The Correlated Gaussian Sparse Histogram Mechanism

TL;DR

This paper introduces a differentially private sparse histogram mechanism using correlated Gaussian noise, reducing noise magnitude and threshold requirements, especially in very sparse data settings, with extensions to unknown sparsity and discrete Gaussian noise.

Contribution

It adapts correlated Gaussian noise techniques to sparse histograms, enabling lower noise and thresholds, and extends the approach to unknown sparsity and discrete Gaussian mechanisms.

Findings

Correlated Gaussian noise reduces noise magnitude in sparse histograms.

Lower thresholds are achievable, up to half of previous bounds.

The method extends to unknown sparsity and discrete Gaussian noise.

Abstract

We consider the problem of releasing a sparse histogram under $(\varepsilon, \delta)$-differential privacy. The stability histogram independently adds noise from a Laplace or Gaussian distribution to the non-zero entries and removes those noisy counts below a threshold. Thereby, the introduction of new non-zero values between neighboring histograms is only revealed with probability at most $\delta$, and typically, the value of the threshold dominates the error of the mechanism. We consider the variant of the stability histogram with Gaussian noise. Recent works ([Joseph and Yu, COLT '24] and [Lebeda, SOSA '25]) reduced the error for private histograms using correlated Gaussian noise. However, these techniques can not be directly applied in the very sparse setting. Instead, we adopt Lebeda's technique and show that adding correlated noise to the non-zero counts only allows us to reduce the magnitude of noise when we have a sparsity bound. This, in turn, allows us to use a lower threshold by up to a factor of $1/2$ compared to the non-correlated noise mechanism. We then extend our mechanism to a setting without a known bound on sparsity. Additionally, we show that correlated noise can give a similar improvement for the more practical discrete Gaussian mechanism.