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 μ-GDP satisfies (μ2/2)-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 μzCDP(ρ)-GDP, where μzCDP(ρ)=2Φ−1((1+e−tρ)−1) and tρ uniquely solves the equation ρ=tρtanh(tρ/2). We also obtain simplified bounds √(πρ)≤μzCDP(ρ)≤√(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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BibTeX
@misc{chen2026optimal,
title = {{Optimal Conversion from Zero-Concentrated to Gaussian Differential Privacy}},
author = {Yu-Wei Chen and Aaradhya Pandey and Jordan Awan},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01218},
eprint = {2611.01218},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01218},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal Conversion from Zero-Concentrated to Gaussian Differential Privacy
AU - Yu-Wei Chen
AU - Aaradhya Pandey
AU - Jordan Awan
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01218
UR - https://arxiv.org/abs/2611.01218
AB - 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_zCDP(ρ)-GDP, where μ_zCDP(ρ)=2Φ^−1((1+e^−t_ρ)^−1) and t_ρ uniquely solves the equation ρ=t_ρtanh(t_ρ/2). We also obtain simplified bounds √(πρ)≤μ_zCDP(ρ)≤√(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.
ER -