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On the Complexity Analysis of Randomized Block-Coordinate Descent Methods

Zhaosong Lu(zhaosong***at***sfu.ca)
Lin Xiao(lin.xiao***at***microsoft.com)

Abstract: In this paper we analyze the randomized block-coordinate descent (RBCD) methods for minimizing the sum of a smooth convex function and a block-separable convex function. In particular, we extend Nesterov's technique (SIOPT 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 Richtarik and Takac (Math Programming 2012). Also, we obtain a better high-probability type of iteration complexity, which improves upon the one by Richtarik and Takac by at least the amount $O(n/\epsilon)$, where $\epsilon$ is the target solution accuracy and $n$ is the number of problem blocks. 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 (SIOPT 2012) and establish a sharper expected-value type of convergence rate.

Keywords: randomized coordinate descent, accelerated coordinate descent, iteration complexity, convergence rate, composite minimization

Category 1: Convex and Nonsmooth Optimization

Citation: Microsoft Research Technical Report, MSR-TR-2013-53, May 2013.

Download: [PDF]

Entry Submitted: 05/20/2013
Entry Accepted: 05/20/2013
Entry Last Modified: 05/20/2013

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