New Lower Bounds for Private Estimation and a Generalized Fingerprinting Lemma

We prove new lower bounds for statistical estimation tasks under the constraint of $(\varepsilon, δ)$-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 $Ω(d^2)$ samples, and in spectral norm requires $Ω(d^{3/2})$ 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 $Ω(d/(α^2 \varepsilon))$ lower bound for estimating the mean of a distribution with bounded covariance to $α$-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.