR

This paper explores how constraints influence the computational difficulty of min-max optimization problems, revealing that constraints can make even convex-concave cases PPAD-hard, with new simplified proofs and broader implications.

Contribution

It provides a new, simpler proof of PPAD-hardness for constrained min-max problems, extends hardness results to general constraints, and discusses related quasi-variational inequalities.

Findings

Constraints are crucial for the hardness of nonconvex-nonconcave min-max problems.

Even convex-concave min-max problems become PPAD-hard with general constraints.

A new proof technique simplifies understanding the role of constraints in complexity.

Abstract

We investigate the role of constraints in the computational complexity of min-max optimization. The work of Daskalakis, Skoulakis, and Zampetakis [2021] was the first to study min-max optimization through the lens of computational complexity, showing that min-max problems with nonconvex-nonconcave objectives are PPAD-hard. However, their proof hinges on the presence of joint constraints between the maximizing and minimizing players. The main goal of this paper is to understand the role of these constraints in min-max optimization. The first contribution of this paper is a fundamentally new proof of their main result, which improves it in multiple directions: it holds for degree 2 polynomials, it is essentially tight in the parameters, and it is much simpler than previous approaches, clearly highlighting the role of constraints in the hardness of the problem. Second, we show that with general constraints (i.e., the min player and max player have different constraints), even convex-concave min-max optimization becomes PPAD-hard. Along the way, we also provide PPAD-membership of a general problem related to quasi-variational inequalities, which has applications beyond our problem.