Implementable confidence sets in high dimensional regression

We consider the setting of linear regression in high dimension. We focus on the problem of constructing adaptive and honest confidence sets for the sparse parameter θ, i.e. we want to construct a confidence set for θ that contains θ with high probability, and that is as small as possible. The l2 diameter of a such confidence set should depend on the sparsity S of θ the larger S, the wider the confidence set. However, in practice, S is unknown. This paper focuses on constructing a confidence set for θ which contains θ with high probability, whose diameter is adaptive to the unknown sparsity S, and which is implementable in practice.