A Private and Computationally-Efficient Estimator for Unbounded Gaussians

TL;DR

This paper introduces the first efficient, differentially private estimator for the mean and covariance of any Gaussian distribution in high dimensions, overcoming previous limitations of nonconstructiveness and parameter bounds.

Contribution

It presents a polynomial-time, polynomial-sample differentially private estimator for Gaussian parameters without requiring prior bounds, using a novel private preconditioning technique.

Findings

Estimator is polynomial-time and polynomial-sample

Works for arbitrary Gaussian distributions in high dimensions

Introduces a new private preconditioning method

Abstract

We give the first polynomial-time, polynomial-sample, differentially private estimator for the mean and covariance of an arbitrary Gaussian distribution $\mathcal{N}(\mu,\Sigma)$ in $\mathbb{R}^d$. All previous estimators are either nonconstructive, with unbounded running time, or require the user to specify a priori bounds on the parameters $\mu$ and $\Sigma$. The primary new technical tool in our algorithm is a new differentially private preconditioner that takes samples from an arbitrary Gaussian $\mathcal{N}(0,\Sigma)$ and returns a matrix $A$ such that $A \Sigma A^T$ has constant condition number.