Noise Reduction for Pufferfish Privacy: A Practical Noise Calibration Method

TL;DR

This work tackles privacy for correlated data under the pufferfish framework by examining the $1$-Wasserstein mechanism, which imposes a strict pointwise condition that often yields excessive noise. It introduces a practical relaxed noise-calibration mechanism and a Brent-based root-finding algorithm to compute a smaller noise scale $\hat{\theta}$ while preserving $(\epsilon,\mathbb{S})$-pufferfish privacy, with guarantees that $\hat{\theta}<\theta_1$ for all $\epsilon$. Theoretical results establish the existence and increasing gain of noise reduction as privacy tightens ($\epsilon$ decreases), and demonstrate equivalence to $\ell_1$-sensitivity in the worst-case $W_1$ scenario. Empirical validation on three real-world datasets shows substantial data-utility improvements (47%–87% relative) and confirms the practical impact of the proposed approach.

Abstract

This paper introduces a relaxed noise calibration method to enhance data utility while attaining pufferfish privacy. This work builds on the existing $1$-Wasserstein (Kantorovich) mechanism by alleviating the existing overly strict condition that leads to excessive noise, and proposes a practical mechanism design algorithm as a general solution. We prove that a strict noise reduction by our approach always exists compared to $1$-Wasserstein mechanism for all privacy budgets $ε$ and prior beliefs, and the noise reduction (also represents improvement on data utility) gains increase significantly for low privacy budget situations--which are commonly seen in real-world deployments. We also analyze the variation and optimality of the noise reduction with different prior distributions. Moreover, all the properties of the noise reduction still exist in the worst-case $1$-Wasserstein mechanism we introduced, when the additive noise is largest. We further show that the worst-case $1$-Wasserstein mechanism is equivalent to the $\ell_1$-sensitivity method. Experimental results on three real-world datasets demonstrate $47\%$ to $87\%$ improvement in data utility.

Figures (4)

• Figure 1: Visualization of strict and relaxed condition: \ref{['fig:relaxed_source_strict']} shows that $I(x,x';\theta_1)\leq 0$ holds for all $(x,x')\in\mathcal{X}^2$ satisfying Eq. \ref{['eq:w_1_mechanism']}. Consequently, $\sum_x I(x,x';\theta_1) \leq 0$ for all $x'\in\mathcal{X}$ satisfying $(\epsilon,\mathbb{S})$-pufferfish privacy. \ref{['fig:relaxed_source_relaxed']} shows that while $\sum_x I(x,x';\theta_1) \leq 0$ still holds for all $x'\in\mathcal{X}$ satisfying pufferfish privacy, not all $I(x,x';\theta_1)$ are non-positive--the relaxation of the strict condition. The horizontal axis indexes the ordered pairs $(x,x')$, sorted primarily by $x'$ and then by $x$. The index starts at $1$ for $(0,0)$, increases sequentially with $x$, and wraps to the next value of $x'$ after every $100$ steps. For instance, $(99,0)$ has index $100$ and $(0,1)$ has index $101$.
• Figure 2: Experimental results using prior distributions in Table \ref{['tab:dataset_1_1']}. \ref{['fig:dataset_example_leq_1_pi']} shows the corresponding Kantorovich optimal transport plan $\pi^*$. \ref{['fig:dataset_example_leq_1_res']} shows the noise parameter of Laplace noise: $\theta_1$ calibrated by $W_1$ mechanism and $\hat{\theta}$ by our approach in Algorithm \ref{['alg:brent_method']}.
• Figure 3: Experimental results in the worst case $W_1$ mechanism. \ref{['fig:dataset_w1_equal_dp_pi']} shows the corresponding Kantorovich optimal transport plan $\pi^*$ where $\pi^*(0,3) \neq 0$ satisfying condition \ref{['eq:W1_worst']}. \ref{['fig:dataset_w1_equal_dp_res']} shows the noise parameter of Laplace noise: $\theta_1$ calibrated by $W_1$ mechanism, $\theta_{\ell}$ by $\ell_1$-sensitivity method, and $\hat{\theta}$ by our proposed practical relaxed mechanism in Algorithm \ref{['alg:brent_method']}.
• Figure 4: In Student Performance: \ref{['fig:dataset_student_probability']} shows the prior distributions of ‘Romantic’ conditioned on ‘higher-yes’ and 'higher-no' events. \ref{['fig:dataset_student_pi']} illustrates the corresponding Kantorovich optimal transport plan $\pi^*$. \ref{['fig:dataset_student_res']} and \ref{['fig:dataset_student_res_ell']} show the Laplace noise parameter, $\theta_1$ by $W_1$ mechanism, $\theta_\ell$ by $\ell_1$-sensitivity mechanism and $\hat{\theta}$ by Algorithm \ref{['alg:brent_method']}. In Bank Marketing: \ref{['fig:dataset_student_probability']} shows the prior distributions of ‘Marital’ conditioned on ‘loan-yes’ and 'loan-no' events. \ref{['fig:dataset_bank_pi']} shows the corresponding Kantorovich optimal transport plan $\pi^*$. \ref{['fig:dataset_bank_res']} and \ref{['fig:dataset_bank_res_ell']} present the Laplace noise parameter, $\theta_1$ by $W_1$ mechanism, $\theta_\ell$ by $\ell_1$-sensitivity mechanism and $\hat{\theta}$ by Algorithm \ref{['alg:brent_method']}. In Census Income: \ref{['fig:dataset_student_probability']} shows the prior distributions of ‘Workclass’ conditioned on ‘Married-civ-spouse’ and 'Never-married' events. \ref{['fig:dataset_census_pi']} illustrates the corresponding Kantorovich optimal transport plan $\pi^*$. \ref{['fig:dataset_census_res']} and \ref{['fig:dataset_census_res_ell']} show the Laplace noise parameter, $\theta_1$ by $W_1$ mechanism, $\theta_\ell$ by $\ell_1$-sensitivity mechanism and $\hat{\theta}$ by Algorithm \ref{['alg:brent_method']}.

Theorems & Definitions (6)

• Proposition 3.2: Relaxed Mechanism
• Proposition 3.3
• Theorem 4.1: Strict Noise Reduction
• Lemma 4.2
• Proposition 5.1
• Remark 5.2