  We provide improved convergence rates for constrained convex-concave min-max
problems and monotone variational inequalities with higher-order smoothness. In
min-max settings where the $p^{th}$-order derivatives are Lipschitz continuous,
we give an algorithm HigherOrderMirrorProx that achieves an iteration
complexity of $O(1/T^{\frac{p+1}{2}})$ when given access to an oracle for
finding a fixed point of a $p^{th}$-order equation. We give analogous rates for
the weak monotone variational inequality problem. For $p>2$, our results
improve upon the iteration complexity of the first-order Mirror Prox method of
Nemirovski [2004] and the second-order method of Monteiro and Svaiter [2012].
We further instantiate our entire algorithm in the unconstrained $p=2$ case.
