Min-Max Optimization Requires Exponentially Many Queries

2026-05-13

AI Summary

This work addresses the computational complexity of finding an ε-approximate stationary point in nonconvex–nonconcave minimax optimization problems defined over the unit hypercube. Under the standard oracle model where only function values and gradients are accessible, the paper establishes the first exponential lower bound on query complexity: the number of required oracle queries grows exponentially either in the inverse accuracy parameter 1/ε or in the problem dimension. This result rigorously demonstrates the intrinsic difficulty of nonconvex–nonconcave minimax optimization, implying that no polynomial-time algorithm can exist for this general setting.

Abstract

We study the query complexity of min-max optimization of a nonconvex-nonconcave function $f$ over $[0,1]^d \times [0,1]^d$. We show that, given oracle access to $f$ and to its gradient $\nabla f$, any algorithm that finds an $\varepsilon$-approximate stationary point must make a number of queries that is exponential in $1/\varepsilon$ or $d$.

Problem

• min-max optimization
• nonconvex-nonconcave
• query complexity
• stationary point
• oracle access

Innovation

• min-max optimization
• nonconvex-nonconcave
• query complexity
• stationary point
• oracle lower bound