Optimal Conversion from Zero-Concentrated to Gaussian Differential Privacy

Abstract

Gaussian differential privacy (GDP) and zero-concentrated differential privacy (zCDP) are two competing privacy accounting frameworks, but they share similar minimax lower bounds in various estimation problems. Moreover, it is known that a mechanism satisfying $\mu$-GDP satisfies $({\mu }^2/2)$-zCDP, whereas the converse fails to hold at the same calibration. In this paper, we establish the optimal reverse conversion, showing that every $\rho$-zCDP mechanism satisfies ${\mu }_{\mathrm{zCDP}}(\rho )$-GDP, where ${\mu }_{\mathrm{zCDP}}(\rho )=2{\Phi }^{-1}((1+e^{-t_{\rho }}{)}^{-1})$ and $t_{\rho }$ uniquely solves the equation $\rho =t_{\rho }\tanh (t_{\rho }/2)$. We also obtain simplified bounds $\sqrt{\pi \rho }\le {\mu }_{\mathrm{zCDP}}(\rho )\le \sqrt{8\rho }$, with sharp constants, which imply equivalence of the two frameworks up to constants. This explains why their minimax risks agree up to a constant rescaling of the privacy budget. We show that the squared conversion is strictly convex, which implies that converting zCDP to GDP before applying composition or group privacy yields a tighter GDP guarantee than converting afterward. We also derive the sharp conversion from bounded-range and logistic DP mechanisms (e.g., exponential mechanism) to GDP, where converting before composition but after group privacy gives the tighter guarantee.