# Analyzing Random Permutations for Cyclic Coordinate Descent

## TL;DR

This paper investigates the performance of random permutation cyclic coordinate descent (RPCD) on convex quadratic problems, showing it can outperform standard cyclic methods and behave similarly to fully-random approaches, supported by theoretical analysis.

## Contribution

It introduces a class of convex quadratic problems where RPCD outperforms CCD and provides a convergence analysis explaining this empirical advantage.

## Key findings

- RPCD outperforms CCD on certain convex quadratic problems.
- RPCD exhibits convergence similar to RCD in these cases.
- Theoretical analysis supports empirical observations.

## Abstract

We consider coordinate descent methods on convex quadratic problems, in which exact line searches are performed at each iteration. (This algorithm is identical to Gauss-Seidel on the equivalent symmetric positive definite linear system.) We describe a class of convex quadratic problems for which the random-permutations version of cyclic coordinate descent (RPCD) outperforms the standard cyclic coordinate descent (CCD) approach, yielding convergence behavior similar to the fully-random variant (RCD). A convergence analysis is developed to explain the empirical observations.