A large-step inexact Halpern iteration with an anchored tensor method yields tilde-O(epsilon^{-1/p}) p-th order oracle complexity for smooth monotone variational inequalities for all p >= 2.
Lower bounds for higher-order convex optimization
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Halpern Iteration Achieves $\tilde{\mathcal{O}}(\epsilon^{-1/p})$ $p$th-Order Oracle Complexity for Monotone Variational Inequalities
A large-step inexact Halpern iteration with an anchored tensor method yields tilde-O(epsilon^{-1/p}) p-th order oracle complexity for smooth monotone variational inequalities for all p >= 2.