R\'enyi Pufferfish Privacy with Gaussian-based Priors: From Single Gaussian to Mixture Model

TL;DR

This paper develops Gaussian-based mechanisms for Rényi Pufferfish Privacy, improving privacy-utility trade-offs for correlated data by deriving exact divergence measures and introducing Gaussian mixture approximations.

Contribution

It provides exact Rényi divergence calculations for Gaussian priors, relaxes conditions for privacy, and extends to Gaussian mixtures for better utility in correlated data privacy.

Findings

Achieves an average noise reduction of 48.9% over baseline methods.

Provides closed-form conditions for privacy guarantees with Gaussian priors.

Demonstrates improved privacy-utility trade-offs on UCI datasets.

Abstract

R\'{e}nyi Pufferfish Privacy (RPP) provides a R\'{e}nyi divergence-based privacy framework for correlated data, but existing $\infty$-Wasserstein mechanisms are often conservative and sacrifice data utility. We study Gaussian mechanisms for RPP under Gaussian and Gaussian-mixture priors. For single Gaussian priors, we derive the exact R\'{e}nyi divergence after Gaussian perturbation, obtain a relaxed closed-form sufficient condition for $(\alpha,\epsilon)$-RPP, and characterize the monotonicity of the calibrated noise with respect to the privacy budget $\epsilon$ and the R\'{e}nyi order $\alpha$. To handle more general non-Gaussian and multimodal priors, we approximate secret-conditioned outputs with Gaussian mixture models and introduce an optimal-transport-based sufficient condition for RPP. Experiments on three UCI datasets with statistical (\textsc{RAW}, \textsc{MEAN}) and model-output (\textsc{BNN}, \textsc{GP}) queries show that our prior-aware mechanisms consistently require less noise than a recent RPP additive-noise baseline, achieving an average noise reduction of 48.9\%. These results show that our mechanisms can substantially improve the privacy-utility trade-off under RPP.