On Privacy Mechanism Design Under a Bounded α-Lift Leakage Constraint with χ2-Utility
Abstract
We study the design of a statistical privacy mechanism under a bounded α-lift leakage constraint. An agent observes useful data Y, which is arbitrarily correlated with private data X, and releases U. The goal is to maximize Pearson χ2-information χ2(Y;U) while satisfying the point-wise privacy constraint. This utility is an f-information that is nonnegative, obeys data processing, and measures the aggregate normalized change from the prior PY to the posteriors PY|U=u. The main benefit of this choice is that χ2(Y;U) is exactly quadratic in the posterior perturbations; hence, no local approximation of the utility is required. For the nonlinear α-lift constraint, we derive a uniform Taylor-remainder bound that remains valid for the optimizing, ε-dependent perturbation. When PX|Y is invertible, this yields a closed-form quadratic surrogate, an explicitly scaled binary mechanism that satisfies the original α-lift constraint, a computable sufficient range for ε, and an O(ε3/2) bound on the utility gap. For α=2, the solution is exact over an explicit range. We transfer these results to point-wise Rényi-divergence privacy, including an exact-feasible mechanism, an explicit privacy-budget range, and matching gap bounds. We also give an exact formulation for arbitrary leakage matrices and evaluate exact-feasible mechanisms numerically.
Related
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BibTeX
@misc{zamani2026privacy,
title = {{On Privacy Mechanism Design Under a Bounded $α$-Lift Leakage Constraint with $χ^2$-Utility}},
author = {Amirreza Zamani and Mohammad Amin Zarrabian and Parastoo Sadeghi and Mikael Skoglund},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13685},
eprint = {2610.13685},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13685},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Privacy Mechanism Design Under a Bounded α-Lift Leakage Constraint with χ^2-Utility
AU - Amirreza Zamani
AU - Mohammad Amin Zarrabian
AU - Parastoo Sadeghi
AU - Mikael Skoglund
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13685
UR - https://arxiv.org/abs/2610.13685
AB - We study the design of a statistical privacy mechanism under a bounded α-lift leakage constraint. An agent observes useful data Y, which is arbitrarily correlated with private data X, and releases U. The goal is to maximize Pearson χ^2-information χ^2(Y;U) while satisfying the point-wise privacy constraint. This utility is an f-information that is nonnegative, obeys data processing, and measures the aggregate normalized change from the prior P_Y to the posteriors P_Y|U=u. The main benefit of this choice is that χ^2(Y;U) is exactly quadratic in the posterior perturbations; hence, no local approximation of the utility is required. For the nonlinear α-lift constraint, we derive a uniform Taylor-remainder bound that remains valid for the optimizing, ε-dependent perturbation. When P_X|Y is invertible, this yields a closed-form quadratic surrogate, an explicitly scaled binary mechanism that satisfies the original α-lift constraint, a computable sufficient range for ε, and an O(ε^3/2) bound on the utility gap. For α=2, the solution is exact over an explicit range. We transfer these results to point-wise Rényi-divergence privacy, including an exact-feasible mechanism, an explicit privacy-budget range, and matching gap bounds. We also give an exact formulation for arbitrary leakage matrices and evaluate exact-feasible mechanisms numerically.
ER -