分段线性正则解路径
Piecewise linear regularized solution paths
摘要：
我们考虑一般的正则化最优化问题$\hat{\mathsfβ}(λ)=\arg \min_βL({\sf{y}},X{\sfβ})+λJ({\sfβ})$。Efron，Hastie，Johnstone and Tibshirani [Ann. Statistist. 32（2004）407- 499]已经表明，对于LASSO--即，如果$L$是平方误差损失，而$J(β)=\|β\|_1$是$β$的$\ell_1$范数--最优系数路径是分段线性的，即，$\partial \hatβ(λ)/\partial λ$是分段常数。我们推导出一个一般的特性（损失$L$，惩罚$J$）对，给出分段线性系数路径。这样的对允许有效地生成完全正则化的系数路径。我们调查的有效路径跟踪算法的性质出现。我们使用我们的研究结果，建议强大的版本的LASSO回归和分类，并开发新的，有效的算法，现有的文献中的问题，包括曼门和范德吉尔的局部自适应回归样条。
英文摘要：
We 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.
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