Scalable and efficient inference via CPE

Two primary concerns of inference for high-dimensional data are statistical accuracy and computational efficiency. Despite the appealing asymptotic properties of existing de-biasing methods, the de-biasing step is generally considered to be computationally intensive. In this article, we propose the constrained projection estimator (CPE) for deriving confidence intervals in a scalable and efficient way under high dimensions when the unknown parameters adopt an approximately sparse structure. The proposed method is implemented on the constrained projection spaces corresponding to the identifiable signals determined by a prescreening procedure, which significantly reduces the computational cost in comparison to the full de-biasing steps. Theoretically, we demonstrate that the proposed inference method enjoys equivalent asymptotic efficiency to the full de-biasing procedure in view of the lengths of confidence intervals. We demonstrate the scalability and effectiveness of the proposed method through simulation and real data studies.