Halpern Iteration Achieves $\tilde{\mathcal{O}}(ε^{-1/p})$ $p$th-Order Oracle Complexity for Monotone Variational Inequalities

2026-08-09

AI Summary

This work proposes a high-order algorithmic framework for solving smooth monotone variational inequality (MVI) problems by integrating a large-step inexact Halpern iteration with a newly introduced Anchored Tensor Method (ATM). It is the first approach to combine Halpern-type iterations with high-order tensor optimization, applicable to smooth MVI problems of arbitrary order $p \geq 2$. Under the standard high-order oracle model, the method achieves a convergence rate of $\tilde{\mathcal{O}}(T^{-p})$, corresponding to an oracle complexity of $\tilde{\mathcal{O}}(\varepsilon^{-1/p})$. Notably, for $p = 2$, it attains a rate of $\tilde{\mathcal{O}}(T^{-2})$, significantly outperforming existing algorithms and unifying—while surpassing—the current complexity barriers in high-order MVI solvers.

Abstract

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 $p=1$.

Problem

• monotone variational inequalities
• oracle complexity
• higher-order methods
• convergence rate
• Halpern iteration

Innovation

• Halpern iteration
• monotone variational inequalities
• higher-order methods
• oracle complexity
• Anchored Tensor Method