Noname manuscript No. (will be inserted by the editor) Coordinate Descent Algorithms

Abstract Coordinate descent algorithms solve optimization problems by suc-cessively 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 anal-ysis, 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 con-vex objectives. We pay particular attention to a certain problem structure that arises frequently in machine learning applications, showing that efficient im-plementations 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. Keywords coordinate descent · randomized algorithms · parallel numerical computing