Randomness and Permutations in Coordinate Descent Methods

TL;DR

This paper analyzes coordinate descent methods with exact line search on convex quadratic problems, showing that random permutations improve worst-case performance over random sampling, especially with diagonally dominant Hessians, but fixed orders outperform permutations.

Contribution

It provides a theoretical comparison of permutation-based, random sampling, and deterministic coordinate descent methods on convex quadratic problems, highlighting the advantages of permutations under certain conditions.

Findings

Random permutations improve worst-case performance over random sampling.

Performance gains increase with greater diagonal dominance of the Hessian.

Fixed deterministic order outperforms random permutations for the studied class.

Abstract

We consider coordinate descent (CD) methods with exact line search on convex quadratic problems. Our main focus is to study the performance of the CD method that use random permutations in each epoch and compare it to the performance of the CD methods that use deterministic orders and random sampling with replacement. We focus on a class of convex quadratic problems with a diagonally dominant Hessian matrix, for which we show that using random permutations instead of random with-replacement sampling improves the performance of the CD method in the worst-case. Furthermore, we prove that as the Hessian matrix becomes more diagonally dominant, the performance improvement attained by using random permutations increases. We also show that for this problem class, using any fixed deterministic order yields a superior performance than using random permutations. We present detailed theoretical analyses with respect to three different convergence criteria that are used in the literature and support our theoretical results with numerical experiments.