We develop $(\epsilon,\delta)$-differentially private projection-depth-based medians using the propose-test-release (PTR) and exponential mechanisms. Under general conditions on the input parameters and the population measure, (e.g. we do not assume any moment bounds), we quantify the probability the test in PTR fails, as well as the cost of privacy via finite sample deviation bounds. We demonstrate our main result on the canonical projection-depth-based median. In the Gaussian setting, we show that the resulting deviation bound matches the known lower bound for private Gaussian mean estimation, up to a polynomial function of the condition number of the covariance matrix. In the Cauchy setting, we show that the ``outlier error amplification'' effect resulting from the heavy tails outweighs the cost of privacy. This result is then verified via numerical simulations. Additionally, we present results on general PTR mechanisms and a uniform concentration result on the projected spacings of order statistics.

翻译：我们采用提出-测试-发布（PTR）机制和指数机制，构建了满足$(\epsilon,\delta)$-差分隐私的投影深度中位数估计量。在输入参数与总体测度的一般条件下（例如不假设任何矩约束），我们量化了PTR中测试失败的概率，并通过有限样本偏差界刻画了隐私代价。我们以经典投影深度中位数为例展示了主要结果。在高斯设定下，我们证明所得偏差界与已知的私有高斯均值估计下界相匹配，仅相差协方差矩阵条件数的多项式函数。在柯西设定下，我们发现重尾分布引发的"离群误差放大"效应超过了隐私保护的成本。这一结论通过数值模拟得到验证。此外，我们给出了关于广义PTR机制的结果，以及关于次序统计量投影间距的一致性集中结果。