Debiasing the lasso: Optimal sample size for Gaussian designs

Abstract

Performing statistical inference in high-dimensional models is challenging because of the lack of precise information on the distribution of high-dimensional regularized estimators. Here, we consider linear regression in the high-dimensional regime p n and the Lasso estimator: we would like to perform inference on the parameter vector θ∗ ∈ Rp. Important progress has been achieved in computing confidence intervals and p-values for single coordinates θi∗, i ∈ {1, . . ., p}. A key role in these new inferential methods is played by a certain debiased estimator θd. Earlier work establishes that, under suitable assumptions on the design matrix, the coordinates of θd are asymptotically Gaussian provided the true parameters vector θ∗ is s0-sparse with s0 = o(n/log p). The condition s0 = o(n/log p) is considerably stronger than the one for consistent estimation, namely s0 = o(n/log p). In this paper, we consider Gaussian designs with known or unknown population covariance. When the covariance is known, we prove that the debiased estimator is asymptotically Gaussian under the nearly optimal condition s0 = o(n/(log p)2). The same conclusion holds if the population covariance is unknown but can be estimated sufficiently well. For intermediate regimes, we describe the trade-off between sparsity in the coefficients θ∗, and sparsity in the inverse covariance of the design. We further discuss several applications of our results beyond high-dimensional inference. In particular, we propose a thresholded Lasso estimator that is minimax optimal up to a factor 1 + on(1) for i.i.d. Gaussian designs.