A guarantee for the whole batch
We study this question without imposing bounds on the mean or positive definite covariance. The guarantee concerns the whole batch: its joint distribution must be close to that of independent Gaussian draws, even though the released samples may be dependent.
Under approximate differential privacy, we determine the required input size up to logarithmic factors for any fixed error level between zero and one. Our algorithm and matching lower bound show when generating a batch from shared data is more efficient than running separate private samplers.
Optimal input size, up to logarithmic factors
n input records, dimension d, and batch size m. For fixed 0 < α < 1, 0 ≤ ε ≤ 1, and 0 < δ < 1. Here α is the total-variation error tolerance. The hidden constants may depend on α.
Accuracy is measured for the whole batch against m independent Gaussian draws.
Related paper
Minimax-Optimal Private Multi-Sampling from Gaussian Distributions
BibTeX
@unpublished{raoof_private_sampling,
title = {{Minimax-Optimal Private Multi-Sampling from Gaussian Distributions}},
author = {Zare Moayedi, Raoof and Yassaee, Mohammad Hossein and Ebrahimi, Javad B.},
year = {2026},
note = {Preprint}
}