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 non-smooth; $(2)$ The distance between the estimator $\widehat{\boldsymbol{\theta}}$ and the true parameters vector $\boldsymbol{\theta}^*$ 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 non-singular covariance structure. This is expressed in terms of a simpler ``fixed-design'' model. We establish non-asymptotic bounds on the distance between the distribution of various quantities in the two models, which hold uniformly over signals $\boldsymbol{\theta}^*$ 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.

翻译：Lasso是一种高维回归方法,目前常用于协变量数量$p$与观测数量$n$同阶或大于后者的情形。由于两个根本原因,经典渐近正态理论不适用于该模型:$(1)$正则化风险非光滑;$(2)$估计量$\widehat{\boldsymbol{\theta}}$与真实参数向量$\boldsymbol{\theta}^*$之间的距离不可忽略。因此,作为渐近正态性传统基础的微扰论证方法在此失效。另一方面,当$n$和$p$都很大且$n/p$为常数阶时,Lasso估计量可被精确刻画。该刻画最初在独立同分布协变量的高斯设计下获得:本文将其推广至具有非奇异协方差结构的高斯相关设计情形。这一推广通过一个更简单的"固定设计"模型来表达。我们建立了两个模型中各种量分布距离的非渐近界,该界在适当稀疏类中的信号$\boldsymbol{\theta}^*$及正则化参数取值上一致成立。作为应用,我们研究了去偏Lasso的分布,并证明在计算有效置信区间时需要进行自由度校正。