Estimating the density of a distribution from samples is a fundamental problem in statistics. In many practical settings, the Wasserstein distance is an appropriate error metric for density estimation. For example, when estimating population densities in a geographic region, a small Wasserstein distance means that the estimate is able to capture roughly where the population mass is. In this work we study differentially private density estimation in the Wasserstein distance. We design and analyze instance-optimal algorithms for this problem that can adapt to easy instances. For distributions $P$ over $\mathbb{R}$, we consider a strong notion of instance-optimality: an algorithm that uniformly achieves the instance-optimal estimation rate is competitive with an algorithm that is told that the distribution is either $P$ or $Q_P$ for some distribution $Q_P$ whose probability density function (pdf) is within a factor of 2 of the pdf of $P$. For distributions over $\mathbb{R}^2$, we use a different notion of instance optimality. We say that an algorithm is instance-optimal if it is competitive with an algorithm that is given a constant-factor multiplicative approximation of the density of the distribution. We characterize the instance-optimal estimation rates in both these settings and show that they are uniformly achievable (up to polylogarithmic factors). Our approach for $\mathbb{R}^2$ extends to arbitrary metric spaces as it goes via hierarchically separated trees. As a special case our results lead to instance-optimal private learning in TV distance for discrete distributions.

翻译：从样本中估计分布的密度是统计学中的一个基本问题。在许多实际场景中，Wasserstein距离是密度估计的合适误差度量。例如，在估计地理区域的人口密度时，较小的Wasserstein距离意味着估计能够大致捕捉人口质量的分布位置。本工作中，我们研究Wasserstein距离下的差分私有密度估计问题。我们为此问题设计并分析了能够适应简单实例的实例最优算法。对于定义在$\mathbb{R}$上的分布$P$，我们采用一种强实例最优性概念：若某算法能一致达到实例最优估计速率，则其性能可与一个事先获知分布为$P$或$Q_P$的算法相竞争，其中$Q_P$的概率密度函数（pdf）与$P$的pdf相差不超过2倍。对于$\mathbb{R}^2$上的分布，我们采用不同的实例最优性定义：若某算法与一个获得分布密度常数因子近似值的算法性能相当，则称该算法是实例最优的。我们刻画了这两种设定下的实例最优估计速率，并证明它们均可一致实现（至多相差多对数因子）。我们针对$\mathbb{R}^2$的方法通过层次分离树结构可推广至任意度量空间。作为特例，我们的结果可导出离散分布在TV距离下的实例最优私有学习。