On Privacy Mechanism Design Under a Bounded $\alpha$-Lift Leakage Constraint with ${\chi }^2$-Utility

We study the design of a statistical privacy mechanism under a bounded $\alpha$-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 ${\chi }^2$-information ${\chi }^2(Y;U)$ 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 $P_{Y\mathrm{\mid }U=u}$. The main benefit of this choice is that ${\chi }^2(Y;U)$ is exactly quadratic in the posterior perturbations; hence, no local approximation of the utility is required. For the nonlinear $\alpha$-lift constraint, we derive a uniform Taylor-remainder bound that remains valid for the optimizing, $\epsilon$-dependent perturbation. When $P_{X\mathrm{\mid }Y}$ is invertible, this yields a closed-form quadratic surrogate, an explicitly scaled binary mechanism that satisfies the original $\alpha$-lift constraint, a computable sufficient range for $\epsilon$, and an $O({\epsilon }^{3\mathrm{/}2})$ bound on the utility gap. For $\alpha =2$, 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.