New Lower Bounds for Private Estimation and a Generalized Fingerprinting Lemma

Part of Advances in Neural Information Processing Systems 35 (NeurIPS 2022) Main Conference Track

Bibtex Paper Supplemental

Authors

Gautam Kamath, Argyris Mouzakis, Vikrant Singhal

Abstract

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.