Confidence sets in sparse regression

TL;DR

This paper investigates the construction of confidence sets in high-dimensional linear models, establishing conditions under which adaptive inference is possible depending on sparsity and design assumptions.

Contribution

It provides necessary and sufficient conditions for adaptive confidence sets in high-dimensional sparse regression, extending understanding of when honest inference can be achieved.

Findings

Adaptive confidence sets exist when the sparse estimation rate is at most n^{-1/4}.

Confidence sets can adapt to fixed sparsity levels under certain minimal separation conditions.

Design assumptions include common coherence and sub-Gaussian conditions, covering correlated designs.

Abstract

The problem of constructing confidence sets in the high-dimensional linear model with $n$ response variables and $p$ parameters, possibly $p\ge n$, is considered. Full honest adaptive inference is possible if the rate of sparse estimation does not exceed $n^{-1/4}$, otherwise sparse adaptive confidence sets exist only over strict subsets of the parameter spaces for which sparse estimators exist. Necessary and sufficient conditions for the existence of confidence sets that adapt to a fixed sparsity level of the parameter vector are given in terms of minimal $\ell^2$-separation conditions on the parameter space. The design conditions cover common coherence assumptions used in models for sparsity, including (possibly correlated) sub-Gaussian designs.