On the Worst-Case Analysis of Cyclic Coordinate-Wise Algorithms on Smooth Convex Functions

TL;DR

This paper introduces a unified framework for analyzing cyclic coordinate algorithms on smooth convex functions, providing exact worst-case bounds and demonstrating that cyclic methods can outperform existing bounds and differ from randomized approaches.

Contribution

It offers the first exact worst-case bounds for cyclic coordinate descent, surpasses previous bounds by an order of magnitude, and highlights differences between cyclic and randomized schemes.

Findings

New upper bounds for cyclic coordinate descent.

Cyclic algorithms can outperform previous bounds.

Deterministic cyclic schemes may be less efficient than accelerated randomized methods.

Abstract

We propose a unifying framework for the automated computer-assisted worst-case analysis of cyclic block coordinate algorithms in the unconstrained smooth convex optimization setup. We compute exact worst-case bounds for the cyclic coordinate descent and the alternating minimization algorithms over the class of smooth convex functions, and provide sublinear upper and lower bounds on the worst-case rate for the standard class of functions with coordinate-wise Lipschitz gradients. We obtain in particular a new upper bound for cyclic coordinate descent that outperforms the best available ones by an order of magnitude. We also demonstrate the flexibility of our approach by providing new numerical bounds using simpler and more natural assumptions than those normally made for the analysis of block coordinate algorithms. Finally, we provide numerical evidence for the fact that a standard scheme that provably accelerates random coordinate descent to a $O(1/k^2)$ complexity is actually inefficient when used in a (deterministic) cyclic algorithm.