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

TLDR

The block coordinate descent (BCD) method is widely used for minimizing continuous functions of several block variables, but requiring each block subproblem to reach a unique global optimum is often too restrictive for practical applications. This paper investigates an inexact BCD approach that updates variable blocks by successively minimizing a sequence of approximations of the objective that are either locally tight upper bounds or strictly convex local approximations. At each iteration, a single block is optimized while the others are fixed, using these approximations instead of solving each subproblem to its global optimum. The authors characterize convergence conditions for this broad class of methods, particularly for nondifferentiable or nonconvex objectives, and unify and extend existing convergence results for algorithms such as BCD, DC, EM, and block forward‑backward splitting.

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 of each block variable needs to be solved to its unique global optimal. Unfortunately, this requirement is 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 $f$. The main contributions of this work include the characterizations of the convergence conditions for a fairly wide class of such methods, especially for the cases where the objective functions are either nondifferentiable 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 block forward-backward splitting algorithm, all of which are popular for large scale optimization problems involving big data.