On Minimax Exponents of Sparse Testing

Understand

We consider exact asymptotics of the minimax risk for global testing against sparse alternatives in the context of high dimensional linear regression.

• Our results characterize the leading order behavior of this minimax risk in several regimes, uncovering new phase transitions in its behavior.
• This complements a vast literature characterizing asymptotic consistency in this problem, and provides a useful benchmark, against which the performance of specific tests may be compared.
• Finally, we provide some preliminary evidence that popular sparsity adaptive procedures might be sub-optimal in terms of the minimax risk.