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

What the math gives to ML

The paper's transferable asset is an anchored fixed-point principle for monotone variational inequalities: rather than repeatedly applying an unanchored high-order solver, it blends each approximate solution with a fixed initial anchor through Halpern iteration. The reported rate improves from the base pth-order rate O(T^{-(p-1)}) to approximately O~(T^{-p}), suggesting a principled stabilization and acceleration mechanism for gradient-based minimax training. A practical neural-network adaptation is to apply the method to convex-concave game objectives using first- and second-order operator information, while retaining the anchor to suppress rotational or cycling dynamics. The first experiment should use a p=2 version on small GAN and bilinear-game benchmarks, comparing convergence, gradient evaluations, and stability against extragradient and optimistic Adam.