Perhaps the single most important use case for differential privacy is to privately answer numerical queries, which is usually achieved by adding noise to the answer vector. The central question, therefore, is to understand which noise distribution optimizes the privacy-accuracy trade-off, especially when the dimension of the answer vector is high. Accordingly, extensive literature has been dedicated to the question and the upper and lower bounds have been matched up to constant factors [BUV18, SU17]. In this paper, we take a novel approach to address this important optimality question. We first demonstrate an intriguing central limit theorem phenomenon in the high-dimensional regime. More precisely, we prove that a mechanism is approximately Gaussian Differentially Private [DRS21] if the added noise satisfies certain conditions. In particular, densities proportional to $\mathrm{e}^{-\|x\|_p^\alpha}$, where $\|x\|_p$ is the standard $\ell_p$-norm, satisfies the conditions. Taking this perspective, we make use of the Cramer--Rao inequality and show an "uncertainty principle"-style result: the product of the privacy parameter and the $\ell_2$-loss of the mechanism is lower bounded by the dimension. Furthermore, the Gaussian mechanism achieves the constant-sharp optimal privacy-accuracy trade-off among all such noises. Our findings are corroborated by numerical experiments.
翻译:也许对于不同隐私而言,最重要的单一应用是私下回答数字查询,通常是通过在回答矢量中增加噪音来实现的。因此,中心问题是,了解哪个噪音分布优化了隐私-准确性交换,特别是当答案矢量的尺寸较高时。因此,大量文献都专门论述这个问题,而上下界限与经常系数[BUV18,SU17]相对应。在本文件中,我们采取了新颖的方法来解决这一重要的最佳性问题。我们首先展示了高维系统中令人感兴趣的中心限制理论现象。更确切地说,我们证明一个机制大约是Gaussian differental-clocal-creal-creal-creal-creal-creative-creat-creat-creative[DRS21],如果添加的噪音满足了某些条件。特别是密度与$mathrm{e ⁇ - ⁇ x ⁇ ⁇ ⁇ p ⁇ pä alpha}。在本文中, $xprocreal-creal-dealalisalal rolismalalalalal yalal roupal resmal resmal resutus exal resulate maus。我们使用Calutal exal exal exal exal exmal exmal ex ex ex ex ex exalutus ex ex exal ex ex exal exalupal exalupal exal exalupalupal 。我们使用Cal 。我们所有的固定的固定的固定机制, exbal ex ex exal ex exal exal exal exal exal exal exal exalbal exalal 。 。。通过 。通过 ex exalal 。 和低 。。我们通过平价制的固定的固定的产产产的产的产是低。我们的固定的产的固定的固定的产的产的固定的固定的产的产的产的产的产的产的产的产的产的产。 。 。 。 。 。 。我们的固定的