A Unified Convergence Analysis of Block Successive Minimization Methods for Nonsmooth Optimization

TL;DR

This paper provides a unified convergence analysis for inexact block coordinate descent methods applied to nonsmooth, nonconvex optimization problems, broadening the understanding of their theoretical guarantees.

Contribution

It extends convergence results to a wide class of inexact BCD methods using local approximations, applicable to nonsmooth and nonconvex functions, unifying several classical algorithms.

Findings

Convergence is guaranteed for inexact BCD with local approximations.

Results unify and extend classical algorithms like EM and DC methods.

Applicable to nonsmooth and nonconvex optimization problems.

Abstract

The block coordinate descent (BCD) method is widely used for minimizing a continuous function f of several block variables. At each iteration of this method, a single block of variables is optimized, while the remaining variables are held fixed. To ensure the convergence of the BCD method, the subproblem to be optimized in each iteration needs to be solved exactly to its unique optimal solution. Unfortunately, these requirements are often too restrictive for many practical scenarios. In this paper, we study an alternative inexact BCD approach which updates the variable blocks by successively minimizing a sequence of approximations of f which are either locally tight upper bounds of f or strictly convex local approximations of f. We focus on characterizing the convergence properties for a fairly wide class of such methods, especially for the cases where the objective functions are either non-differentiable or nonconvex. Our results unify and extend the existing convergence results for many classical algorithms such as the BCD method, the difference of convex functions (DC) method, the expectation maximization (EM) algorithm, as well as the alternating proximal minimization algorithm.