We prove the conjecture stated in Appendix F.3 of \citet{zhu2022optimalaccountingdifferentialprivacy}: among all conversion rules that map a Rényi Differential Privacy (RDP) profile $τ\mapsto ρ(τ)$ to a valid hypothesis-testing trade-off $f$, the rule based on the intersection of single-order RDP privacy regions is optimal. This optimality holds simultaneously for all valid RDP profiles and for all Type I error levels $α$. Concretely, we show that in the space of trade-off functions, the tightest possible bound is $f_{ρ(\cdot)}(α) = \sup_{τ\geq 0.5} f_{τ,ρ(τ)}(α)$: the pointwise maximum of the single-order bounds for each RDP privacy region. Our proof unifies and sharpens the insights of \citet{balle2019hypothesistestinginterpretationsrenyi}, \citet{asoodeh2021variantsdifferentialprivacylossless}, and \citet{zhu2022optimalaccountingdifferentialprivacy}. Our analysis relies on a precise geometric characterization of the RDP privacy region, leveraging its convexity and the fact that its boundary is determined exclusively by Bernoulli mechanisms. Our results establish that the \enquote{intersection-of-RDP-privacy-regions} rule is not only valid, but optimal: no other black-box conversion can uniformly dominate it in the Blackwell sense, marking the fundamental limit of what can be inferred about a mechanism's privacy solely from its RDP guarantees.

翻译：我们证明了\citet{zhu2022optimalaccountingdifferentialprivacy}附录F.3中提出的猜想：在所有将Rényi差分隐私（RDP）折线图$τ\mapsto ρ(τ)$映射为有效假设检验权衡函数$f$的转换规则中，基于单阶RDP隐私区域交集的方法是 最优的。该最优性对所有有效RDP折线图及所有第一类错误水平$α$同时成立。具体而言，我们证明在权衡函数空间中，最tight界为$f_{ρ(\cdot)}(α) = \sup_{τ\geq 0.5} f_{τ,ρ(τ)}(α)$：即每个RDP隐私区域单阶界的逐点最大值。我们的证明统一并深化了\citet{balle2019hypothesistestinginterpretationsrenyi}、\citet{asoodeh2021variantsdifferentialprivacylossless}和\citet{zhu2022optimalaccountingdifferentialprivacy}的见解。分析依赖于对RDP隐私区域的精确几何刻画，利用其凸性及边界由伯努利机制唯一确定的性质。我们的结果确立了"RDP隐私区域交集"规则不仅有效，而且最优：在Blackwell意义下，没有其他黑盒转换能一致地优于它，这标志着仅从RDP保证推断机制隐私性的基本极限。