On Differentially Private Sampling from Gaussian and Product Distributions

TL;DR

This paper develops new differentially private algorithms for sampling from Gaussian and product distributions, achieving near-optimal sample complexity and extending the capabilities of private sampling methods.

Contribution

The paper introduces novel DP sampling algorithms for Gaussian distributions with various covariance assumptions and provides the first pure-DP algorithm for product distributions.

Findings

Achieves near-optimal sample complexity for Gaussian sampling.

Introduces a pure-DP algorithm for product distributions.

Extends private sampling techniques to broader distribution classes.

Abstract

Given a dataset of $n$ i.i.d. samples from an unknown distribution $P$, we consider the problem of generating a sample from a distribution that is close to $P$ in total variation distance, under the constraint of differential privacy (DP). We study the problem when $P$ is a multi-dimensional Gaussian distribution, under different assumptions on the information available to the DP mechanism: known covariance, unknown bounded covariance, and unknown unbounded covariance. We present new DP sampling algorithms, and show that they achieve near-optimal sample complexity in the first two settings. Moreover, when $P$ is a product distribution on the binary hypercube, we obtain a pure-DP algorithm whereas only an approximate-DP algorithm (with slightly worse sample complexity) was previously known.