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

• URL: http://arxiv.org/abs/2602.04562v1
• Date: Wed, 04 Feb 2026 13:49:51 GMT
• Title: Optimal conversion from Rényi Differential Privacy to $f$-Differential Privacy
• Authors: Anneliese Riess, Juan Felipe Gomez, Flavio du Pin Calmon, Julia Anne Schnabel, Georgios Kaissis,
• Abstract summary: We prove the conjecture stated in [Zhu et al. (2022) among all conversion rules that map a Rényi Differential Privacy (RDP) profile $mapsto ()$ to a valid hypothesis-testing trade-off $f$.<n>This optimality holds simultaneously for all valid profiles and for all Type I error levels $$.
• Score: 10.084485389183802
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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