Accelerated Mirror Descent for Non-Euclidean Star-convex Functions

TL;DR

This paper introduces an accelerated mirror descent algorithm for minimizing weakly smooth star-convex functions under arbitrary norms, achieving near-optimal convergence rates and extending previous Euclidean-based methods.

Contribution

It generalizes acceleration techniques to weakly smooth star-convex functions with arbitrary norms, refining complexity analysis and achieving sharp convergence rates.

Findings

Achieves near-optimal convergence rates for star-convex functions with $oldsymbol{ extit{ extalpha}}$-Holder continuous gradients.

Extends acceleration methods from Euclidean to arbitrary norms.

Provides refined complexity analysis for the proposed algorithms.

Abstract

Acceleration for non-convex functions is a fundamental challenge in optimisation. We revisit star-convex functions, which are strictly unimodal on all lines through a minimizer. [1] accelerate unconstrained star-convex minimization of functions that are smooth with respect to the Euclidean norm. To do so, they add a certain binary search step to gradient descent. In this paper, we accelerate unconstrained star-convex minimization of functions that are weakly smooth with respect to an arbitrary norm. We add a binary search step to mirror descent, generalize the approach and refine its complexity analysis. We prove that our algorithms have sharp convergence rates for star-convex functions with $\alpha$-Holder continuous gradients and demonstrate that our rates are nearly optimal for $p$-norms. [1] Near-Optimal Methods for Minimizing Star-Convex Functions and Beyond, Hinder Oliver and Sidford Aaron and Sohoni Nimit