DEBIASING THE LASSO: OPTIMAL SAMPLE SIZE FOR GAUSSIAN DESIGNS

• 成果类型: Article
• 署名作者: Javanmard, Adel; Montanari, Andrea
• 署名单位: University of Southern California; Stanford University; Stanford University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1630
• 发表日期: 2018
• 页码: 2593-2622
• 关键词: high-dimensional inference; confidence-intervals; variable selection; phase-transitions; DANTZIG SELECTOR; regression; prediction; sparsity; RECOVERY; regions

摘要

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 theta*is an element of R-p. Important progress has been achieved in computing confidence intervals and p-values for single coordinates. theta(i)*, i is an element of{1,..., p}. A key role in these new inferential methods is played by a certain debiased estimator (theta) over cap (d). Earlier work establishes that, under suitable assumptions on the design matrix, the coordinates of (theta) over cap (d) are asymptotically Gaussian provided the true parameters vector theta* is s(0)-sparse with s(0) = o(root n/log p). The condition s(0) = o(root n/log p) is considerably stronger than the one for consistent estimation, namely s(0) = 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 s(0) = 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 theta*, 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 + o(n)(1) for i.i.d. Gaussian designs.