# Random Permutations Fix a Worst Case for Cyclic Coordinate Descent

arXiv: https://arxiv.org/abs/1607.08320 · HTML: https://mlsanity.com/paper/1607.08320/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jul 2016.

## Built on by (most cited first, 5 of 5)

- [Cyclic Block Coordinate Descent With Variance Reduction for Composite Nonconvex Optimization](https://mlsanity.com/paper/2212.05088/) (2022)
- [Empirical Risk Minimization with Shuffled SGD: A Primal-Dual Perspective and Improved Bounds](https://mlsanity.com/paper/2306.12498/) (2023)
- [Cyclic Coordinate Dual Averaging with Extrapolation](https://mlsanity.com/paper/2102.13244/) (2021)
- [Accelerated Cyclic Coordinate Dual Averaging with Extrapolation for Composite Convex Optimization](https://mlsanity.com/paper/2303.16279/) (2023)
- [Continuous Optimization for p-adic Models](https://mlsanity.com/paper/2609.25501/) (2026)

## Abstract (arXiv)

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 citeSunY16a 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.