Document Open Access Logo

Laplace Transform Interpretation of Differential Privacy

Authors Rishav Chourasia , Uzair Javaid , Biplab Sikdar

PDF

File

Thumbnail PDF
PDF
LIPIcs.FORC.2025.11.pdf
  • Filesize: 1.29 MB
  • 23 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Security and privacy
Keywords
  • Differential Privacy
  • Composition Theorem
  • Laplace Transform

Metrics

Abstract

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.

Cite As Get BibTex

Rishav Chourasia, Uzair Javaid, and Biplab Sikdar. Laplace Transform Interpretation of Differential Privacy. In 6th Symposium on Foundations of Responsible Computing (FORC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 329, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.FORC.2025.11

Author Details

Rishav Chourasia
  • School of Computing, National University of Singapore, 13 Computing Drive, Singapore
Uzair Javaid
  • Betterdata, PIXEL at One North, 10 Central Exchange Green, Singapore
Biplab Sikdar
  • Department of ECE, 4 Engineering Drive 3, National University of Singapore, Singapore

Funding

This work was supported in part by the Asian Institute of Digital Finance (AIDF) under grant A-0003504-09-00.

Acknowledgements

We thank Bao Ergute for feedback on earlier versions of this paper.

References

  1. Martin Abadi, Andy Chu, Ian Goodfellow, H Brendan McMahan, Ilya Mironov, Kunal Talwar, and Li Zhang. Deep learning with differential privacy. In Proceedings of the 2016 ACM SIGSAC conference on computer and communications security, pages 308-318, 2016. URL: https://doi.org/10.1145/2976749.2978318.
  2. Shahab Asoodeh, Jiachun Liao, Flavio P Calmon, Oliver Kosut, and Lalitha Sankar. A better bound gives a hundred rounds: Enhanced privacy guarantees via f-divergences. In 2020 IEEE International Symposium on Information Theory (ISIT), pages 920-925. IEEE, 2020. URL: https://doi.org/10.1109/ISIT44484.2020.9174015.
  3. Borja Balle, Gilles Barthe, and Marco Gaboardi. Privacy amplification by subsampling: Tight analyses via couplings and divergences. Advances in neural information processing systems, 31, 2018. Google Scholar
  4. Borja Balle, Gilles Barthe, and Marco Gaboardi. Privacy profiles and amplification by subsampling. Journal of Privacy and Confidentiality, 10(1), 2020. URL: https://doi.org/10.29012/JPC.726.
  5. Borja Balle, Gilles Barthe, Marco Gaboardi, Justin Hsu, and Tetsuya Sato. Hypothesis testing interpretations and renyi differential privacy. In International Conference on Artificial Intelligence and Statistics, pages 2496-2506. PMLR, 2020. URL: http://proceedings.mlr.press/v108/balle20a.html.
  6. Borja Balle and Yu-Xiang Wang. Improving the gaussian mechanism for differential privacy: Analytical calibration and optimal denoising. In International Conference on Machine Learning, pages 394-403. PMLR, 2018. Google Scholar
  7. Mark Bun and Thomas Steinke. Concentrated differential privacy: Simplifications, extensions, and lower bounds. In Theory of Cryptography Conference, pages 635-658. Springer, 2016. URL: https://doi.org/10.1007/978-3-662-53641-4_24.
  8. Clément L Canonne, Gautam Kamath, and Thomas Steinke. The discrete gaussian for differential privacy. Advances in Neural Information Processing Systems, 33:15676-15688, 2020. Google Scholar
  9. Alan M Cohen. Numerical methods for Laplace transform inversion, volume 5. Springer Science & Business Media, 2007. Google Scholar
  10. Jinshuo Dong, Aaron Roth, and Weijie J Su. Gaussian differential privacy. arXiv preprint arXiv:1905.02383, 2019. URL: https://arxiv.org/abs/1905.02383.
  11. Vadym Doroshenko, Badih Ghazi, Pritish Kamath, Ravi Kumar, and Pasin Manurangsi. Connect the dots: Tighter discrete approximations of privacy loss distributions. arXiv preprint arXiv:2207.04380, 2022. Google Scholar
  12. Cynthia Dwork. Differential privacy. In International colloquium on automata, languages, and programming, pages 1-12. Springer, 2006. URL: https://doi.org/10.1007/11787006_1.
  13. Cynthia Dwork, Aaron Roth, et al. The algorithmic foundations of differential privacy. Foundations and Trends® in Theoretical Computer Science, 9(3-4):211-407, 2014. URL: https://doi.org/10.1561/0400000042.
  14. Cynthia Dwork and Guy N Rothblum. Concentrated differential privacy. arXiv preprint arXiv:1603.01887, 2016. URL: https://arxiv.org/abs/1603.01887.
  15. Cynthia Dwork, Guy N Rothblum, and Salil Vadhan. Boosting and differential privacy. In 2010 IEEE 51st annual symposium on foundations of computer science, pages 51-60. IEEE, 2010. URL: https://doi.org/10.1109/FOCS.2010.12.
  16. Sivakanth Gopi, Yin Tat Lee, and Lukas Wutschitz. Numerical composition of differential privacy. Advances in Neural Information Processing Systems, 34:11631-11642, 2021. URL: https://proceedings.neurips.cc/paper/2021/hash/6097d8f3714205740f30debe1166744e-Abstract.html.
  17. Peter Kairouz, Sewoong Oh, and Pramod Viswanath. The composition theorem for differential privacy. In International conference on machine learning, pages 1376-1385. PMLR, 2015. URL: http://proceedings.mlr.press/v37/kairouz15.html.
  18. Antti Koskela, Joonas Jälkö, and Antti Honkela. Computing tight differential privacy guarantees using fft. In International Conference on Artificial Intelligence and Statistics, pages 2560-2569. PMLR, 2020. URL: http://proceedings.mlr.press/v108/koskela20b.html.
  19. Ninghui Li, Wahbeh Qardaji, and Dong Su. On sampling, anonymization, and differential privacy or, k-anonymization meets differential privacy. In Proceedings of the 7th ACM Symposium on Information, Computer and Communications Security, pages 32-33, 2012. URL: https://doi.org/10.1145/2414456.2414474.
  20. Ilya Mironov. Rényi differential privacy. In 2017 IEEE 30th computer security foundations symposium (CSF), pages 263-275. IEEE, 2017. URL: https://doi.org/10.1109/CSF.2017.11.
  21. Jack Murtagh and Salil Vadhan. The complexity of computing the optimal composition of differential privacy. In Theory of Cryptography Conference, pages 157-175. Springer, 2015. Google Scholar
  22. Alan V Oppenhiem, Alan S Willsky, and S Hamid Nawab. Signals and systems, 1996. Google Scholar
  23. David Sommer, Sebastian Meiser, and Esfandiar Mohammadi. Privacy loss classes: The central limit theorem in differential privacy. Cryptology ePrint Archive, 2018. Google Scholar
  24. Thomas Steinke. Composition of differential privacy & privacy amplification by subsampling. arXiv preprint arXiv:2210.00597, 2022. URL: https://doi.org/10.48550/arXiv.2210.00597.
  25. Tim Van Erven and Peter Harremos. Rényi divergence and kullback-leibler divergence. IEEE Transactions on Information Theory, 60(7):3797-3820, 2014. URL: https://doi.org/10.1109/TIT.2014.2320500.
  26. Yuqing Zhu, Jinshuo Dong, and Yu-Xiang Wang. Optimal accounting of differential privacy via characteristic function. In International Conference on Artificial Intelligence and Statistics, pages 4782-4817. PMLR, 2022. URL: https://proceedings.mlr.press/v151/zhu22c.html.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact