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，我们为高斯分布的私有协方差估计提供了紧下界。研究表明，在Frobenius范数下估计协方差矩阵需要$\Omega(d^2)$个样本，在谱范数下需要$\Omega(d^{3/2})$个样本，两者均与上界匹配（仅相差对数因子）。后一个结果验证了高斯协方差谱估计中私有与非私有样本复杂度之间存在一个猜想中的统计差距。我们通过主要技术贡献（指纹方法对指数族的广泛泛化）证明了这些下界。此外，利用Acharya、Sun和Zhang提出的私有Assouad方法，我们证明了在$\ell_2$距离下，对于具有有界协方差的分布均值估计达到$\alpha$误差时，存在紧下界$\Omega(d/(\alpha^2 \varepsilon))$。此前所有这些问题已知的下界要么是多项式弱化的，要么仅在更严格的$(\varepsilon, 0)$-差分隐私条件下成立。