Coordinate descent algorithms for lasso penalized regression

TL;DR

This paper introduces two fast algorithms for lasso penalized regression, including a new greedy coordinate descent method, and explores their extensions to group penalties, with theoretical convergence guarantees and empirical testing.

Contribution

The paper presents a novel greedy coordinate descent algorithm for lasso regression and extends existing algorithms to group penalties, enhancing computational efficiency and applicability.

Findings

The greedy coordinate descent algorithm converges to the minimum of the objective function.

Both algorithms perform well on simulated and real data.

Extensions to group penalties are feasible and effective.

Abstract

Imposition of a lasso penalty shrinks parameter estimates toward zero and performs continuous model selection. Lasso penalized regression is capable of handling linear regression problems where the number of predictors far exceeds the number of cases. This paper tests two exceptionally fast algorithms for estimating regression coefficients with a lasso penalty. The previously known $\ell_2$ algorithm is based on cyclic coordinate descent. Our new $\ell_1$ algorithm is based on greedy coordinate descent and Edgeworth's algorithm for ordinary $\ell_1$ regression. Each algorithm relies on a tuning constant that can be chosen by cross-validation. In some regression problems it is natural to group parameters and penalize parameters group by group rather than separately. If the group penalty is proportional to the Euclidean norm of the parameters of the group, then it is possible to majorize the norm and reduce parameter estimation to $\ell_2$ regression with a lasso penalty. Thus, the existing algorithm can be extended to novel settings. Each of the algorithms discussed is tested via either simulated or real data or both. The Appendix proves that a greedy form of the $\ell_2$ algorithm converges to the minimum value of the objective function.