Performing statistical inference in high-dimension is an outstanding challenge. A major source of difficulty is the absence of precise information on the distribution of high-dimensional estimators. Here, we consider linear regression in the high-dimensional regime @math . In this context, we would like to perform inference on a high-dimensional parameters vector @math . Important progress has been achieved in computing confidence intervals for single coordinates @math . A key role in these new methods is played by a certain debiased estimator @math that is constructed from the Lasso. Earlier work establishes that, under suitable assumptions on the design matrix, the coordinates of @math are asymptotically Gaussian provided @math is @math -sparse with @math . The condition @math is stronger than the one for consistent estimation, namely @math . We study 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 @math . Note that earlier work was limited to @math even for perfectly known covariance. The same conclusion holds if the population covariance is unknown but can be estimated sufficiently well, e.g. under the same sparsity conditions on the inverse covariance as assumed by earlier work. For intermediate regimes, we describe the trade-off between sparsity in the coefficients and in the inverse covariance of the design. We further discuss several applications of our results to high-dimensional inference. In particular, we propose a new estimator that is minimax optimal up to a factor @math for i.i.d. Gaussian designs.
We consider the problem of fitting the parameters of a high-dimensional linear regression model. In the regime where the number of parameters @math is comparable to or exceeds the sample size @math , a successful approach uses an @math -penalized least squares estimator, known as Lasso. Unfortunately, unlike for linear estimators (e.g., ordinary least squares), no well-established method exists to compute confidence intervals or p-values on the basis of the Lasso estimator. Very recently, a line of work javanmard2013hypothesis, confidenceJM, GBR-hypothesis has addressed this problem by constructing a debiased version of the Lasso estimator. In this paper, we study this approach for random design model, under the assumption that a good estimator exists for the precision matrix of the design. Our analysis improves over the state of the art in that it establishes nearly optimal testing power if the sample size @math asymptotically dominates @math , with @math being the sparsity level (number of non-zero coefficients). Earlier work obtains provable guarantees only for much larger sample size, namely it requires @math to asymptotically dominate @math . In particular, for random designs with a sparse precision matrix we show that an estimator thereof having the required properties can be computed efficiently. Finally, we evaluate this approach on synthetic data and compare it with earlier proposals.