Coordinate descent with arbitrary sampling I: algorithms and complexity

Zheng Qu*, Peter Richtárik

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

46 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)829-857
Number of pages29
JournalOptimization Methods and Software
Issue number5
StatePublished - Sep 2 2016
Externally publishedYes


  • accelerated coordinate descent
  • arbitrary sampling
  • complexity analysis
  • coordinate descent

ASJC Scopus subject areas

  • Software
  • Control and Optimization
  • Applied Mathematics


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