# Coordinate Descent for SLOPE

arXiv: https://arxiv.org/abs/2210.14780 · HTML: https://mlsanity.com/paper/2210.14780/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 47 references that are in the map.
- First posted on arXiv 26 Oct 2022.

## Builds on (most cited first, 10 of 10)

- [SLOPE - Adaptive variable selection via convex optimization](https://mlsanity.com/paper/1407.3824/) (2014)
- [Mind the duality gap: safer rules for the Lasso](https://mlsanity.com/paper/1505.03410/) (2015)
- [Benchopt: Reproducible, efficient and collaborative optimization benchmarks](https://mlsanity.com/paper/2206.13424/) (2022)
- [Globally Convergent Type-I Anderson Acceleration for Non-Smooth Fixed-Point Iterations](https://mlsanity.com/paper/1808.03971/) (2018)
- [Local Linear Convergence of Forward-Backward under Partial Smoothness](https://mlsanity.com/paper/1407.5611/) (2014)
- [The Geometry of Uniqueness, Sparsity and Clustering in Penalized Estimation](https://mlsanity.com/paper/2004.09106/) (2020)
- [The Strong Screening Rule for SLOPE](https://mlsanity.com/paper/2005.03730/) (2020)
- [Safe rules for the identification of zeros in the solutions of the SLOPE problem](https://mlsanity.com/paper/2110.11784/) (2021)
- [Newton acceleration on manifolds identified by proximal-gradient methods](https://mlsanity.com/paper/2012.12936/) (2020)
- [Solving the OSCAR and SLOPE Models Using a Semismooth Newton-Based Augmented Lagrangian Method](https://mlsanity.com/paper/1803.10740/) (2018)

## Built on by (most cited first, 4 of 4)

- [Strong Screening Rules for Group-based SLOPE Models](https://mlsanity.com/paper/2405.15357/) (2024)
- [Proximal Operators of Sorted Nonconvex Penalties](https://mlsanity.com/paper/2506.15315/) (2025)
- [Efficient Solvers for SLOPE in R, Python, Julia, and C++](https://mlsanity.com/paper/2511.02430/) (2025)
- [Beyond False Discovery Rate: A Stepdown Group SLOPE Approach for Grouped Variable Selection](https://mlsanity.com/paper/2603.00955/) (2026)

## Abstract (arXiv)

The lasso is the most famous sparse regression and feature selection method. One reason for its popularity is the speed at which the underlying optimization problem can be solved. Sorted L-One Penalized Estimation (SLOPE) is a generalization of the lasso with appealing statistical properties. In spite of this, the method has not yet reached widespread interest. A major reason for this is that current software packages that fit SLOPE rely on algorithms that perform poorly in high dimensions. To tackle this issue, we propose a new fast algorithm to solve the SLOPE optimization problem, which combines proximal gradient descent and proximal coordinate descent steps. We provide new results on the directional derivative of the SLOPE penalty and its related SLOPE thresholding operator, as well as provide convergence guarantees for our proposed solver. In extensive benchmarks on simulated and real data, we show that our method outperforms a long list of competing algorithms.