Optimal conversion from R'enyi Differential Privacy to f-Differential Privacy

We prove the conjecture stated in Appendix F.3 of citetzhu2022optimalaccountingdifferentialprivacy: among all conversion rules that map a R'enyi 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 fρ(·)(α) = supτ ≥ 0.5 fτ,ρ(τ)(α): the pointwise maximum of the single-order bounds for each RDP privacy region. Our proof unifies and sharpens the insights of citetballe2019hypothesistestinginterpretationsrenyi, citetasoodeh2021variantsdifferentialprivacylossless, and citetzhu2022optimalaccountingdifferentialprivacy. 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 enquoteintersection-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.