We introduce a set of useful expressions of Differential Privacy (DP) notions in terms of the Laplace transform of the privacy loss distribution. Its bare form expression appears in several related works on analyzing DP, either as an integral or an expectation. We show that recognizing the expression as a Laplace transform unlocks a new way to reason about DP properties by exploiting the duality between time and frequency domains. Leveraging our interpretation, we connect the $(q, \rho(q))$-R\'enyi DP curve and the $(\epsilon, \delta(\epsilon))$-DP curve as being the Laplace and inverse-Laplace transforms of one another. This connection shows that the R\'enyi divergence is well-defined for complex orders $q = \gamma + i \omega$. Using our Laplace transform-based analysis, we also prove an adaptive composition theorem for $(\epsilon, \delta)$-DP guarantees that is exactly tight (i.e., matches even in constants) for all values of $\epsilon$. Additionally, we resolve an issue regarding symmetry of $f$-DP on subsampling that prevented equivalence across all functional DP notions.

翻译：我们引入了一套基于隐私损失分布拉普拉斯变换的差分隐私（DP）概念的有用表达式。其原始形式表达式在多个关于分析DP的相关工作中出现，或以积分形式，或以期望形式呈现。我们证明，将该表达式识别为拉普拉斯变换，通过利用时域与频域之间的对偶性，为推理DP特性开辟了一条新途径。借助我们的解释，我们将$(q, \rho(q))$-Rényi DP曲线与$(\epsilon, \delta(\epsilon))$-DP曲线联系起来，表明它们互为彼此的拉普拉斯变换与逆拉普拉斯变换。这一联系揭示了Rényi散度对于复阶数$q = \gamma + i \omega$是良定义的。利用我们基于拉普拉斯变换的分析，我们还证明了$(\epsilon, \delta)$-DP保证的自适应组合定理，该定理对于所有$\epsilon$值都是精确紧致的（即即使在常数项上也完全匹配）。此外，我们解决了关于$f$-DP在子采样上的对称性问题，该问题曾阻碍所有函数型DP概念之间的等价性。