Differentially Private Sampling from Distributions

TL;DR

This paper explores the fundamental limits of differentially private sampling from various distribution families, establishing bounds on dataset size and revealing when private sampling is easier or as hard as private learning.

Contribution

It provides tight bounds for private sampling from multiple distribution classes and compares its complexity to private learning, highlighting regimes where sampling is easier or equally difficult.

Findings

Private sampling can require fewer observations than private learning in some cases.

For certain distribution classes, private sampling and private learning have comparable data requirements.

The study characterizes the dataset size needed for private sampling across different distribution families.

Abstract

We initiate an investigation of private sampling from distributions. Given a dataset with $n$ independent observations from an unknown distribution $P$, a sampling algorithm must output a single observation from a distribution that is close in total variation distance to $P$ while satisfying differential privacy. Sampling abstracts the goal of generating small amounts of realistic-looking data. We provide tight upper and lower bounds for the dataset size needed for this task for three natural families of distributions: arbitrary distributions on $\{1,\ldots ,k\}$, arbitrary product distributions on $\{0,1\}^d$, and product distributions on $\{0,1\}^d$ with bias in each coordinate bounded away from 0 and 1. We demonstrate that, in some parameter regimes, private sampling requires asymptotically fewer observations than learning a description of $P$ nonprivately; in other regimes, however, private sampling proves to be as difficult as private learning. Notably, for some classes of distributions, the overhead in the number of observations needed for private learning compared to non-private learning is completely captured by the number of observations needed for private sampling.