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：我们研究了局部差分隐私机制的压缩性质。具体而言，我们针对一个ε-局域差分隐私机制K，推导了其输出分布PK与QK之间散度的紧致上界，该上界由相应输入分布P与Q之间的散度表示。第一项主要技术结果给出了基于χ²(P‖Q)与ε的χ²散度χ²(PK‖QK)的精确上界。同时证明该结果适用于包括KL散度与平方海林格距离在内的一大类散度。第二项主要技术结果给出了基于总变差距离TV(P,Q)与ε的χ²(PK‖QK)上界。我们进而利用这些界限建立了范特里斯不等式、勒卡姆引理、阿苏阿德引理及互信息方法（这些是限制极小极大估计风险的有力工具）的局部隐私版本。结果表明，在熵分布估计、非参数密度估计和假设检验等统计问题中，这些方法相较于现有技术可获得更优的隐私分析。