New Lower Bounds for Private Estimation and a Generalized Fingerprinting Lemma

TL;DR

This paper establishes new tight lower bounds for private covariance and mean estimation in Gaussian distributions, revealing fundamental limits of differential privacy in high-dimensional statistical tasks.

Contribution

It introduces a generalized fingerprinting method for exponential families and tight bounds that confirm a statistical gap in private spectral covariance estimation.

Findings

Covariance estimation in Frobenius norm requires Ω(d^2) samples.

Spectral norm covariance estimation requires Ω(d^{3/2}) samples.

Lower bounds match known upper bounds up to logarithmic factors.

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. 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.