Context-aware Privacy Bounds for Linear Queries

TL;DR

The paper addresses the excess noise induced by context-free differential privacy in releasing linear queries. It adopts pointwise maximal leakage (PML) to incorporate a prior lower bound on class probabilities, deriving a tight context-aware leakage bound for Laplace-released linear queries and proving it strictly improves over the standard DP bound while converging to DP as the prior bound vanishes. Theoretical results include a main bound $\\ell(D_i \\to y) \\le \\max_{\\mathcal{I} \\subseteq [m]} \\\log \left( \frac{e^{-c_{j_*}^{\\mathcal{I}}/b}}{\alpha \sum_{j=1}^k e^{-c_j^{\\mathcal{I}}/b} + (1 - k\alpha) e^{-c_{j^*}^{\\mathcal{I}}/b}} \right)$ and a simplified corollary bound, both tight under certain conditions. Numerical evaluations across histogram, range, and Haar-like workloads show that the PML bounds enable reduced noise scales for the same privacy level, highlighting a practical utility gain from context-aware privacy. Overall, the work advances privacy analysis by integrating prior knowledge into DP via PML, with implications for design of more efficient privacy-preserving data-release mechanisms.

Abstract

Linear queries, as the basis of broad analysis tasks, are often released through privacy mechanisms based on differential privacy (DP), the most popular framework for privacy protection. However, DP adopts a context-free definition that operates independently of the data-generating distribution. In this paper, we revisit the privacy analysis of the Laplace mechanism through the lens of pointwise maximal leakage (PML). We demonstrate that the distribution-agnostic definition of the DP framework often mandates excessive noise. To address this, we incorporate an assumption about the prior distribution by lower-bounding the probability of any single record belonging to any specific class. With this assumption, we derive a tight, context-aware leakage bound for general linear queries, and prove that our derived bound is strictly tighter than the standard DP guarantee and converges to the DP guarantee as this probability lower bound approaches zero. Numerical evaluations demonstrate that by exploiting this prior knowledge, the required noise scale can be reduced while maintaining privacy guarantees.

Figures (2)

• Figure 1: Comparison of privacy bounds for different workload type (leakage vs. prior parameter $\alpha$).
• Figure 2: Noise scale $b$ vs. privacy parameter $\varepsilon$. This figure illustrates the minimum noise scale $b$ required to ensure a target privacy leakage $\varepsilon$ for the difference query workload \ref{['eq:haar_kronecker']}. The solid ($b_{\text{PML(i)}}$), dashed ($b_{\text{PML(ii)}}$), and dash-dotted ($b_{\text{DP}}$) lines denote the minimum noise scales required for Thm. \ref{['thm:linear_queries_pml']}, Cor. \ref{['crl:simplebound']}, and $\varepsilon_{\text{DP}}$ to be at most $\varepsilon$, respectively.

Theorems & Definitions (10)

• Definition 1: Pointwise maximal leakage SaeidianPMLonAlp
• Remark 1
• Definition 2: $\varepsilon$-DP DworkCalibratingNoiseSensitivity
• Definition 3: Laplace mechanism DworkCalibratingNoiseSensitivity
• Theorem 1: DP as a PML Constraint SaeidianRethinking
• Theorem 2
• Corollary 1
• Remark 2
• Lemma 1
• Lemma 2