We introduce general tools for designing efficient private estimation algorithms, in the high-dimensional settings, whose statistical guarantees almost match those of the best known non-private algorithms. To illustrate our techniques, we consider two problems: recovery of stochastic block models and learning mixtures of spherical Gaussians. For the former, we present the first efficient $(ε, δ)$-differentially private algorithm for both weak recovery and exact recovery. Previously known algorithms achieving comparable guarantees required quasi-polynomial time. For the latter, we design an $(ε, δ)$-differentially private algorithm that recovers the centers of the $k$-mixture when the minimum separation is at least $ O(k^{1/t}\sqrt{t})$. For all choices of $t$, this algorithm requires sample complexity $n\geq k^{O(1)}d^{O(t)}$ and time complexity $(nd)^{O(t)}$. Prior work required minimum separation at least $O(\sqrt{k})$ as well as an explicit upper bound on the Euclidean norm of the centers.
Differentially Private Distribution Release of Gaussian Mixture Models via KL-Divergence Minimization
State Key Laboratory of Internet of Things for Smart City and the Department of Electrical and Computer Engineering, University of Macau（物联网智能城市国家重点实验室和澳门大学电子与计算机工程系）；Department of Electrical and Computer Engineering, Cornell Tech, Cornell University（电气与计算机工程系，康奈尔科技，康奈尔大学）；Computing Sciences Research, Lawrence Berkeley National Laboratory（计算科学研究所，劳伦斯伯克利国家实验室）
Bayesian Mixture Models for Heterogeneous Extremes
Unité Biostatistique et Processus Spatiaux, INRAE, France（生物统计与空间过程单元，法国国家农业食品与环境研究院）；School of Mathematics, University of Edinburgh, UK（爱丁堡大学数学学院）；Department of Mathematics, University of Aveiro, Portugal（阿威罗大学数学系）；Department of Environmental Sciences, Informatics and Statistics, Ca’Foscari University of Venice, Italy（威尼斯卡福斯卡里大学环境科学、信息学与统计学系）
Le Cam方法、Fano不等式和Assouad引理是证明统计估计任务下界的三种广泛使用的技术。在中心差分隐私下，我们提出了它们的类似物。我们的结果是简单的，易于应用，我们用它们来建立样本的复杂性界限在几个估计任务。在全变差距离和$\ell_2$距离下，建立了离散分布估计的最优样本复杂度.我们还提供了其他几个分布类的下限，包括产品分布和高斯混合是紧密的对数因子。我们论文的技术部分将分布之间的耦合与差分隐私下估计的样本复杂性联系起来。
