Square-root lasso: pivotal recovery of sparse signals via conic programming

作     者：Belloni, A. Chernozhukov, V. Wang, L. 

作者机构：Duke Univ Fuqua Sch Business Durham NC 27708 USA MIT Dept Econ Cambridge MA 02142 USA MIT Dept Math Cambridge MA 02139 USA 

出 版 物：《BIOMETRIKA》 (生物测量学)

年 卷 期：2011年第98卷第4期

页      面：791-806页

核心收录：

学科分类：0710[理学-生物学] 07[理学] 09[农学] 0714[理学-统计学（可授理学、经济学学位）] 

基　　金：National Science Foundation, U.S.A Direct For Mathematical & Physical Scien Division Of Mathematical Sciences Funding Source: National Science Foundation Direct For Social, Behav & Economic Scie Divn Of Social and Economic Sciences Funding Source: National Science Foundation Divn Of Social and Economic Sciences Direct For Social, Behav & Economic Scie Funding Source: National Science Foundation 

主　　题：Conic programming High-dimensional sparse model Moderate deviation theory 

摘      要：We propose a pivotal method for estimating high-dimensional sparse linear regression models, where the overall number of regressors p is large, possibly much larger than n, but only s regressors are significant. The method is a modification of the lasso, called the square-root lasso. The method is pivotal in that it neither relies on the knowledge of the standard deviation Sigma nor does it need to pre-estimate Sigma. Moreover, the method does not rely on normality or sub-Gaussianity of noise. It achieves near-oracle performance, attaining the convergence rate Sigma{(s/n) log p}(1/2) in the prediction norm, and thus matching the performance of the lasso with known Sigma. These performance results are valid for both Gaussian and non-Gaussian errors, under some mild moment restrictions. We formulate the square-root lasso as a solution to a convex conic programming problem, which allows us to implement the estimator using efficient algorithmic methods, such as interior-point and first-order methods.