  Cyclic coordinate descent is a classic optimization method that has witnessed
a resurgence of interest in machine learning. Reasons for this include its
simplicity, speed and stability, as well as its competitive performance on
$\ell_1$ regularized smooth optimization problems. Surprisingly, very little is
known about its finite time convergence behavior on these problems. Most
existing results either just prove convergence or provide asymptotic rates. We
fill this gap in the literature by proving $O(1/k)$ convergence rates (where
$k$ is the iteration counter) for two variants of cyclic coordinate descent
under an isotonicity assumption. Our analysis proceeds by comparing the
objective values attained by the two variants with each other, as well as with
the gradient descent algorithm. We show that the iterates generated by the
cyclic coordinate descent methods remain better than those of gradient descent
uniformly over time.
