Honest Confidence Sets for High-Dimensional Regression by Projection and Shrinkage

The issue of honesty in constructing confidence sets arises in nonparametric regression. While optimal rate in nonparametric estimation can be achieved and utilized to construct sharp confidence sets, severe degradation of confidence level often happens after estimating the degree of smoothness. Similarly, for high-dimensional regression, oracle inequalities for sparse estimators could be utilized to construct sharp confidence sets. Yet, the degree of sparsity itself is unknown and needs to be estimated, which causes the honesty problem. To resolve this issue, we develop a novel method to construct honest confidence sets for sparse high-dimensional linear regression. The key idea in our construction is to separate signals into a strong and a weak group, and then construct confidence sets for each group separately. This is achieved by a projection and shrinkage approach, the latter implemented via Stein estimation and the associated Stein unbiased risk estimate. Our confidence set is honest over the full parameter space without any sparsity constraints, while its size adapts to the optimal rate of n−1/4 when the true parameter is indeed sparse. Moreover, under some form of a separation assumption between the strong and weak signals, the diameter of our confidence set can achieve a faster rate than existing methods. Through extensive numerical comparisons on both simulated and real data, we demonstrate that our method outperforms other competitors with big margins for finite samples, including oracle methods built upon the true sparsity of the underlying model.