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

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