Nearly Optimal Sample Size in Hypothesis Testing for High-Dimensional Regression
This provides a more efficient method for hypothesis testing in high-dimensional statistics, which is incremental but offers a significant reduction in sample size requirements for practitioners.
The paper tackles the problem of computing confidence intervals and p-values for high-dimensional linear regression using a debiased Lasso estimator, establishing nearly optimal average testing power with a sample size requirement of n asymptotically dominating s_0 (log p)^2, improving over earlier work that required n to dominate (s_0 log p)^2.
We consider the problem of fitting the parameters of a high-dimensional linear regression model. In the regime where the number of parameters $p$ is comparable to or exceeds the sample size $n$, a successful approach uses an $\ell_1$-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 \cite{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 \emph{average} testing power if the sample size $n$ asymptotically dominates $s_0 (\log p)^2$, with $s_0$ being the sparsity level (number of non-zero coefficients). Earlier work obtains provable guarantees only for much larger sample size, namely it requires $n$ to asymptotically dominate $(s_0 \log p)^2$. 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.