Optimal Rates and Tradeoffs in Multiple Testing

TL;DR

This paper establishes the fundamental limits of multiple hypothesis testing by deriving a precise non-asymptotic tradeoff between false discovery rate and false non-discovery rate, and shows that popular algorithms are rate-optimal.

Contribution

It provides the first detailed non-asymptotic analysis of FDR-FNR tradeoffs in a generalized Gaussian model, demonstrating the optimality of existing algorithms across various regimes.

Findings

Derived a non-asymptotic FDR-FNR tradeoff in Gaussian models.

Proved the optimality of Benjamini-Hochberg and Barber-Candès algorithms.

Applicable to sparse and dense regimes with varying parameters.

Abstract

Multiple hypothesis testing is a central topic in statistics, but despite abundant work on the false discovery rate (FDR) and the corresponding Type-II error concept known as the false non-discovery rate (FNR), a fine-grained understanding of the fundamental limits of multiple testing has not been developed. Our main contribution is to derive a precise non-asymptotic tradeoff between FNR and FDR for a variant of the generalized Gaussian sequence model. Our analysis is flexible enough to permit analyses of settings where the problem parameters vary with the number of hypotheses $n$, including various sparse and dense regimes (with $o(n)$ and $\mathcal{O}(n)$ signals). Moreover, we prove that the Benjamini-Hochberg algorithm as well as the Barber-Cand\`{e}s algorithm are both rate-optimal up to constants across these regimes.