Minimax Rate of Testing in Sparse Linear Regression

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 (s/n)log(p/s). We also show that this is the minimax rate of estimation of the l2-norm of the regression parameter.