Finite-Sample Binary Hypothesis Testing via Rényi Divergences: Strong Converse and Local Privacy

Abstract

We study asymmetric simple binary hypothesis testing between $H_0:P_0^{n}$ and $H_1:P_1^{n}$, based on $n$ independent and identically distributed observations. Leveraging a variational representation of Rényi divergence of order $α$, we derive our main result: a finite-sample converse with $α>1$. The bound uses both directions of the divergence $D_α(P_1\|P_0)$ and $D_α(P_0\|P_1)$, tensorises under product measures, and contains familiar data-processing converses as boundary cases. For comparison, we apply the same variational approach to general $f$-divergences and specialise it to total variation, $E_γ$, Hellinger, and Kullback Leibler divergences, thereby recovering familiar converses within a unified framework. Together with an achievability bound involving Rényi divergence with $α\in (0,1)$, the main converse recovers the phase transition of the optimal Type II error under the exponentially decaying Type I error constraint $\varepsilon_n=e^{-nr}$. Under regularity conditions, the optimal Type II error vanishes exponentially when $r D(P_1\|P_0)$. We also derive sample-complexity bounds and extend both the converse and achievability analyses to locally differentially private observations, quantifying the cost of privacy and recovering the non-private achievability bound as the privacy constraint vanishes.