Non-Euclidean High-Order Smooth Convex Optimization

Abstract

We develop algorithms for the optimization of convex objectives that have Hölder continuous $q$-th derivatives by using a $q$-th order oracle, for any $q\ge 1$. Our algorithms work for general norms under mild conditions, including the ${\ell }_p$-settings for $1\le p\le \infty$. We can also optimize structured functions that allow for inexactly implementing a non-Euclidean ball optimization oracle. We do this by developing a non-Euclidean inexact accelerated proximal point method that makes use of an \emph{inexact uniformly convex regularizer}. We show a lower bound for general norms that demonstrates our algorithms are nearly optimal in high-dimensions in the black-box oracle model for ${\ell }_p$-settings and all $q\ge 1$, even in randomized and parallel settings. This new lower bound, when applied to the first-order smooth case, resolves an open question in parallel convex optimization.

randomized and parallel lower bounds (and gen. to all norms), convexity of subproblems, inexactness of unacc. alg., better writing