On the Complexity Analysis of Randomized Block-Coordinate Descent Methods

In this paper we analyze the randomized block-coordinate descent (RBCD) methods proposed in [11, 15] for minimizing the sum of a smooth convex function and a blockseparable convex function, and derive improved bounds on their convergence rates. In particular, we extend Nesterov’s technique developed in [11] 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 [15]. 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 [11] and establish a sharper expected-value type of convergence rate than the one given in [11].