Perseus: A Simple and Optimal High-Order Method for Variational Inequalities
Tianyi Lin, Michael. I. Jordan
TL;DR
Perseus introduces a simple $p$th-order VI method that avoids any line search by integrating an adaptive dual-extrapolation framework with regularized high-order Taylor models. It achieves a global convergence rate of $O\left(\varepsilon^{-2/(p+1)}\right)$ for monotone VIs and a matching lower bound under a generalized linear-span assumption, confirming optimality. Restarting yields linear convergence under uniform/strong monotonicity and can deliver local superlinear convergence for $p\ge 2$ under strong Minty conditions; the method also extends to nonmonotone VIs satisfying Minty-type assumptions with a global rate of $O\left(\varepsilon^{-2/p}\right)$. These results collectively close the gap between theory and practical high-order VI methods by offering a simple, scalable, line-search-free approach with rigorous guarantees and broad applicability to Minty-compliant nonmonotone problems. The work thus has significant implications for large-scale VI problems in ML and economics, enabling high-order accuracy without inner line searches or complex subproblem solvers.
Abstract
This paper settles 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 $x^\star \in \mathcal{X}$ such that $\langle F(x), x - x^\star\rangle \geq 0$ for all $x \in \mathcal{X}$. We consider the setting in which $F$ is smooth with up to $(p-1)^{th}$-order derivatives. For $p = 2$, the cubic regularized Newton method was extended to VIs with a global rate of $O(ε^{-1})$. An improved rate of $O(ε^{-2/3}\log\log(1/ε))$ 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 line-search procedures have been shown to achieve a rate of $O(ε^{-2/(p+1)}\log\log(1/ε))$. As emphasized by Nesterov, however, such procedures do not necessarily imply practical applicability in large-scale applications, and it would be desirable to complement these results with a simple high-order VI method that retains the optimality of the more complex methods. We propose a $p^{th}$-order method that does \textit{not} require any line search procedure and provably converges to a weak solution at a rate of $O(ε^{-2/(p+1)})$. We prove that our $p^{th}$-order method is optimal in the monotone setting by establishing a matching lower bound under a generalized linear span assumption. Our method with restarting attains a linear rate for smooth and uniformly monotone VIs and a local superlinear rate for smooth and strongly monotone VIs. Our method also achieves a global rate of $O(ε^{-2/p})$ for solving smooth and nonmonotone VIs satisfying the Minty condition and when augmented with restarting it attains a global linear and local superlinear rate for smooth and nonmonotone VIs satisfying the uniform/strong Minty condition.