Large-scale Multiple Testing: Fundamental Limits of False Discovery Rate Control and Compound Oracle

The false discovery rate (FDR) and the false non-discovery rate (FNR), defined as the expected false discovery proportion (FDP) and the false non-discovery proportion (FNP), are the most popular benchmarks for multiple testing. Despite the theoretical and algorithmic advances in recent years, the optimal tradeoff between the FDR and the FNR has been largely unknown except for certain restricted classes of decision rules, e.g., separable rules, or for other performance metrics, e.g., the marginal FDR and the marginal FNR (mFDR and mFNR). In this paper, we determine the asymptotically optimal FDR-FNR tradeoff under the two-group random mixture model when the number of hypotheses tends to infinity. Distinct from the optimal mFDR-mFNR tradeoff, which is achieved by separable decision rules, the optimal FDR-FNR tradeoff requires compound rules even in the large-sample limit and for models as simple as the Gaussian location model. This suboptimality of separable rules also holds for other objectives, such as maximizing the expected number of true discoveries. Finally, to address the limitation of the FDR which only controls the expectation but not the fluctuation of the FDP, we also determine the optimal tradeoff when the FDP is controlled with high probability and show it coincides with that of the mFDR and the mFNR. Extensions to models with a fixed non-null proportion are also obtained.