We consider a private discrete distribution estimation problem with one-bit communication constraint. The privacy constraints are imposed with respect to the local differential privacy and the maximal leakage. The estimation error is quantified by the worst-case mean squared error. We completely characterize the first-order asymptotics of this privacy-utility trade-off under the one-bit communication constraint for both types of privacy constraints by using ideas from local asymptotic normality and the resolution of a block design mechanism. These results demonstrate the optimal dependence of the privacy-utility trade-off under the one-bit communication constraint in terms of the parameters of the privacy constraint and the size of the alphabet of the discrete distribution.

中文翻译：

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1 位通信的最优私有离散分布估计


我们考虑具有一位通信约束的私有离散分布估计问题。隐私约束是针对局部差分隐私和最大泄漏施加的。估计误差通过最坏情况均方误差来量化。我们通过使用局部渐近正态性的思想和块设计机制的解决方案，完整地描述了在两种类型的隐私约束的一位通信约束下这种隐私-效用权衡的一阶渐近性。这些结果证明了在隐私约束的参数和离散分布的字母表的大小方面，在一位通信约束下隐私-效用权衡的最佳依赖性。