Efficiency of higher-order algorithms for minimizing composite functions

TL;DR

This paper introduces a higher-order majorization algorithmic framework for composite function minimization, providing convergence guarantees for nonconvex, nonsmooth, and convex problems, with improved rates under certain conditions.

Contribution

It develops a novel higher-order surrogate-based framework for composite optimization, extending convergence guarantees to nonconvex and nonsmooth settings with KL property.

Findings

Convergence to stationary points in nonconvex nonsmooth problems.

Improved convergence rates under Kurdyka-Lojasiewicz property.

Sublinear rates for convex nonsmooth problems.

Abstract

Composite minimization involves a collection of functions which are aggregated in a nonsmooth manner. It covers, as a particular case, smooth approximation of minimax games, minimization of max-type functions, and simple composite minimization problems, where the objective function has a nonsmooth component. We design a higher-order majorization algorithmic framework for fully composite problems (possibly nonconvex). Our framework replaces each component with a higher-order surrogate such that the corresponding error function has a higher-order Lipschitz continuous derivative. We present convergence guarantees for our method for composite optimization problems with (non)convex and (non)smooth objective function. In particular, we prove stationary point convergence guarantees for general nonconvex (possibly nonsmooth) problems and under Kurdyka-Lojasiewicz (KL) property of the objective function we derive improved rates depending on the KL parameter. For convex (possibly nonsmooth) problems we also provide sublinear convergence rates.