Accelerated Stochastic Block Coordinate Descent with Optimal Sampling

Abstract

We study the composite minimization problem where the objective function is the sum of two convex functions: one is the sum of a finite number of strongly convex and smooth functions, and the other is a general convex function that is non-differentiable. Specifically, we consider the case where the non-differentiable function is block separable and admits a simple proximal mapping for each block. This type of composite optimization is common in many data mining and machine learning problems, and can be solved by block coordinate descent algorithms. We propose an accelerated stochastic block coordinate descent (ASBCD) algorithm, which incorporates the incrementally averaged partial derivative into the stochastic partial derivative and exploits optimal sampling. We prove that ASBCD attains a linear rate of convergence. In contrast to uniform sampling, we reveal that the optimal non-uniform sampling can be employed to achieve a lower iteration complexity. Experimental results on different large-scale real data sets support our theory.