Selective Linearization For Multi-Block Convex Optimization

TL;DR

This paper introduces the selective linearization method, an algorithm for efficiently minimizing sums of convex non-smooth functions, with proven convergence and practical applications in structured regularization.

Contribution

The paper presents a novel selective linearization algorithm for multi-block convex optimization, with convergence analysis and rate estimates.

Findings

Global convergence proved.

Iteration complexity of order O(ln(1/ε)/ε).

Effective on structured regularization problems.

Abstract

We consider the problem of minimizing a sum of several convex non-smooth functions. We introduce a new algorithm called the selective linearization method, which iteratively linearizes all but one of the functions and employs simple proximal steps. The algorithm is a form of multiple operator splitting in which the order of processing partial functions is not fixed, but rather determined in the course of calculations. Global convergence is proved and estimates of the convergence rate are derived. Specifically, the number of iterations needed to achieve solution accuracy $\varepsilon$ is of order $\mathcal{O}\big(\ln(1/\varepsilon)/\varepsilon\big)$. We also illustrate the operation of the algorithm on structured regularization problems.