Implementable confidence sets in high dimensional regression

TL;DR

This paper develops practical methods for constructing adaptive, honest confidence sets in high-dimensional linear regression that automatically adjust to the unknown sparsity level of the parameter.

Contribution

It introduces a new approach for creating confidence sets that are both adaptive to sparsity and practically implementable in high-dimensional settings.

Findings

Confidence sets adapt to unknown sparsity levels

High probability coverage of the true parameter

Constructed sets are as small as possible given sparsity

Abstract

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 \theta, i.e. we want to construct a confidence set for theta that contains theta with high probability, and that is as small as possible. The l_2 diameter of a such confidence set should depend on the sparsity S of \theta - the larger S, the wider the confidence set. However, in practice, S is unknown. This paper focuses on constructing a confidence set for \theta which contains \theta with high probability, whose diameter is adaptive to the unknown sparsity S, and which is implementable in practice.