Optimal Methods for Higher-Order Smooth Monotone Variational Inequalities

In this work, we present new simple and optimal algorithms for solving the variational inequality (VI) problem for $p^{th}$-order smooth, monotone operators -- a problem that generalizes convex optimization and saddle-point problems. Recent works (Bullins and Lai (2020), Lin and Jordan (2021), Jiang and Mokhtari (2022)) present methods that achieve a rate of $\tilde{O}(\epsilon^{-2/(p+1)})$ for $p\geq 1$, extending results by (Nemirovski (2004)) and (Monteiro and Svaiter (2012)) for $p=1,2$. A drawback to these approaches, however, is their reliance on a line search scheme. We provide the first $p^{\textrm{th}}$-order method that achieves a rate of $O(\epsilon^{-2/(p+1)}).$ Our method does not rely on a line search routine, thereby improving upon previous rates by a logarithmic factor. Building on the Mirror Prox method of Nemirovski (2004), our algorithm works even in the constrained, non-Euclidean setting. Furthermore, we prove the optimality of our algorithm by constructing matching lower bounds. These are the first lower bounds for smooth MVIs beyond convex optimization for $p > 1$. This establishes a separation between solving smooth MVIs and smooth convex optimization, and settles the oracle complexity of solving $p^{\textrm{th}}$-order smooth MVIs.