THE LASSO WITH GENERAL GAUSSIAN DESIGNS WITH APPLICATIONS TO HYPOTHESIS TESTING

• 成果类型: Article
• 署名作者: Celentano, Michael; Montanari, Andrea; Wei, Yuting
• 署名单位: University of California System; University of California Berkeley; Stanford University; University of Pennsylvania
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/23-AOS2327
• 发表日期: 2023
• 页码: 2194-2220
• 关键词: phase-transitions; confidence-intervals; UNIVERSALITY; geometry; signal

摘要

The Lasso is a method for high-dimensional regression, which is now commonly used when the number of covariates p is of the same order or larger than the number of observations n. Classical asymptotic normality theory does not apply to this model due to two fundamental reasons: (1) The regularized risk is nonsmooth; (2) The distance between the estimator 0 ⠂and the true parameters vector 0* cannot be neglected. As a consequence, standard perturbative arguments that are the traditional basis for asymptotic normality fail. On the other hand, the Lasso estimator can be precisely characterized in the regime in which both n and p are large and n/p is of order one. This characterization was first obtained in the case of Gaussian designs with i.i.d. covariates: here we generalize it to Gaussian correlated designs with nonsingular covariance structure. This is expressed in terms of a simpler fixeddesign model. We establish nonasymptotic bounds on the distance between the distribution of various quantities in the two models, which hold uniformly over signals 0* in a suitable sparsity class and over values of the regularization parameter. As an application, we study the distribution of the debiased Lasso and show that a degrees-of-freedom correction is necessary for computing valid confidence intervals.