We provide improved lower bounds for two well-known high-dimensional private estimation tasks. First, we prove that for estimating the covariance of a Gaussian up to spectral error $\alpha$ with approximate differential privacy, one needs $\tilde{\Omega}\left(\frac{d^{3/2}}{\alpha \varepsilon} + \frac{d}{\alpha^2}\right)$ samples for any $\alpha \le O(1)$, which is tight up to logarithmic factors. This improves over previous work which established this for $\alpha \le O\left(\frac{1}{\sqrt{d}}\right)$, and is also simpler than previous work. Next, we prove that for estimating the mean of a heavy-tailed distribution with bounded $k$th moments with approximate differential privacy, one needs $\tilde{\Omega}\left(\frac{d}{\alpha^{k/(k-1)} \varepsilon} + \frac{d}{\alpha^2}\right)$ samples. This matches known upper bounds and improves over the best known lower bound for this problem, which only hold for pure differential privacy, or when $k = 2$. Our techniques follow the method of fingerprinting and are generally quite simple. Our lower bound for heavy-tailed estimation is based on a black-box reduction from privately estimating identity-covariance Gaussians. Our lower bound for covariance estimation utilizes a Bayesian approach to show that, under an Inverse Wishart prior distribution for the covariance matrix, no private estimator can be accurate even in expectation, without sufficiently many samples.

翻译：我们为两个著名的高维私有估计任务提供了改进的下界。首先，我们证明：在近似差分隐私下，对于高斯分布协方差矩阵的谱误差估计，当$\alpha \le O(1)$时，需要$\tilde{\Omega}\left(\frac{d^{3/2}}{\alpha \varepsilon} + \frac{d}{\alpha^2}\right)$个样本，该结果在忽略对数因子时是紧的。这改进了先前仅限于$\alpha \le O\left(\frac{1}{\sqrt{d}}\right)\)的结果，且比先前工作更简单。其次，我们证明：在近似差分隐私下，对于具有有界$k$阶矩的厚尾分布均值估计，需要$\tilde{\Omega}\left(\frac{d}{\alpha^{k/(k-1)} \varepsilon} + \frac{d}{\alpha^2}\right)$个样本。该结果匹配已知上界，并改进了该问题的最佳已知下界——此前下界仅适用于纯差分隐私或$k = 2$的情形。我们的技术遵循指纹编码方法，整体上相当简洁。厚尾估计的下界基于一个黑盒归约：从私有估计单位协方差高斯分布问题归约而来。协方差估计的下界采用贝叶斯方法，证明在协方差矩阵服从逆威沙特先验分布时，若无足够样本，任何私有估计器即使期望意义下也无法达到准确度。