Block-Activated Algorithms for Multicomponent Fully Nonsmooth Minimization

TL;DR

This paper explores block-activated algorithms for solving multicomponent nonsmooth convex minimization problems, focusing on their implementation, features, and performance through theoretical analysis and experiments.

Contribution

It introduces and analyzes block-activated proximal algorithms for fully nonsmooth multicomponent minimization, an area with limited existing methods.

Findings

Algorithms effectively handle fully nonsmooth problems

Block-activated methods reduce computational complexity

Experimental results demonstrate competitive performance

Abstract

Under consideration are multicomponent minimization problems involving a separable nonsmooth convex function penalizing the components individually, and nonsmooth convex coupling terms penalizing linear mixtures of the components. We investigate block-activated proximal algorithms for solving such problems, i.e., algorithms which, at each iteration, need to use only a block of the underlying functions, as opposed to all of them as in standard methods. For smooth coupling functions, several block-activated algorithms exist and they are well understood. By contrast, in the fully nonsmooth case, few block-activated methods are available and little effort has been devoted to assessing them. Our goal is to shed more light on the implementation, the features, and the behavior of these algorithms, compare their merits, and provide machine learning and image recovery experiments illustrating their performance.