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

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/.