Halpern迭代实现单调变分不等式的$\tilde{\mathcal{O}}(ε^{-1/p})$ $p$阶预言复杂度

中文摘要 AI 辅助

我们研究求解光滑单调变分不等式（MVI）的二阶及更高阶方法。Monteiro和Svaiter（SIAM J. Optim., 2012）证明了二阶方法NPE的收敛速率为$\mathcal{O}(T^{-1.5})$。对于MVI问题的一个子类——凸凹极小极大优化，Chen、Liu、Luo和Zhang（COLT 2025）近期将复杂度改进至$\tilde{\tilde{\mathcal{O}}}(T^{-1.75})$。然而，MVI的推测复杂度是否能被提升仍是开放问题。本文中，我们通过大步长非精确Halpern迭代，提出了一种新颖的Halpern-NPE方法，求解MVI时可达到$\tilde{\tilde{\mathcal{O}}}(T^{-2})$的更快速率。我们还给出了方法的$p$阶推广：首先引入锚定张量方法（Anchored Tensor Method, ATM），其收敛速率为$\mathcal{O}(T^{-(p-1)})$，随后将其与Halpern迭代结合，实现了$\tilde{\tilde{\mathcal{O}}}(T^{-p})$的更快收敛速率。该结果改进了所有$p \ge 2$时的已有结论，且在$p=1$时与经典外梯度法结果一致。

英文摘要

We study second- and higher-order methods for solving smooth monotone variational inequalities (MVI). Monteiro and Svaiter (SIAM J. Optim., 2012) showed that a second-order method, NPE, converges at the rate of $\mathcal{O}(T^{-1.5})$. For convex-concave minimax optimization, a subset of MVI problems, Chen, Liu, Luo, and Zhang (COLT 2025) recently improved the complexity to $\tilde{\mathcal{O}}( T^{-1.75})$ . However, it is open whether the conjectured complexity for MVI can be improved. In this paper, by using a large-step inexact Halpern iteration, we propose a novel Halpern-NPE method that achieves an even faster rate of $\tilde{\mathcal{O}}(T^{-2})$ for solving MVIs. We also provide the $p$th-order generalization of our method. We first introduce an Anchored Tensor Method (ATM) that achieves the rate of $\mathcal{O}(T^{-(p-1)})$, and then combine it with the Halpern iteration to achieve a faster convergence rate of $\tilde{\mathcal{O}}(T^{-p})$. This improves all prior results for $p \ge 2$ and matches the classical extragradient method for $p=1$.

发表机构

• IIIS, Tsinghua University（清华大学智能产业研究院）
• College of AI, Tsinghua University（清华大学人工智能学院）
• Apex Intelligence（顶点智能科技）
• School of Mathematical Sciences, Tongji University（同济大学数学科学学院）
• Department of Artificial Intelligence, Westlake University（西湖大学人工智能系）

机构由 AI 辅助整理，请以论文原文为准。