We investigate unbiased high-dimensional mean estimators in differential privacy. We consider differentially private mechanisms whose expected output equals the mean of the input dataset, for every dataset drawn from a fixed convex domain $K$ in $\mathbb{R}^d$. In the setting of concentrated differential privacy, we show that, for every input such an unbiased mean estimator introduces approximately at least as much error as a mechanism that adds Gaussian noise with a carefully chosen covariance. This is true when the error is measured with respect to $\ell_p$ error for any $p \ge 2$. We extend this result to local differential privacy, and to approximate differential privacy, but for the latter the error lower bound holds either for a dataset or for a neighboring dataset. We also extend our results to mechanisms that take i.i.d.~samples from a distribution over $K$ and are unbiased with respect to the mean of the distribution.

翻译：我们研究了差分隐私中的无偏高维均值估计问题。考虑差分隐私机制，其期望输出等于输入数据集的均值，其中每个数据集取自$\mathbb{R}^d$中固定的凸域$K$。在集中差分隐私设置下，我们证明：对于每个输入，这类无偏均值估计器引入的误差几乎至少与添加精心选择协方差的高斯噪声的机制相当。当误差以任意$p \ge 2$的$\ell_p$范数衡量时，这一结论成立。我们将此结果推广到局部差分隐私和近似差分隐私，但后者中误差下界要么针对某个数据集，要么针对其相邻数据集。我们还进一步将结果扩展到从$K$上的分布中独立同分布采样、且关于分布均值无偏的机制。