Abstract: We develop a new method to estimate the projection direction in the debiased Lasso estimator. The basic idea is to decompose the overall bias into two terms, corresponding to strong and weak signals, respectively. We propose estimating the projection direction by balancing the squared biases associated with the strong and weak signals and the variance of the projection-based estimator. A standard quadratic programming solver can solve the resulting optimization problem efficiently. We show theoretically that the unknown set of strong signals can be estimated consistently, and that the projection-based estimator enjoys asymptotic normality under suitable assumptions. A slight modification of our procedure leads to an estimator with a potentially smaller order of bias than that of the original debiased Lasso. We further generalize our method to conduct an inference for a sparse linear combination of the regression coefficients. Numerical studies demonstrate the advantage of the proposed approach in terms of coverage accuracy over several existing alternatives.

Key words and phrases: Confidence interval, high-dimensional linear models, Lasso, quadratic programming.