Natural coordinate descent algorithm for L1-penalised regression in generalised linear models

TL;DR

This paper introduces a new coordinate descent algorithm for L1-penalised generalized linear models, which guarantees convergence and is efficient for large-scale datasets, especially when computing solutions at fixed penalty values.

Contribution

It presents a novel coordinate descent method based on a soft-thresholding mechanism for L1-penalised GLMs, with guaranteed convergence and improved efficiency for large datasets.

Findings

Efficient algorithm for L1-penalised logistic regression.

Demonstrated on large-scale cancer gene expression data.

Outperforms existing methods when computing solutions at fixed penalties.

Abstract

The problem of finding the maximum likelihood estimates for the regression coefficients in generalised linear models with an L1 sparsity penalty is shown to be equivalent to minimising the unpenalised maximum log-likelihood function over a box with boundary defined by the L1-penalty parameter. In one-parameter models or when a single coefficient is estimated at a time, this result implies a generic soft-thresholding mechanism which leads to a novel coordinate descent algorithm for generalised linear models that is entirely described in terms of the natural formulation of the model and is guaranteed to converge to the true optimum. A prototype implementation for logistic regression tested on two large-scale cancer gene expression datasets shows that this algorithm is efficient, particularly so when a solution is computed at set values of the L1-penalty parameter as opposed to along a regularisation path. Source code and test data are available from http://tmichoel.github.io/glmnat/.