  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.
