We prove the conjecture stated in Appendix F.3 of [Zhu et al. (2022)]: 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 [Balle et al. (2019)], [Asoodeh et al. (2021)], and [Zhu et al. (2022)]. 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 "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[Zhu et al. (2022)]附录F.3中提出的猜想：在所有将Rényi差分隐私（RDP）轮廓$τ\mapsto ρ(τ)$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隐私区域交集的规则是最优的。这种最优性对所有有效的RDP轮廓和所有I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，最紧的界为$f_{ρ(\cdot)}(α) = \sup_{τ\geq 0.5} f_{τ,ρ(τ)}(α)$：即各阶RDP隐私区域对应单阶界的逐点最大值。我们的证明统一并强化了[Balle et al. (2019)]、[Asoodeh et al. (2021)]和[Zhu et al. (2022)]的核心见解。分析基于对RDP隐私区域的精确几何刻画，利用其凸性及边界完全由伯努利机制决定的特性。研究结果表明，“RDP隐私区域交集”规则不仅是有效的，而且是最优的：在布莱克威尔意义下，没有任何其他黑盒转换能一致地优于该规则，这标志着仅从RDP保证推断机制隐私性的根本极限。