Piecewise linear regularized solution paths

Saharon Rosset*, Ji Zhu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We consider the generic regularized optimization problem β̂(λ) = arg minβ L(y, Xβ) + λJ(β). 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||l is the ℓl norm of β - the optimal coefficient path is piecewise linear, that is, ∂β(λ)/∂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.

Original languageEnglish
Pages (from-to)1012-1030
Number of pages19
JournalAnnals of Statistics
Issue number3
StatePublished - Jul 2007
Externally publishedYes


  • Polynomial splines
  • Regularization
  • Solution paths
  • Sparsity
  • Total variation
  • ℓ-norm penalty


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