Nonconvex Optimization via MM Algorithms: Convergence Theory

TL;DR

This paper provides a comprehensive convergence analysis of majorization-minimization (MM) algorithms, highlighting their advantages in nonconvex optimization problems and unifying various existing results for broad applications.

Contribution

It offers a unified convergence theory for MM algorithms, including non-smooth and non-asymptotic cases, applicable to large-scale machine learning problems.

Findings

Unified convergence analysis for MM algorithms.

Applicability to non-smooth and non-asymptotic settings.

Demonstrates advantages in stability and problem convexification.

Abstract

The majorization-minimization (MM) principle is an extremely general framework for deriving optimization algorithms. It includes the expectation-maximization (EM) algorithm, proximal gradient algorithm, concave-convex procedure, quadratic lower bound algorithm, and proximal distance algorithm as special cases. Besides numerous applications in statistics, optimization, and imaging, the MM principle finds wide applications in large scale machine learning problems such as matrix completion, discriminant analysis, and nonnegative matrix factorizations. When applied to nonconvex optimization problems, MM algorithms enjoy the advantages of convexifying the objective function, separating variables, numerical stability, and ease of implementation. However, compared to the large body of literature on other optimization algorithms, the convergence analysis of MM algorithms is scattered and problem specific. This survey presents a unified treatment of the convergence of MM algorithms. With modern applications in mind, the results encompass non-smooth objective functions and non-asymptotic analysis.