Halpern Iteration Achieves tildeO(ε-1/p) pth-Order Oracle Complexity for Monotone Variational Inequalities

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 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 tildeO( 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 tildeO(T-2) for solving MVIs. We also provide the pth-order generalization of our method. We first introduce an Anchored Tensor Method (ATM) that achieves the rate of O(T-(p-1)), and then combine it with the Halpern iteration to achieve a faster convergence rate of tildeO(T-p). This improves all prior results for p ≥ 2 and matches the classical extragradient method for p=1.