← 研踪 PaperTrailarXiv:2611.01218 · v1
Optimal Conversion from Zero-Concentrated to Gaussian Differential Privacy
首次提交 2026-11-01 03:39
版本更新 2026-11-01 03:39
雷达观测 2026-11-04 23:43
摘要
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 $μ$-GDP satisfies $(μ^2/2)$-zCDP, whereas the converse fails to hold at the same calibration. In this paper, we establish the optimal reverse conversion, showing that every $ρ$-zCDP mechanism satisfies $μ_{\mathrm{zCDP}}(ρ)$-GDP, where $μ_{\mathrm{zCDP}}(ρ)=2Φ^{-1}((1+e^{-t_ρ})^{-1})$ and $t_ρ$ uniquely solves the equation $ρ=t_ρ\tanh(t_ρ/2)$. We also obtain simplified bounds $\sqrt{πρ}\leqμ_{\mathrm{zCDP}}(ρ)\leq\sqrt{8ρ}$, 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.
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v1 · 2026-11-01 03:39 · 最新观测版本
Optimal Conversion from Zero-Concentrated to Gaussian Differential Privacy
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 $μ$-GDP satisfies $(μ^2/2)$-zCDP, whereas the converse fails to hold at the same calibration. In this paper, we establish the optimal reverse conversion, showing that every $ρ$-zCDP mechanism satisfies $μ_{\mathrm{zCDP}}(ρ)$-GDP, where $μ_{\mathrm{zCDP}}(ρ)=2Φ^{-1}((1+e^{-t_ρ})^{-1})$ and $t_ρ$ uniquely solves the equation $ρ=t_ρ\tanh(t_ρ/2)$. We also obtain simplified bounds $\sqrt{πρ}\leqμ_{\mathrm{zCDP}}(ρ)\leq\sqrt{8ρ}$, 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.
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