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.

翻译：我们考虑从数据集中差分隐私地估计多个分位数（MQ）的问题，这是现代数据分析中的关键构建模块。我们将近期提出的非平滑逆敏感度（IS）机制应用于这一特定问题。研究表明，该方法与近期发表的专用算法JointExp密切相关，两者在计算复杂度与效率上具有相似性。我们证明了这两种算法对连续分布具有统计一致性。此外，我们从理论和实验两个层面揭示了该方法在尖峰分布情况下性能显著下降的问题，尤其在存在原子分布时可能产生灾难性影响。其平滑版本（即在其输出密度上应用最大核）可解决此问题，但实现仍存在开放性挑战。作为替代方案，我们提出一种简单且数值高效的方法——启发式平滑联合指数机制（HSJointExp），该方法在广泛分布类别中具备性能保证，并在问题数据集上实现了数量级提升的效果。