We investigate the contraction properties of locally differentially private mechanisms. More specifically, we derive tight upper bounds on the divergence between $PK$ and $QK$ output distributions of an $\epsilon$-LDP mechanism $K$ in terms of a divergence between the corresponding input distributions $P$ and $Q$, respectively. Our first main technical result presents a sharp upper bound on the $\chi^2$-divergence $\chi^2(PK}\|QK)$ in terms of $\chi^2(P\|Q)$ and $\varepsilon$. We also show that the same result holds for a large family of divergences, including KL-divergence and squared Hellinger distance. The second main technical result gives an upper bound on $\chi^2(PK\|QK)$ in terms of total variation distance $\mathsf{TV}(P, Q)$ and $\epsilon$. We then utilize these bounds to establish locally private versions of the van Trees inequality, Le Cam's, Assouad's, and the mutual information methods, which are powerful tools for bounding minimax estimation risks. These results are shown to lead to better privacy analyses than the state-of-the-arts in several statistical problems such as entropy and discrete distribution estimation, non-parametric density estimation, and hypothesis testing.

翻译：我们研究局部差分隐私机制的收缩性质。具体而言，针对一个ε-局部差分隐私机制K，我们基于相应输入分布P和Q之间的散度，推导出输出分布PK与QK之间散度的紧致上界。第一个主要技术结果给出了χ²散度χ²(PK‖QK)关于χ²(P‖Q)和ε的精确上界。我们同时证明该结果适用于一大类散度，包括KL散度和平方Hellinger距离。第二个主要技术结果给出了χ²(PK‖QK)关于全变差距离TV(P,Q)和ε的上界。随后我们利用这些界建立了局部私有版本的范特里斯不等式、勒卡姆引理、阿苏瓦德引理及互信息方法，这些是约束极小化极大估计风险的有力工具。在熵与离散分布估计、非参数密度估计及假设检验等若干统计问题中，这些结果相比现有技术能实现更优的隐私分析。