Optimal Differentially Private Sampling of Unbounded Gaussians
Valentio Iverson, Gautam Kamath, Argyris Mouzakis
TL;DR
This work addresses private sampling from high-dimensional Gaussians with unknown and potentially unbounded covariance matrices under $(\varepsilon,\delta)$-DP. It combines covariance-aware mean estimation primitives from Brown, Hopkins, and Smith with a privacy-preserving sampling framework inspired by Ghazi, Hu, Kumar, and Manurangsi, and augments it with a Propose-Test-Release step to certify data quality. The result is a near-optimal sampler with $\widetilde{O}(d)$ samples that achieves $d_{TV}$-closeness to the target Gaussian, significantly reducing previous dependence on the ambient dimension. This advances private generative-model applications by enabling efficient, private sampling without fully learning the covariance, and it tightens the gap between private sampling and private distribution learning in the unbounded-covariance regime.
Abstract
We provide the first $\widetilde{\mathcal{O}}\left(d\right)$-sample algorithm for sampling from unbounded Gaussian distributions under the constraint of $\left(\varepsilon, δ\right)$-differential privacy. This is a quadratic improvement over previous results for the same problem, settling an open question of Ghazi, Hu, Kumar, and Manurangsi.