Random Permutations Fix a Worst Case for Cyclic Coordinate Descent

TL;DR

This paper analyzes the convergence behavior of different coordinate descent methods, showing that random permutations improve performance on certain quadratic functions where cyclic methods are slow.

Contribution

It provides a tight analysis explaining why random-permutations cyclic coordinate descent outperforms cyclic coordinate descent on specific quadratic functions.

Findings

RPCD outperforms CCD on certain quadratic functions

RPCD performs better than RCD in some regimes

The paper offers a theoretical explanation for RPCD's effectiveness

Abstract

Variants of the coordinate descent approach for minimizing a nonlinear function are distinguished in part by the order in which coordinates are considered for relaxation. Three common orderings are cyclic (CCD), in which we cycle through the components of $x$ in order; randomized (RCD), in which the component to update is selected randomly and independently at each iteration; and random-permutations cyclic (RPCD), which differs from CCD only in that a random permutation is applied to the variables at the start of each cycle. Known convergence guarantees are weaker for CCD and RPCD than for RCD, though in most practical cases, computational performance is similar among all these variants. There is a certain type of quadratic function for which CCD is significantly slower than for RCD; a recent paper by \cite{SunY16a} has explored the poor behavior of CCD on functions of this type. The RPCD approach performs well on these functions, even better than RCD in a certain regime. This paper explains the good behavior of RPCD with a tight analysis.