The Complexity of Min-Max Optimization with Product Constraints

TL;DR

The paper proves that computing local min-max equilibria for smooth $f:[0,1]^d\times[0,1]^d\to[0,1]$ is PPAD-hard even under natural product constraints on the hypercube, by focusing on the GDA problem: find $(x,y)$ such that $\max_{x'\in[0,1]^d}\langle \nabla_x f(x,y),x'-x\rangle\le\epsilon$ and $\max_{y'\in[0,1]^d}-\langle \nabla_y f(x,y),y'-y\rangle\le\epsilon$. The authors build a reduction from End-of-the-Line through two PPAD-complete problems, Pure-Circuit and LinVI, employing a gate-based construction with copies of variables and a quadratic regularizer to control a noise term $\Delta_q(x,y)$. A central idea is to design a linking function $H_q$ and a gating mechanism so that any GDA solution either yields a LinVI solution or induces a valid Pure-Circuit assignment, thereby solving the End-of-the-Line instance. The resulting smooth objective $f(x,y)=\sum_q s_q(x,y)H_q(x,y)+\varphi(x,y)$ encodes both gadgets, with $f$ being Lipschitz and $L$-smooth, and the analysis shows that the hardness persists with polynomially bounded parameters. This resolves an open question about the complexity of min-max optimization under product constraints and highlights fundamental limits for poly$(1/\epsilon)$ algorithms in this setting, while leaving open questions about constants, black-box access, and constant-dimension regimes.

Abstract

We study the computational complexity of the problem of computing local min-max equilibria of games with a nonconvex-nonconcave utility function $f$. From the work of Daskalakis, Skoulakis, and Zampetakis [DSZ21], this problem was known to be hard in the restrictive case in which players are required to play strategies that are jointly constrained, leaving open the question of its complexity under more natural constraints. In this paper, we settle the question and show that the problem is PPAD-hard even under product constraints and, in particular, over the hypercube.

Figures (3)

• Figure 1: Dependency graph of a simple Pure-Circuit instance and influence graph of the variable $x_q$ on the utility function $f$ defined in \ref{['eq:firstutil']}.
• Figure 2: Illustration of the smooth functions $g$, $\ell$ and $\lambda$.
• Figure 3: Given an instance $I_{\textnormal{E}OTL}$ of End-of-the-Line we build an instance $I_{PC}$ of Pure-Circuit and $I_{\textnormal{VI}}$ of LinVI that we combine in an instance $I_{\textnormal{G}DA}$ of GDA. From a solution $(x,y)$ of GDA, we check if any of the $x_i^q$ are a solution to LinVI; if not, we build an assignment $b:V\to\{0,1,\bot\}$ to Pure-Circuit according to \ref{['eq:purecirc']}. \ref{['lm:main']} assures that either one of the $x_i^q$ is a solution to the LinVI instance $I_{\textnormal{V}I}$ (Case A) or that $b$ is a valid assignment to the instance $I_{\textnormal{P}C}$ of Pure-Circuit (Case B). In either case, we can then build a solution to the original End-of-the-Line instance $I_{\textnormal{E}OTL}$.

Theorems & Definitions (23)

• Remark 2.1
• Remark 3.1
• Theorem 3.2: deligkas2022pure
• Theorem 3.3: bernasconi2024roleconstraintscomplexityminmax
• Lemma 5.0
• Remark 5.1
• Lemma 6.1: Dichotomy Lemma
• Theorem 6.2
• proof
• Lemma 7.1