We prove new lower bounds for statistical estimation tasks under the constraint of $(\varepsilon, \delta)$-differential privacy. First, we provide tight lower bounds for private covariance estimation of Gaussian distributions. We show that estimating the covariance matrix in Frobenius norm requires $\Omega(d^2)$ samples, and in spectral norm requires $\Omega(d^{3/2})$ samples, both matching upper bounds up to logarithmic factors. The latter bound verifies the existence of a conjectured statistical gap between the private and the non-private sample complexities for spectral estimation of Gaussian covariances. We prove these bounds via our main technical contribution, a broad generalization of the fingerprinting method to exponential families. Additionally, using the private Assouad method of Acharya, Sun, and Zhang, we show a tight $\Omega(d/(\alpha^2 \varepsilon))$ lower bound for estimating the mean of a distribution with bounded covariance to $\alpha$-error in $\ell_2$-distance. Prior known lower bounds for all these problems were either polynomially weaker or held under the stricter condition of $(\varepsilon,0)$-differential privacy.

翻译：我们证明了在$(\varepsilon, \delta)$-差分隐私约束下统计估计任务的新下界。首先，我们为高斯分布的隐私协方差估计提供了严格的下界。研究表明，在Frobenius范数下估计协方差矩阵需要$\Omega(d^2)$个样本，而在谱范数下则需要$\Omega(d^{3/2})$个样本，两者均与上界匹配至对数因子。后者验证了高斯协方差谱估计中隐私与非隐私样本复杂度之间存在的猜想性统计差距。这些下界通过我们的主要技术贡献——指纹方法在指数族上的广泛推广得以证明。此外，利用Acharya、Sun和Zhang的私有Assouad方法，我们展示了在$\ell_2$距离中估计有界协方差分布均值达到$\alpha$-误差时，存在一个严格的$\Omega(d/(\alpha^2 \varepsilon))$下界。此前所有这些问题已知的下界要么呈多项式弱化，要么仅在$(\varepsilon,0)$-差分隐私的更严格条件下成立。