Optimal private density estimation for instances in the Wasserstein distance

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 can capture approximately where the population mass is located. In this work we study the differential estimation of private density in the Wasserstein distance. We design and analyze instance-optimal algorithms for this problem that can be adapted to easy instances.

For distributions P<annotation encoding="application/x-tex”>PP on R<annotation encoding="application/x-tex”>\mathbb{R}RWe consider a strong notion of instance optimization: an algorithm that uniformly achieves the optimal instance estimation rate is competitive with an algorithm that is told that the distribution is P<annotation encoding="application/x-tex”>PP either qP<annotation encoding="application/x-tex”>Q_PqP​ for some distribution qP<annotation encoding="application/x-tex”>Q_PqP​ whose probability density function (pdf) is within a factor of 2 of the pdf of P<annotation encoding="application/x-tex”>PP. For higher distributions R2<annotation encoding="application/x-tex”>\mathbb{R}^2R2

we use a different notion of instance optimization. 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 optimal instance estimation rates in both environments and show that they can be achieved consistently (up to polylogarithmic factors). Our approach to R2<annotation encoding="application/x-tex”>\mathbb{R}^2R2 it extends to arbitrary metric spaces as it passes through hierarchically separated trees. As a special case, our results lead to optimal private learning for instances in remote television for discrete distributions.