Coordinate descent algorithms

• Published in: Mathematical programming Vol. 151; no. 1; pp. 3 - 34
• Main Author: Wright, Stephen J.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2015; Springer Nature B.V
• Subjects: Algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Full Length Paper; Linear equations; Machine learning; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Optimization; Optimization algorithms; Studies; Theoretical; United States > US; Parallel numerical computing; 49M20; Randomized algorithms; 90C25; Coordinate descent
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-015-0892-3

Coordinate descent algorithms solve optimization problems by successively performing approximate minimization along coordinate directions or coordinate hyperplanes. They have been used in applications for many years, and their popularity continues to grow because of their usefulness in data analysis, machine learning, and other areas of current interest. This paper describes the fundamentals of the coordinate descent approach, together with variants and extensions and their convergence properties, mostly with reference to convex objectives. We pay particular attention to a certain problem structure that arises frequently in machine learning applications, showing that efficient implementations of accelerated coordinate descent algorithms are possible for problems of this type. We also present some parallel variants and discuss their convergence properties under several models of parallel execution.