The Correlated Gaussian Sparse Histogram Mechanism
This work addresses privacy-preserving data release for sparse datasets, offering incremental improvements over existing techniques in differential privacy.
The paper tackles the problem of releasing sparse histograms under differential privacy by introducing correlated Gaussian noise to non-zero entries, which reduces noise magnitude and allows a lower threshold by up to a factor of 1/2 compared to non-correlated methods, with extensions to unbounded sparsity and discrete Gaussian mechanisms.
We consider the problem of releasing a sparse histogram under $(\varepsilon, δ)$-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 $δ$, 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.