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arXiv 1312.0655cs.CR
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差分隐私的最优机制:多维设置

The Optimal Mechanism in Differential Privacy: Multidimensional Setting

Quan Geng, Pramod Viswanath

摘要:

我们推导出最佳的$ε$ -差分私有机制下的效用最大化(或成本最小化)的$\ell^1$成本函数的框架下的一般二维实值(直方图)查询函数。我们表明,最佳的噪声概率分布有一个相关的多维阶梯形的概率密度函数。与拉普拉斯机制相比,我们发现,在高隐私制度(如$ε\to 0$),拉普拉斯机制是近似最优的;在低隐私制度(如$ε\to +\infty$),最优的成本是$Θ(e^{-\fracε{3}})$,而拉普拉斯机制的成本是$\frac{2Δ}ε$,其中$Δ$是查询函数的敏感度。我们的结论是,收益是更明显的低隐私制度。我们猜想,阶梯机制的最优性适用于任意维度的向量值(直方图)查询函数,并适用于许多其他类别的成本函数。

英文摘要:

We derive the optimal $ε$-differentially private mechanism for a general two-dimensional real-valued (histogram-like) query function under a utility-maximization (or cost-minimization) framework for the $\ell^1$ cost function. We show that the optimal noise probability distribution has a correlated multidimensional staircase-shaped probability density function. Compared with the Laplacian mechanism, we show that in the high privacy regime (as $ε\to 0$), the Laplacian mechanism is approximately optimal; and in the low privacy regime (as $ε\to +\infty$), the optimal cost is $Θ(e^{-\fracε{3}})$, while the cost of the Laplacian mechanism is $\frac{2Δ}ε$, where $Δ$ is the sensitivity of the query function. We conclude that the gain is more pronounced in the low privacy regime. We conjecture that the optimality of the staircase mechanism holds for vector-valued (histogram-like) query functions with arbitrary dimension, and holds for many other classes of cost functions as well.

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