The Lasso with general Gaussian designs with applications to hypothesis testing

Abstract

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 $\hat{\mathit{\theta }}$ and the true parameters vector ${\mathit{\theta }}^{\ast }$ 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 $\mathit{n}/\mathit{p}$ 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 nonasymptotic bounds on the distance between the distribution of various quantities in the two models, which hold uniformly over signals ${\mathit{\theta }}^{\ast }$ 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.