Coordinate Descent for SLOPE

TL;DR

This paper introduces a fast, convergent algorithm combining proximal gradient and coordinate descent for efficiently solving the SLOPE optimization problem, significantly improving high-dimensional performance.

Contribution

A novel, efficient algorithm for SLOPE optimization that outperforms existing methods and includes new theoretical insights on the SLOPE penalty and thresholding operator.

Findings

Our algorithm achieves faster convergence in high dimensions.

It outperforms existing SLOPE solvers on simulated data.

It demonstrates superior performance on real datasets.

Abstract

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.