On the Complexity Analysis of Randomized Block-Coordinate Descent Methods

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 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/ε), where ε 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.