Sharp multiple testing boundary for sparse sequences

This work investigates multiple testing from the point of view of minimax separation rates in the sparse sequence model, when the testing risk is measured as the sum FDR+FNR (False Discovery Rate plus False Negative Rate). First using the popular beta-min separation condition, with all nonzero signals separated from $0$ by at least some amount, we determine the sharp minimax testing risk asymptotically and thereby explicitly describe the transition from "achievable multiple testing with vanishing risk" to "impossible multiple testing". Adaptive multiple testing procedures achieving the corresponding optimal boundary are provided: the Benjamini--Hochberg procedure with properly tuned parameter, and an empirical Bayes $\ell$-value ('local FDR') procedure. We prove that the FDR and FNR have non-symmetric contributions to the testing risk for most procedures, the FNR part being dominant at the boundary. The optimal multiple testing boundary is then investigated for classes of arbitrary sparse signals. A number of extensions, including results for classification losses, are also discussed.