Perseus: A Simple and Optimal High-Order Method for Variational Inequalities

This paper settles an open and challenging question pertaining to the design of simple high-order regularization methods for solving smooth and monotone variational inequalities (VIs). A VI involves finding $x^\star \in \XCal$ such that $\langle F(x), x - x^\star\rangle \geq 0$ for all $x \in \XCal$ and we consider the setting where $F: \br^d \mapsto \br^d$ is smooth with up to $(p-1)^{\textnormal{th}}$-order derivatives. High-order methods based on similar binary search procedures have been further developed and shown to achieve a rate of $O(\epsilon^{-2/(p+1)}\log(1/\epsilon))$~\citep{Bullins-2020-Higher,Lin-2021-Monotone,Jiang-2022-Generalized}. However, such search procedure can be computationally prohibitive in practice~\citep{Nesterov-2018-Lectures} and the problem of finding a simple high-order regularization methods remains as an open and challenging question in the optimization theory. We propose a $p^{\textnormal{th}}$-order method that does \textit{not} require any binary search procedure and prove that it can converge to a weak solution at a global rate of $O(\epsilon^{-2/(p+1)})$. A lower bound of $\Omega(\epsilon^{-2/(p+1)})$ is also established under a linear span assumption to show that our $p^{\textnormal{th}}$-order method is optimal in the monotone setting. A version with restarting attains a global linear and local superlinear convergence rate for smooth and strongly monotone VIs. Our method can achieve a global rate of $O(\epsilon^{-2/p})$ for solving smooth and non-monotone VIs satisfying the Minty condition. The restarted version again attains a global linear and local superlinear convergence rate if the strong Minty condition holds.