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arXiv · 2609.14235

Optimal Deterministic First-Order Oracle Complexity for Nonconvex-Concave Minimax Optimization

Siyu Pan·Taoli Zheng·Jiajin Li
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Abstract

We study the deterministic first-order oracle complexity of smooth nonconvex-concave minimax optimization over a bounded convex dual domain. Let $\ell$ denote the joint smoothness constant, $D_{\mathcal{Y}}$ the diameter of the dual domain, and $Δ$ the initial gap. We prove that every deterministic first-order algorithm requires $Ω(\ell^2D_{\mathcal{Y}}Δ/ε^3)$ oracle queries in the worst case to find an $ε$-optimization-stationary point whenever $ε\lesssim\min\{\ell D_{\mathcal{Y}},\sqrt{\ellΔ}\}$. We then develop Tracked-FOAM, a first-order method that attains a matching upper bound, removing the logarithmic factor from previous upper bounds. Together, these results establish the optimal dependence on all problem parameters in the stated regime.

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Siyu Pan, Taoli Zheng, Jiajin Li. 2026-09-13. Optimal Deterministic First-Order Oracle Complexity for Nonconvex-Concave Minimax Optimization. https://arxiv.org/abs/2609.14235

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