On the complexity analysis of randomized block-coordinate descent methods

Zhaosong Lu, Lin Xiao

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81 Scopus citations

Abstract

In this paper we analyze the randomized block-coordinate descent (RBCD) methods proposed in Nesterov (SIAM J Optim 22(2):341–362, 2012), Richtárik and Takáč (Math Program 144(1–2):1–38, 2014) for minimizing the sum of a smooth convex function and a block-separable convex function, and derive improved bounds on their convergence rates. In particular, we extend Nesterov’s technique developed in Nesterov (SIAM J Optim 22(2):341–362, 2012) for analyzing the RBCD method for minimizing a smooth convex function over a block-separable closed convex set to the aforementioned more general problem and obtain a sharper expected-value type of convergence rate than the one implied in Richtárik and Takáč (Math Program 144(1–2):1–38, 2014). As a result, we also obtain a better high-probability type of iteration complexity. In addition, for unconstrained smooth convex minimization, we develop a new technique called randomized estimate sequence to analyze the accelerated RBCD method proposed by Nesterov (SIAM J Optim 22(2):341–362, 2012) and establish a sharper expected-value type of convergence rate than the one given in Nesterov (SIAM J Optim 22(2):341–362, 2012).

Original languageEnglish (US)
Pages (from-to)615-642
Number of pages28
JournalMathematical Programming
Volume152
Issue number1-2
DOIs
StatePublished - Aug 24 2015
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2014, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

Keywords

  • Accelerated coordinate descent
  • Composite minimization
  • Convergence rate
  • Iteration complexity
  • Randomized block-coordinate descent

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