Accelerated Mirror Descent for Non-Euclidean Star-convex Functions

Acceleration for non-convex functions has been an important problem in optimisation. We revisit star-convex functions, which are strictly unimodal on all lines through a minimizer. In [1], the authors accelerate gradient descent for star-convex functions with gradients that are Lipschitz with respect to the Euclidean norm in an unconstrained domain. In this paper, we introduce a new assumption about the regularity of the derivative of a general norm and we accelerate mirror descent for this class of normed spaces. We show that, under it, our algorithms show sharp convergence rates for star-convex functions with -H"older continuous gradients. We also prove that our convergence rate is near optimal for -norms. [1] Near-Optimal Methods for Minimizing Star-Convex Functions and Beyond, Hinder Oliver and Sidford Aaron and Sohoni Nimit