On the Finite Time Convergence of Cyclic Coordinate Descent Methods

TL;DR

This paper establishes finite-time convergence rates for cyclic coordinate descent methods in smooth optimization problems, demonstrating their competitive performance and stability compared to gradient descent.

Contribution

It provides the first $O(1/k)$ convergence rate analysis for cyclic coordinate descent under an isotonicity assumption, filling a key gap in the literature.

Findings

Cyclic coordinate descent achieves $O(1/k)$ convergence rates.

Iterates of cyclic coordinate descent outperform gradient descent.

The analysis compares two variants of cyclic coordinate descent.

Abstract

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.