A better convergence analysis of the block coordinate descent method for large scale machine learning

Abstract

This paper considers the problems of unconstrained minimization of large scale smooth convex functions having block-coordinate-wise Lipschitz continuous gradients. The block coordinate descent (BCD) method are among the first optimization schemes suggested for solving such problems \cite{nesterov2012efficiency}. We obtain a new lower (to our best knowledge the lowest currently) bound that is $16p^3$ times smaller than the best known on the information-based complexity of BCD method based on an effective technique called Performance Estimation Problem (PEP) proposed by Drori and Teboulle \cite{drori2012performance} recently for analyzing the performance of first-order black box optimization methods. Numerical test confirms our analysis.