  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.
