Stochastic Coordinate Descent Methods for Regularized Smooth and Nonsmooth Losses

Abstract

Stochastic Coordinate Descent (SCD) methods are among the first optimization schemes suggested for efficiently solving large scale problems. However, until now, there exists a gap between the convergence rate analysis and practical SCD algorithms for general smooth losses and there is no primal SCD algorithm for nonsmooth losses. In this paper, we discuss these issues using the recently developed structural optimization techniques. In particular, we first present a principled and practical SCD algorithm for regularized smooth losses, in which the one-variable subproblem is solved using the proximal gradient method and the adaptive componentwise Lipschitz constant is obtained employing the line search strategy. When the loss is nonsmooth, we present a novel SCD algorithm, in which the one-variable subproblem is solved using the dual averaging method. We show that our algorithms exploit the regularization structure and achieve several optimal convergence rates that are standard in the literature. The experiments demonstrate the expected efficiency of our SCD algorithms in both smooth and nonsmooth cases.