We introduce a set of useful expressions of Differential Privacy (DP) notions in terms of Laplace transformations. The underlying bare-form expressions for these transforms appear in several works on analyzing DP, either as an integral or an expectation. We show that recognizing these expressions as Laplace transformations 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, ρ(q))-Rényi DP curve and the (ε, δ(ε))-DP curve as being the Laplace and inverse-Laplace transforms of one another. Using our Laplace transform-based analysis, we also prove an adaptive composition theorem for (ε, δ)-DP guarantees that is exactly-tight (i.e., matches even in constants) for all values of ε. Additionally, we resolve an issue regarding symmetry of f-DP on subsampling that prevented equivalence across all functional DP notions.
@InProceedings{chourasia_et_al:LIPIcs.FORC.2025.11, author = {Chourasia, Rishav and Javaid, Uzair and Sikdar, Biplab}, title = {{Laplace Transform Interpretation of Differential Privacy}}, booktitle = {6th Symposium on Foundations of Responsible Computing (FORC 2025)}, pages = {11:1--11:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-367-6}, ISSN = {1868-8969}, year = {2025}, volume = {329}, editor = {Bun, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2025.11}, URN = {urn:nbn:de:0030-drops-231387}, doi = {10.4230/LIPIcs.FORC.2025.11}, annote = {Keywords: Differential Privacy, Composition Theorem, Laplace Transform} }
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