Laplace Transform Interpretation of Differential Privacy

TL;DR

This paper introduces a novel Laplace transform perspective on Differential Privacy, enabling new analytical tools, tighter composition theorems, and resolving symmetry issues in subsampling for various DP notions.

Contribution

It establishes a Laplace transform framework for DP, linking R\'enyi and $(\epsilon, \delta)$-DP curves, and proves a tight adaptive composition theorem.

Findings

Laplace transform interpretation of DP notions

Connection between R\'enyi and $(\epsilon, \delta)$-DP curves via Laplace transforms

Proved a tight adaptive composition theorem for $(\epsilon, \delta)$-DP

Abstract

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.