We study the problem of privately estimating the parameters of $d$-dimensional Gaussian Mixture Models (GMMs) with $k$ components. For this, we develop a technique to reduce the problem to its non-private counterpart. This allows us to privatize existing non-private algorithms in a blackbox manner, while incurring only a small overhead in the sample complexity and running time. As the main application of our framework, we develop an $(\varepsilon, \delta)$-differentially private algorithm to learn GMMs using the non-private algorithm of Moitra and Valiant [MV10] as a blackbox. Consequently, this gives the first sample complexity upper bound and first polynomial time algorithm for privately learning GMMs without any boundedness assumptions on the parameters.

翻译：我们研究使用$k$个分量对$d$维高斯混合模型（GMMs）参数进行隐私估计的问题。为此，我们开发了一种将该问题简化为非隐私对应问题的方法。这使我们能够以黑盒方式对现有的非隐私算法进行隐私化，同时仅增加少量样本复杂度和运行时间开销。作为我们框架的主要应用，我们以Moitra和Valiant [MV10]的非隐私算法为黑盒，开发了一种$(\varepsilon, \delta)$-差分隐私算法来学习GMMs。因此，这在不假设参数有界性的条件下，首次给出了隐私学习GMMs的样本复杂度上界和第一个多项式时间算法。