Minimax Quantile Bounds via Information Measures

Abstract: We develop a unified information-theoretic framework for lower bounding minimax quantiles. The starting point is a loss-adapted Neyman--Pearson metaconverse that bounds the minimax success probability at every loss radius and confidence level. The bound separates the small-ball geometry induced by the prior and loss from the statistical distinguishability of the experiment, and is optimised over an auxiliary output distribution. Different relaxations of this single testing bound yield converses based on (f)-informativity, Sibson information (I_α), Maximal Leakage, and Amemiya norms. Classical Fano and Le Cam lower bounds are recovered as special cases. The framework also clarifies why different information measures are suited to different recovery criteria. Maximal Leakage is exact for a class of symmetric exact-recovery problems. We use this identity to derive finite-sample bounds on the full minimax exact-recovery risk in the balanced Gaussian weighted stochastic block model, as well as two-sided finite-sample minimax-quantile bounds for low-rank matrix estimation under isotropic bounded-energy noise. For approximate Hamming recovery, we exhibit a heterogeneous binary model in which an optimised finite Sibson order yields a strong converse while the Maximal Leakage specialisation is trivial. Finally, for one-coordinate Poisson localisation, a Bennett-type Young function used through its Amemiya norm recovers the exact success-probability scale, whereas classical Fano and fixed-power relaxations are strictly weaker. These results show that sharp minimax-quantile converses require adapting the information control to both the recovery resolution and the likelihood-ratio tail.