# Cyclic Coordinate Update Algorithms for Fixed-Point Problems: Analysis and Applications

arXiv: https://arxiv.org/abs/1611.02456 · HTML: https://mlsanity.com/paper/1611.02456/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Nov 2016.

## Built on by (most cited first, 6 of 6)

- [Improved Analysis and Rates for Variance Reduction under Without-replacement Sampling Orders](https://mlsanity.com/paper/2104.12112/) (2021)
- [Cyclic Block Coordinate Descent With Variance Reduction for Composite Nonconvex Optimization](https://mlsanity.com/paper/2212.05088/) (2022)
- [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)
- [A Class of Accelerated Fixed-Point-Based Methods with Delayed Inexact Oracles and Its Applications](https://mlsanity.com/paper/2512.13547/) (2025)
- [Adaptive Delayed-Update Cyclic Algorithm for Variational Inequalities](https://mlsanity.com/paper/2603.29128/) (2026)

## Abstract (arXiv)

Many problems reduce to the fixed-point problem of solving x=T(x). To this problem, we apply the coordinate-update algorithms, which update only one or a few components of x at each step. When each update is cheap, these algorithms are faster than the full fixed-point iteration (which updates all the components). In this paper, we focus on the coordinate-update algorithms based on the cyclic selection rules, where the ordering of coordinates in each cycle is arbitrary. These algorithms are fast, but their convergence is unknown in the fixed-point setting. When T is a nonexpansive operator and has a fixed point, we show that the sequence of coordinate-update iterates converges to a fixed point under proper step sizes. This result applies to the primal-dual coordinate-update algorithms, which have applications to optimization problems with nonseparable nonsmooth objectives, as well as global linear constraints. Numerically, we apply coordinate-update algorithms with the cyclic, shuffled cyclic, and random selection rules to ℓ1 robust least squares, a CT image reconstruction problem, as well as nonnegative matrix factorization. They converge much faster than the standard fixed-point iteration. Among the three rules, cyclic and shuffled cyclic rules are overall faster than the random rule.