  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, causing 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 diameter
adapts to the optimal rate of $n^{-1/4}$ when the true parameter is indeed
sparse. Through extensive numerical comparisons, 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.
