We investigate the contraction properties of locally differentially private mechanisms. More specifically, we derive tight upper bounds on the divergence between $P\mathsf{K}$ and $Q\mathsf{K}$ output distributions of an $\varepsilon$-LDP mechanism $\mathsf{K}$ 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 $\chi^2(P\mathsf{K}\|Q\mathsf{K})$ 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 $\chi^2(P\mathsf{K}\|Q\mathsf{K})$ in terms of total variation distance $\mathsf{TV}(P, Q)$ and $\varepsilon$. 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收缩性质。具体而言，我们基于相应输入分布$P$和$Q$之间的散度，推导了$\varepsilon$-LDP机制$\mathsf{K}$输出分布$P\mathsf{K}$与$Q\mathsf{K}$之间散度的紧致上界。我们的第一个主要技术结果给出了$\chi^2$-散度$\chi^2(P\mathsf{K}\|Q\mathsf{K})$关于$\chi^2(P\|Q)$和$\varepsilon$的尖锐上界。同时证明该结果适用于包括KL散度和平方Hellinger距离在内的广泛散度族。第二个主要技术结果给出了$\chi^2(P\mathsf{K}\|Q\mathsf{K})$关于全变差距离$\mathsf{TV}(P, Q)$和$\varepsilon$的上界。我们利用这些界限建立了van Trees不等式、Le Cam方法、Assouad方法以及互信息方法的局部隐私版本，这些方法是约束极小极大估计风险的有力工具。在熵估计、离散分布估计、非参数密度估计和假设检验等多个统计问题中，这些结果被证明能比现有最优方法实现更优的隐私分析。