Provable Benefit of Random Permutations over Uniform Sampling in Stochastic Coordinate Descent

TL;DR

This paper provides the first theoretical proof that random-permutation coordinate descent (RPCD) consistently outperforms random coordinate descent (RCD) in convergence rates for a class of quadratic functions, explaining empirical advantages.

Contribution

The authors prove that RPCD has a strictly better contraction rate than RCD for certain quadratic functions, filling a key theoretical gap in understanding their performance.

Findings

RPCD's contraction rate is always smaller than RCD's for specific quadratic functions.

Conjecture that the identified function class includes worst-case RPCD examples.

Results extend to all positive-definite quadratic functions under the conjecture.

Abstract

We analyze the convergence rates of two popular variants of coordinate descent (CD): random CD (RCD), in which the coordinates are sampled uniformly at random, and random-permutation CD (RPCD), in which random permutations are used to select the update indices. Despite abundant empirical evidence that RPCD outperforms RCD in various tasks, the theoretical gap between the two algorithms' performance has remained elusive. Even for the benign case of positive-definite quadratic functions with permutation-invariant Hessians, previous efforts have failed to demonstrate a provable performance gap between RCD and RPCD. To this end, we present novel results showing that, for a class of quadratics with permutation-invariant structures, the contraction rate upper bound for RPCD is always strictly smaller than the contraction rate lower bound for RCD for every individual problem instance. Furthermore, we conjecture that this function class contains the worst-case examples of RPCD among all positive-definite quadratics. Combined with our RCD lower bound, this conjecture extends our results to the general class of positive-definite quadratic functions.