Solving Convex-Concave Problems with $\tilde{\mathcal{O}}(ε^{-4/(3p+1)})$ $p$th-Order Oracle Complexity
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866911611902492672 |
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| author | Chen, Lesi Zhang, Xinliang Liu, Chengchang Li, Junru Luo, Luo Zhang, Jingzhao |
| author_facet | Chen, Lesi Zhang, Xinliang Liu, Chengchang Li, Junru Luo, Luo Zhang, Jingzhao |
| contents | When the objective has Lipschitz continuous $p$th-order derivatives, it is known that convex-concave minimax problems can be solved with $\mathcal{O}(ε^{-2/(p+1)})$ $p$th-order oracle calls. This complexity upper bound was speculated to be optimal as it is achieved by a natural generalization of the optimal first-order method. In this work, we show an improved upper bound of $\tilde{\mathcal{O}}(ε^{-4/(3p+1)})$ by applying the Monteiro-Svaiter acceleration. We also establish a lower complexity bound of $Ω(ε^{-2/(3p-1)})$, suggesting a gap still exists for $p \ge 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19462 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Solving Convex-Concave Problems with $\tilde{\mathcal{O}}(ε^{-4/(3p+1)})$ $p$th-Order Oracle Complexity Chen, Lesi Zhang, Xinliang Liu, Chengchang Li, Junru Luo, Luo Zhang, Jingzhao Optimization and Control When the objective has Lipschitz continuous $p$th-order derivatives, it is known that convex-concave minimax problems can be solved with $\mathcal{O}(ε^{-2/(p+1)})$ $p$th-order oracle calls. This complexity upper bound was speculated to be optimal as it is achieved by a natural generalization of the optimal first-order method. In this work, we show an improved upper bound of $\tilde{\mathcal{O}}(ε^{-4/(3p+1)})$ by applying the Monteiro-Svaiter acceleration. We also establish a lower complexity bound of $Ω(ε^{-2/(3p-1)})$, suggesting a gap still exists for $p \ge 2$. |
| title | Solving Convex-Concave Problems with $\tilde{\mathcal{O}}(ε^{-4/(3p+1)})$ $p$th-Order Oracle Complexity |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2604.19462 |