  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.
