Coordinate Descent Converges Faster with the Gauss-Southwell Rule Than Random Selection

TL;DR

This paper demonstrates that the Gauss-Southwell rule for coordinate descent generally converges faster than random selection, especially when costs are comparable, and introduces variants that further improve convergence.

Contribution

It provides a simple analysis showing faster convergence of Gauss-Southwell over random selection, and proposes new rules and variants to enhance performance.

Findings

Gauss-Southwell rule outperforms random selection in convergence speed.

Exact coordinate optimization benefits sparse problems.

Lipschitz-aware Gauss-Southwell rule accelerates convergence.

Abstract

There has been significant recent work on the theory and application of randomized coordinate descent algorithms, beginning with the work of Nesterov [SIAM J. Optim., 22(2), 2012], who showed that a random-coordinate selection rule achieves the same convergence rate as the Gauss-Southwell selection rule. This result suggests that we should never use the Gauss-Southwell rule, as it is typically much more expensive than random selection. However, the empirical behaviours of these algorithms contradict this theoretical result: in applications where the computational costs of the selection rules are comparable, the Gauss-Southwell selection rule tends to perform substantially better than random coordinate selection. We give a simple analysis of the Gauss-Southwell rule showing that---except in extreme cases---its convergence rate is faster than choosing random coordinates. Further, in this work we (i) show that exact coordinate optimization improves the convergence rate for certain sparse problems, (ii) propose a Gauss-Southwell-Lipschitz rule that gives an even faster convergence rate given knowledge of the Lipschitz constants of the partial derivatives, (iii) analyze the effect of approximate Gauss-Southwell rules, and (iv) analyze proximal-gradient variants of the Gauss-Southwell rule.