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arXiv · 2506.08362

Solving Convex-Concave Problems with $\tilde{\mathcal{O}}(\epsilon^{-4/7})$ Second-Order Oracle Complexity

Lesi Chen·Chengchang Liu·Luo Luo·Jingzhao Zhang
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Abstract

Previous algorithms can solve convex-concave minimax problems $\min_{x \in \mathcal{X}} \max_{y \in \mathcal{Y}} f(x,y)$ with $\mathcal{O}(\epsilon^{-2/3})$ second-order oracle calls using Newton-type methods. This result has been speculated to be optimal because the upper bound is achieved by a natural generalization of the optimal first-order method. In this work, we show an improved upper bound of $\tilde{\mathcal{O}}(\epsilon^{-4/7})$ by generalizing the optimal second-order method for convex optimization to solve the convex-concave minimax problem. We further apply a similar technique to lazy Hessian algorithms and show that our proposed algorithm can also be seen as a second-order ``Catalyst'' framework (Lin et al., JMLR 2018) that could accelerate any globally convergent algorithms for solving minimax problems.

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Lesi Chen, Chengchang Liu, Luo Luo, Jingzhao Zhang. 2025-06-10. Solving Convex-Concave Problems with $\tilde{\mathcal{O}}(\epsilon^{-4/7})$ Second-Order Oracle Complexity. https://arxiv.org/abs/2506.08362

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