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Piecewise linear regularized solution paths. (English) Zbl 1194.62094
Summary: We consider the generic regularized optimization problem $\stackrel{^}{\beta }\left(\lambda \right)=arg{min}_{\beta }L\left(y,X\beta \right)+\lambda J\left(\beta \right)$. B. Efron, T. Hastie, I. Johnstone and R. Tibshirani [Ann. Stat. 32, No. 2, 407–499 (2004; Zbl 1091.62054)] have shown that for the LASSO – that is, if $L$ is the squared error loss and $J\left(\beta \right)={\parallel \beta \parallel }_{1}$ is the ${\ell }_{1}$ norm of $\beta$ – the optimal coefficient path is piecewise linear, that is, $\partial \stackrel{^}{\beta }\left(\lambda \right)/\partial \lambda$ 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.
##### MSC:
 62J99 Linear statistical inference 65C60 Computational problems in statistics 90C90 Applications of mathematical programming 62H30 Classification and discrimination; cluster analysis (statistics)