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. 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：以 $( varepsilon,\ delta) 不同隐私为限制的统计估计任务,我们证明新的较低界限。 首先,我们为Gaussian 分布的私人共变估算提供严格的较低界限。 我们显示,对Frobenius 规范中的共变矩阵进行估算需要$\ Omega( d ⁇ 2) 美元样本,而在光谱规范中,则需要$\ Omega( d ⁇ 3/2}) 美元样本,同时将上界与对数系数相匹配。 我们通过我们的主要技术贡献来证明这些界限,将指纹方法广泛概括到指数式家庭。 此外,使用 Acharya、 Sun 和 Zhang 的私人Asouad 方法,我们显示,在估算与 $( alpha2) 和 varepsilon 的捆绑定的分布平均值时,我们所知道的所有这些问题的较低界限要么是聚性较弱,要么是处于 $0 ( varlll) 的严格条件之下。