Optimal conversion from Rényi Differential Privacy to $f$-Differential Privacy

We prove the conjecture stated in Appendix F.3 of Zhu et al.: among all conversion rules that map a Rényi Differential Privacy (RDP) profile $\tau \mapsto \rho(\tau)$ to a valid hypothesis-testing trade-off $f$ (or equivalently, an $(\varepsilon,\delta)$-Differential Privacy curve), 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 $\alpha$. Concretely, we show that in the space of trade-off functions, the tightest possible bound is $f_{\rho(\cdot)}(\alpha) = \sup_{\tau \geq 0.5} f_{\tau,\rho(\tau)}(\alpha)$: the pointwise maximum of the single-order bounds for each RDP privacy region. Our proof unifies and sharpens the insights of Balle et al., Asoodeh et al., and Zhu et al.. 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.

Differential privacy is a mathematical framework that provides rigorous guarantees that no single person's data has too much influence on what an algorithm reveals. Over the years, researchers have developed several different languages for expressing differential privacy, each with its own strengths. Two of the most important are Rényi Differential Privacy (RDP), convenient for calculations during iterative algorithms, and f-Differential Privacy (f-DP), which gives the most complete and interpretable picture of privacy loss. In practice, engineers might measure privacy using RDP but need to report results in f-DP terms, requiring a conversion between the two. Our paper proves that a particular conversion rule is not just good, but optimal — no other rule relying solely on the RDP guarantee can ever do better. This definitively closes a long-standing open problem in the privacy accounting literature.