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Less is More: Revisiting the Gaussian Mechanism for Differential Privacy

4 June 2023
Tianxi Ji
Pan Li
ArXiv (abs)PDFHTML
Abstract

Differential privacy via output perturbation has been a \textit{de facto} standard for releasing query or computation results on sensitive data. However, we identify that all existing Gaussian mechanisms suffer from the curse of full-rank covariance matrices, and hence the expected accuracy losses of these mechanisms equal the trace of the covariance matrix of the noise. To lift this curse, we design a Rank-1 Singular Multivariate Gaussian (R1SMG) mechanism. It achieves (ϵ,δ)(\epsilon,\delta)(ϵ,δ)-DP on query results in RM\mathbb{R}^MRM by perturbing the results with noise following a singular multivariate Gaussian distribution, whose covariance matrix is a \textbf{randomly} generated rank-1 positive semi-definite matrix. In contrast, the classic Gaussian mechanism and its variants all consider \textbf{deterministic} full-rank covariance matrices. Our idea is motivated by a clue from Dwork et al.'s seminal work on the classic Gaussian mechanism that has been ignored: when projecting multivariate Gaussian noise with a full-rank covariance matrix onto a set of orthonormal basis in RM\mathbb{R}^MRM, only the coefficient of a single basis can contribute to the privacy guarantee. We make the following contributions. The R1SMG mechanisms achieves (ϵ,δ)(\epsilon,\delta)(ϵ,δ)-DP guarantee on query results in RM\R^MRM, while its expected accuracy loss is lower bounded by CR(Δ2f)2C_R(\Delta_2f)^2CR​(Δ2​f)2, where CR=2ϵψC_R = \frac{2}{\epsilon \psi}CR​=ϵψ2​ and ψ=(δΓ(M−12)πΓ(M2))2M−2\psi = \Big(\frac{\delta\Gamma(\frac{M-1}{2})}{\sqrt{\pi}\Gamma(\frac{M}{2})}\Big)^{\frac{2}{M-2}}ψ=(π​Γ(2M​)δΓ(2M−1​)​)M−22​. We show that CRC_RCR​ has a decreasing trend as MMM increases, and converges to 2ϵ\frac{2}{\epsilon}ϵ2​ as MMM approaches infinity. Compared with other mechanisms, the R1SMG mechanism is more stable and less likely to generate noise with large magnitude that overwhelms the query results.

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