Piecewise linear regularized solution paths

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dc.creatorRosset, Saharon
dc.creatorZhu, Ji
dc.date2007-08-16
dc.date.accessioned2026-07-07T08:24:38Z
dc.date.available2026-07-07T08:24:38Z
dc.descriptionWe consider the generic regularized optimization problem $\hat{\mathsfβ}(λ)=\arg \min_βL({\sf{y}},X{\sfβ})+λJ({\sfβ})$. Efron, Hastie, Johnstone and Tibshirani [Ann. Statist. 32 (2004) 407--499] have shown that for the LASSO--that is, if $L$ is squared error loss and $J(β)=\|β\|_1$ is the $\ell_1$ norm of $β$--the optimal coefficient path is piecewise linear, that is, $\partial \hatβ(λ)/\partial λ$ is piecewise constant. We derive a general characterization of the properties of (loss $L$, penalty $J$) pairs which give piecewise linear coefficient paths. Such pairs allow for efficient generation of the full regularized coefficient paths. We investigate the nature of efficient path following algorithms which arise. We use our results to suggest robust versions of the LASSO for regression and classification, and to develop new, efficient algorithms for existing problems in the literature, including Mammen and van de Geer's locally adaptive regression splines.
dc.descriptionPublished at http://dx.doi.org/10.1214/009053606000001370 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
dc.identifierhttps://arxiv.org/abs/0708.2197
dc.identifierhttp://arxiv.org/abs/0708.2197
dc.identifierAnnals of Statistics 2007, Vol. 35, No. 3, 1012-1030
dc.identifierdoi:10.1214/009053606000001370
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/136407
dc.subjectStatistics Theory
dc.subjectMachine Learning
dc.subject62J07 (Primary); 62F35, 62G08, 62H30 (Secondary)
dc.titlePiecewise linear regularized solution paths
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