  We consider the differentially private estimation of multiple quantiles (MQ)
of a distribution from a dataset, a key building block in modern data analysis.
We apply the recent non-smoothed Inverse Sensitivity (IS) mechanism to this
specific problem. We establish that the resulting method is closely related to
the recently published ad hoc algorithm JointExp. In particular, they share the
same computational complexity and a similar efficiency. We prove the
statistical consistency of these two algorithms for continuous distributions.
Furthermore, we demonstrate both theoretically and empirically that this method
suffers from an important lack of performance in the case of peaked
distributions, which can degrade up to a potentially catastrophic impact in the
presence of atoms. Its smoothed version (i.e. by applying a max kernel to its
output density) would solve this problem, but remains an open challenge to
implement. As a proxy, we propose a simple and numerically efficient method
called Heuristically Smoothed JointExp (HSJointExp), which is endowed with
performance guarantees for a broad class of distributions and achieves results
that are orders of magnitude better on problematic datasets.
