Hypothesis Testing in High-Dimensional Regression Under the Gaussian Random Design Model: Asymptotic Theory

We consider linear regression in the high-dimensional regime where the number of observations n is smaller than the number of parameters p. A very successful approach in this setting uses l(1)-penalized least squares (also known as the Lasso) to search for a subset of s(0) < n parameters that best explain the data, while setting the other parameters to zero. Considerable amount of work has been devoted to characterizing the estimation and model selection problems within this approach. In this paper, we consider instead the fundamental, but far less understood, question of statistical significance. More precisely, we address the problem of computing p-values for single regression coefficients. On one hand, we develop a general upper bound on the minimax power of tests with a given significance level. We show that rigorous guarantees for earlier methods do not allow to achieve this bound, except in special cases. On the other, we prove that this upper bound is (nearly) achievable through a practical procedure in the case of random design matrices with independent entries. Our approach is based on a debiasing of the Lasso estimator. The analysis builds on a rigorous characterization of the asymptotic distribution of the Lasso estimator and its debiased version. Our result holds for optimal sample size, i.e., when n is at least on the order of s(0) log(p/s(0)). We generalize our approach to random design matrices with independent identically distributed Gaussian rows x(i) similar to N(0, Sigma). In this case, we prove that a similar distributional characterization (termed standard distributional limit) holds for n much larger than s(0)(log p)(2). Our analysis assumes Sigma is known. To cope with unknown Sigma, we suggest a plug-in estimator for sparse covariances Sigma and validate the method through numerical simulations. Finally, we show that for optimal sample size, n being at least of order s(0) log(p/s(0)), the standard distributional limit for general Gaussian designs can be derived from the replica heuristics in statistical physics. This derivation suggests a stronger conjecture than the result we prove, and near-optimality of the statistical power for a large class of Gaussian designs.