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.

我们使用提出 - 测试 - 发布（PTR）和指数机制开发了 ε，δ 差分隐私投影深度中位数。我们对输入参数和人群测度的一般条件（例如，我们不假设任何时刻界限）进行了量化，以及通过有限样本偏离界限评估隐私成本。我们在典型的投影深度中位数上展示了主要结果。在高斯环境中，我们展示了所得的偏离界限与用于私有高斯均值估计的已知下界相匹配，最多是协方差矩阵条件数的多项式函数。在柯西环境中，我们展示了由重尾部引起的 “异常错误放大” 效应比隐私成本更大。通过数值模拟验证了这一结果。此外，我们还提出了关于一般 PTR 机制的结果，以及关于有序统计量投影间距的均匀集中结果。