Exact zCDP Characterizations for Fundamental Differentially Private Mechanisms
Charlie Harrison, Pasin Manurangsi
TL;DR
This work advances the understanding of zero-concentrated DP by deriving exact, tight zCDP bounds for several fundamental ε-DP mechanisms, including the Laplace and discrete Laplace mechanisms, as well as local mechanisms like k-RR (for small k) and RAPPOR, and the bounding-range family. The authors establish that the tight zCDP bound for the ε-DP Laplace mechanism is exactly $\varepsilon + e^{-\varepsilon} - 1$, and they provide closed-form tight bounds for the discrete Laplace, k-RR, RAPPOR, and η-BR mechanisms, plus asymptotics as $\varepsilon\to 0$. Their analysis hinges on connecting zCDP to KL divergence (via α→1 limits) and on meticulous RDP-to-zCDP conversions, including derivative-based and convexity arguments, to certify tightness. The results enable precise privacy accounting under composition and improve practical guarantees in large-composition regimes, while also outlining open problems for larger k in k-RR and other mechanisms. Overall, the paper supplies a comprehensive set of exact zCDP characterizations that tighten prior bounds and enhance DP mechanism design and analysis.
Abstract
Zero-concentrated differential privacy (zCDP) is a variant of differential privacy (DP) that is widely used partly thanks to its nice composition property. While a tight conversion from $ε$-DP to zCDP exists for the worst-case mechanism, many common algorithms satisfy stronger guarantees. In this work, we derive tight zCDP characterizations for several fundamental mechanisms. We prove that the tight zCDP bound for the $ε$-DP Laplace mechanism is exactly $ε+ e^{-ε} - 1$, confirming a recent conjecture by Wang (2022). We further provide tight bounds for the discrete Laplace mechanism, $k$-Randomized Response (for $k \leq 6$), and RAPPOR. Lastly, we also provide a tight zCDP bound for the worst case bounded range mechanism.