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Less is More: Revisiting Gaussian Mechanism for Differential Privacy

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

In this paper, we identify that the classic Gaussian mechanism and its variants for differential privacy all suffer from \textbf{the curse of full-rank covariance matrices}, and hence the expected accuracy losses of these mechanisms applied to high dimensional query results, e.g., in RM\mathbb{R}^MRM, all increase linearly with MMM. To lift this curse, we design a Rank-1 Singular Multivariate Gaussian Mechanism (R1SMG). 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 work on Gaussian mechanism that has been ignored in the literature: 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. This paper makes the following technical contributions. (i) R1SMG achieves (ϵ,δ)(\epsilon,\delta)(ϵ,δ)-DP guarantee on query results in RM\mathbb{R}^MRM, while the magnitude of the additive noise decreases with MMM. Therefore, \textbf{less is more}, i.e., less amount of noise is able to sanitize higher dimensional query results. When M→∞M\rightarrow \inftyM→∞, the expected accuracy loss converges to 2(Δ2f)2/ϵ{2(\Delta_2f)^2}/{\epsilon}2(Δ2​f)2/ϵ, where Δ2f\Delta_2fΔ2​f is the l2l_2l2​ sensitivity of the query function fff. (ii) Compared with other mechanisms, R1SMG is less likely to generate noise with large magnitude that overwhelms the query results, because the kurtosis and skewness of the nondeterministic accuracy loss introduced by R1SMG is larger than that introduced by other mechanisms.

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