Lower Bounds for Private Estimation of Gaussian Covariance Matrices under All Reasonable Parameter Regimes

TL;DR

This paper establishes fundamental lower bounds on the sample complexity for privately estimating Gaussian covariance matrices, matching known upper bounds across various parameter regimes, using advanced mathematical identities.

Contribution

It introduces new lower bounds for private covariance estimation that align with existing upper bounds, utilizing the Stein-Haff identity for the first time in this context.

Findings

Lower bounds match existing upper bounds in broad parameter regimes

Uses Stein-Haff identity to derive bounds

Advances understanding of sample complexity in private covariance estimation

Abstract

We prove lower bounds on the number of samples needed to privately estimate the covariance matrix of a Gaussian distribution. Our bounds match existing upper bounds in the widest known setting of parameters. Our analysis relies on the Stein-Haff identity, an extension of the classical Stein's identity used in previous fingerprinting lemma arguments.