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.

翻译：我们研究了在$(\varepsilon, \delta)$-差分隐私约束下发布稀疏直方图的问题。稳定性直方图机制独立地对非零条目添加拉普拉斯或高斯分布的噪声，并移除低于阈值的噪声计数值。因此，相邻直方图之间新非零值的引入最多以$\delta$的概率被揭示，且通常阈值的选择主导了机制的误差。我们考虑采用高斯噪声的稳定性直方图变体。近期研究（[Joseph and Yu, COLT '24] 与 [Lebeda, SOSA '25]）通过使用相关高斯噪声降低了私有直方图的误差。然而，这些技术无法直接应用于高度稀疏的场景。我们采用Lebeda的技术，证明仅对非零计数添加相关噪声可以在已知稀疏性上界时降低噪声幅度。这进而允许我们使用比非相关噪声机制低至$1/2$因子的阈值。随后，我们将机制扩展至稀疏性上界未知的场景。此外，我们证明相关噪声可为更实用的离散高斯机制带来类似的改进。