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

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  • Michoel, Tom

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/.

Suggested Citation

  • Michoel, Tom, 2016. "Natural coordinate descent algorithm for L1-penalised regression in generalised linear models," Computational Statistics & Data Analysis, Elsevier, vol. 97(C), pages 60-70.
    • Handle: RePEc:eee:csdana:v:97:y:2016:i:c:p:60-70
    DOI: 10.1016/j.csda.2015.11.009
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    References listed on IDEAS

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    1. P. Tseng, 2001. "Convergence of a Block Coordinate Descent Method for Nondifferentiable Minimization," Journal of Optimization Theory and Applications, Springer, vol. 109(3), pages 475-494, June.
    2. Friedman, Jerome H. & Hastie, Trevor & Tibshirani, Rob, 2010. "Regularization Paths for Generalized Linear Models via Coordinate Descent," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 33(i01).
    3. Hui Zou & Trevor Hastie, 2005. "Addendum: Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 768-768, November.
    4. Robert Tibshirani & Jacob Bien & Jerome Friedman & Trevor Hastie & Noah Simon & Jonathan Taylor & Ryan J. Tibshirani, 2012. "Strong rules for discarding predictors in lasso‐type problems," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 74(2), pages 245-266, March.
    5. Hui Zou & Trevor Hastie, 2005. "Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 301-320, April.
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    Cited by:

    1. Margherita Giuzio, 2017. "Genetic algorithm versus classical methods in sparse index tracking," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 40(1), pages 243-256, November.

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