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

We settle an open and challenging question pertaining to the design of simple and optimal high-order methods for solving smooth and monotone variational inequalities (VIs). A VI involves finding such that for all and we consider the setting in which is smooth with up to -order derivatives. For , the cubic regularized Newton's method has been extended to VIs with a global rate of . An improved rate of can be obtained via an alternative second-order method, but this method requires a nontrivial line-search procedure as an inner loop. Similarly, high-order methods based on similar line-search procedures have been shown to achieve a rate of , but the inner loop requires fine-tuning of parameters and can be computationally complex. As highlighted by Nesterov, it would be desirable to develop a simple high-order VI method that retains the optimality of the more complex methods. We propose a -order method that does \textit{not} require any search procedure and provably converges to a weak solution at a rate of . We prove that our -order method is optimal in the monotone setting by establishing a lower bound of under a linear span assumption. A version with restarting attains a global linear and local superlinear convergence rate for smooth and strongly monotone VIs. Furthermore, our method achieves a global rate of for solving smooth and nonmonotone VIs satisfying the Minty condition. The restarted version again attains a global linear and local superlinear convergence rate if the strong Minty condition is satisfied.
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