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 $(\epsilon, \delta)$-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 $(\epsilon, \delta)$-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.

翻译：我们引入了在高维环境中设计高效隐私估计算法的通用工具，其统计保证几乎与已知最优的非隐私算法相匹配。为阐述我们的技术，我们考虑两个问题：随机块模型的恢复与球面高斯混合模型的学习。针对前者，我们提出了首个针对弱恢复与精确恢复的高效$(\epsilon, \delta)$-差分隐私算法。此前已知能达到类似保证的算法需要拟多项式时间。针对后者，我们设计了一个$(\epsilon, \delta)$-差分隐私算法，当最小间隔至少为$O(k^{1/t}\sqrt{t})$时，该算法能够恢复$k$-混合模型的中心。对于所有$t$的取值，该算法所需的样本复杂度为$n\geq k^{O(1)}d^{O(t)}$，时间复杂度为$(nd)^{O(t)}$。先前的研究要求最小间隔至少为$O(\sqrt{k})$，并且需要对中心的欧几里得范数给出显式上界。