Minimax rate of testing in sparse linear regression

TL;DR

This paper establishes the optimal rate for testing sparse signals in high-dimensional linear regression, revealing fundamental limits and matching estimation rates.

Contribution

It derives the non-asymptotic minimax testing rate in Gaussian linear regression with sparse parameters, connecting testing and estimation limits.

Findings

Minimax testing rate: sqrt((s/n) log(1 + sqrt(p)/s))

Matching minimax rate for l2-norm estimation

Results hold for p < n in Gaussian models

Abstract

We consider the problem of testing the hypothesis that the parameter of linear regression model is 0 against an s-sparse alternative separated from 0 in the l2-distance. We show that, in Gaussian linear regression model with p < n, where p is the dimension of the parameter and n is the sample size, the non-asymptotic minimax rate of testing has the form sqrt((s/n) log(1 + sqrt(p)/s )). We also show that this is the minimax rate of estimation of the l2-norm of the regression parameter.