Coordinate descent with arbitrary sampling I: algorithms and complexity†. (2nd September 2016)

• Record Type: Journal Article
• Authors: Qu, Zheng; Richtárik, Peter
• Is Part Of: Optimization methods and software. Volume 31:Number 5(2016)
• Page Start: 829
• Page End: 857
• Publication Date: 2016-09-02
• Subjects: coordinate descent -- accelerated coordinate descent -- arbitrary sampling -- complexity analysis; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2016.1190360
• Languages: English
• ISSNs: 1055-6788

Abstract

Abstract : We study the problem of minimizing the sum of a smooth convex function and a convex block-separable regularizer and propose a new randomized coordinate descent method, which we call ALPHA. Our method at every iteration updates a random subset of coordinates, following an arbitrary distribution . No coordinate descent methods capable to handle an arbitrary sampling have been studied in the literature before for this problem. ALPHA is a very flexible algorithm: in special cases, it reduces to deterministic and randomized methods such as gradient descent, coordinate descent, parallel coordinate descent and distributed coordinate descent—both in nonaccelerated and accelerated variants. The variants with arbitrary (or importance) sampling are new. We provide a complexity analysis of ALPHA, from which we deduce as a direct corollary complexity bounds for its many variants, all matching or improving best known bounds.