We consider the generic regularized optimization problem
$\hat{\mathsf{\beta}}(\lambda)=\arg
\min_{\beta}L({\sf{y}},X{\sf{\beta}})+\lambda J({\sf{\beta}})$. 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 $J(\beta)=\|\beta\|_1$ is
the $\ell_1$ norm of $\beta$--the optimal coefficient path is piecewise linear,
that is, $\partial \hat{\beta}(\lambda)/\partial \lambda$ 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.

对一种正则化优化问题，引入一种求系数路径的方法及其性质，为回归和分类的 LASSO 提供了鲁棒性解决方案，同时提出了现有问题的新型算法，包括 Mammen 与 van de Geer 的局部自适应回归样条。