Non-Euclidean High-Order Smooth Convex Optimization

TL;DR

This paper introduces algorithms for optimizing convex functions with high-order smoothness using a $q$-th order oracle, applicable to general norms, and establishes nearly optimal lower bounds in high-dimensional settings.

Contribution

It develops a unified framework for high-order convex optimization with inexact non-Euclidean oracles, extending to structured functions and providing near-optimal complexity bounds.

Findings

Algorithms work for general norms including $ ext{l}_p$-settings.

Established a nearly optimal lower bound in high-dimensional black-box models.

Resolved an open question in parallel convex optimization for smooth functions.

Abstract

We develop algorithms for the optimization of convex objectives that have H\"older continuous $q$-th derivatives by using a $q$-th order oracle, for any $q \geq 1$. Our algorithms work for general norms under mild conditions, including the $\ell_p$-settings for $1\leq p\leq \infty$. 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 $q \geq 1$, 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.