Document Open Access Logo

How to Make Your Approximation Algorithm Private: A Black-Box Differentially-Private Transformation for Tunable Approximation Algorithms of Functions with Low Sensitivity

Authors Jeremiah Blocki, Elena Grigorescu, Tamalika Mukherjee, Samson Zhou

PDF

File

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2023.59.pdf
  • Filesize: 0.78 MB
  • 24 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Security and privacy → Usability in security and privacy
Keywords
  • Differential privacy
  • approximation algorithms

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We develop a framework for efficiently transforming certain approximation algorithms into differentially-private variants, in a black-box manner. Specifically, our results focus on algorithms A that output an approximation to a function f of the form (1-α)f(x)-κ ≤ A(x) ≤ (1+α)f(x)+κ, where κ ∈ ℝ_{≥ 0} denotes additive error and α ∈ [0,1) denotes multiplicative error can be"tuned" to small-enough values while incurring only a polynomial blowup in the running time/space. We show that such algorithms can be made differentially private without sacrificing accuracy, as long as the function f has small "global sensitivity". We achieve these results by applying the "smooth sensitivity" framework developed by Nissim, Raskhodnikova, and Smith (STOC 2007). 
Our framework naturally applies to transform non-private FPRAS and FPTAS algorithms into ε-differentially private approximation algorithms where the former case requires an additional postprocessing step. We apply our framework in the context of sublinear-time and sublinear-space algorithms, while preserving the nature of the algorithm in meaningful ranges of the parameters. Our results include the first (to the best of our knowledge) ε-edge differentially-private sublinear-time algorithm for estimating the number of triangles, the number of connected components, and the weight of a minimum spanning tree of a graph whose accuracy holds with high probability. In the area of streaming algorithms, our results include ε-DP algorithms for estimating L_p-norms, distinct elements, and weighted minimum spanning tree for both insertion-only and turnstile streams. Our transformation also provides a private version of the smooth histogram framework, which is commonly used for converting streaming algorithms into sliding window variants, and achieves a multiplicative approximation to many problems, such as estimating L_p-norms, distinct elements, and the length of the longest increasing subsequence.

Cite As Get BibTex

Jeremiah Blocki, Elena Grigorescu, Tamalika Mukherjee, and Samson Zhou. How to Make Your Approximation Algorithm Private: A Black-Box Differentially-Private Transformation for Tunable Approximation Algorithms of Functions with Low Sensitivity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 59:1-59:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.59

Author Details

Jeremiah Blocki
  • Purdue University, West Lafayette, IN, USA
Elena Grigorescu
  • Purdue University, West Lafayette, IN, USA
Tamalika Mukherjee
  • Purdue University, West Lafayette, IN, USA
Samson Zhou
  • University of California Berkeley, CA, USA
  • Rice University, Houston, TX, USA

Funding

  • Blocki, Jeremiah: Supported in part by NSF CCF-1910659, NSF CNS-1931443. and NSF CAREER award CNS-2047272.
  • Grigorescu, Elena: Supported in part by NSF CCF-1910659, NSF CCF-1910411, and NSF CCF-2228814.
  • Mukherjee, Tamalika: Supported in part by NSF CCF-1910659 and and NSF CCF-2228814.
  • Zhou, Samson: Work done in part while at Carnegie Mellon University and supported in part by a Simons Investigator Award and by NSF CCF-1815840.

References

  1. Jayadev Acharya, Ziteng Sun, and Huanyu Zhang. Differentially private testing of identity and closeness of discrete distributions. In Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, Nicolò Cesa-Bianchi, and Roman Garnett, editors, Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, Canada, pages 6879-6891, 2018. URL: https://proceedings.neurips.cc/paper/2018/hash/7de32147a4f1055bed9e4faf3485a84d-Abstract.html.
  2. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Analyzing graph structure via linear measurements. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 459-467, 2012. Google Scholar
  3. Noga Alon, Wenceslas Fernandez de la Vega, Ravi Kannan, and Marek Karpinski. Random sampling and approximation of max-csps. J. Comput. Syst. Sci., 67(2):212-243, 2003. URL: https://doi.org/10.1016/S0022-0000(03)00008-4.
  4. Noga Alon, Eldar Fischer, Ilan Newman, and Asaf Shapira. A combinatorial characterization of the testable graph properties: It’s all about regularity. SIAM J. Comput., 39(1):143-167, 2009. Google Scholar
  5. Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. J. Comput. Syst. Sci., 58(1):137-147, 1999. Google Scholar
  6. Gunnar Andersson and Lars Engebretsen. Property testers for dense constraint satisfaction programs on finite domains. Random Struct. Algorithms, 21(1):14-32, 2002. Google Scholar
  7. Petra Berenbrink, Bruce Krayenhoff, and Frederik Mallmann-Trenn. Estimating the number of connected components in sublinear time. Inf. Process. Lett., 114(11):639-642, 2014. URL: https://doi.org/10.1016/j.ipl.2014.05.008.
  8. Jaroslaw Blasiok. Optimal streaming and tracking distinct elements with high probability. ACM Trans. Algorithms, 16(1):3:1-3:28, 2020. Google Scholar
  9. Jeremiah Blocki, Avrim Blum, Anupam Datta, and Or Sheffet. The johnson-lindenstrauss transform itself preserves differential privacy. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 410-419, 2012. Google Scholar
  10. Jeremiah Blocki, Elena Grigorescu, and Tamalika Mukherjee. Privately estimating graph parameters in sublinear time. In 49th International Colloquium on Automata, Languages, and Programming, ICALP, pages 26:1-26:19, 2022. Google Scholar
  11. Jeremiah Blocki, Elena Grigorescu, Tamalika Mukherjee, and Samson Zhou. How to make your approximation algorithm private: A black-box differentially-private transformation for tunable approximation algorithms of functions with low sensitivity, 2022. URL: https://arxiv.org/abs/2210.03831.
  12. Vladimir Braverman, Petros Drineas, Cameron Musco, Christopher Musco, Jalaj Upadhyay, David P. Woodruff, and Samson Zhou. Near optimal linear algebra in the online and sliding window models. In 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 517-528, 2020. Google Scholar
  13. Vladimir Braverman, Joel Manning, Zhiwei Steven Wu, and Samson Zhou. Private data stream analysis for universal symmetric norm estimation. In 3rd Annual Symposium on Foundations of Responsible Computing FORC, 2022. Google Scholar
  14. Vladimir Braverman and Rafail Ostrovsky. Effective computations on sliding windows. SIAM J. Comput., 39(6):2113-2131, 2010. Google Scholar
  15. Vladimir Braverman, Rafail Ostrovsky, and Carlo Zaniolo. Optimal sampling from sliding windows. J. Comput. Syst. Sci., 78(1):260-272, 2012. Google Scholar
  16. Vladimir Braverman, Viska Wei, and Samson Zhou. Symmetric norm estimation and regression on sliding windows. In Computing and Combinatorics - 27th International Conference, COCOON, Proceedings, pages 528-539, 2021. Google Scholar
  17. Zhiqi Bu, Sivakanth Gopi, Janardhan Kulkarni, Yin Tat Lee, Judy Hanwen Shen, and Uthaipon Tantipongpipat. Fast and memory efficient differentially private-sgd via JL projections. CoRR, abs/2102.03013, 2021. URL: https://arxiv.org/abs/2102.03013.
  18. Mark Bun, Jonathan R. Ullman, and Salil P. Vadhan. Fingerprinting codes and the price of approximate differential privacy. SIAM J. Comput., 47(5):1888-1938, 2018. URL: https://doi.org/10.1137/15M1033587.
  19. Bernard Chazelle, Ronitt Rubinfeld, and Luca Trevisan. Approximating the minimum spanning tree weight in sublinear time. SIAM J. Comput., 34(6):1370-1379, 2005. URL: https://doi.org/10.1137/S0097539702403244.
  20. Cynthia Dwork. Differential privacy. In Automata, Languages and Programming, 33rd International Colloquium, ICALP, Proceedings, Part II, pages 1-12, 2006. Google Scholar
  21. Cynthia Dwork and Jing Lei. Differential privacy and robust statistics. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC, pages 371-380. ACM, 2009. Google Scholar
  22. Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam D. Smith. Calibrating noise to sensitivity in private data analysis. In Shai Halevi and Tal Rabin, editors, Theory of Cryptography, Third Theory of Cryptography Conference, TCC, Proceedings, pages 265-284, 2006. Google Scholar
  23. Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam D. Smith. Calibrating noise to sensitivity in private data analysis. J. Priv. Confidentiality, 7(3):17-51, 2016. Google Scholar
  24. Talya Eden, Amit Levi, Dana Ron, and C. Seshadhri. Approximately counting triangles in sublinear time. SIAM J. Comput., 46(5):1603-1646, 2017. URL: https://doi.org/10.1137/15M1054389.
  25. Alessandro Epasto, Mohammad Mahdian, Vahab S. Mirrokni, and Peilin Zhong. Improved sliding window algorithms for clustering and coverage via bucketing-based sketches. In Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 3005-3042, 2022. Google Scholar
  26. Alessandro Epasto, Jieming Mao, Andres Muñoz Medina, Vahab Mirrokni, Sergei Vassilvitskii, and Peilin Zhong. Differentially private continual releases of streaming frequency moment estimations. In 14th Innovations in Theoretical Computer Science Conference, ITCS 2023, January 10-13, 2023, MIT, Cambridge, Massachusetts, USA, volume 251 of LIPIcs, pages 48:1-48:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. Google Scholar
  27. Nimrod Fiat and Dana Ron. On efficient distance approximation for graph properties. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 1618-1637. SIAM, 2021. Google Scholar
  28. Eldar Fischer and Ilan Newman. Testing versus estimation of graph properties. SIAM J. Comput., 37(2):482-501, 2007. Google Scholar
  29. Sumit Ganguly and David P. Woodruff. High probability frequency moment sketches. In 45th International Colloquium on Automata, Languages, and Programming, ICALP, pages 58:1-58:15, 2018. Google Scholar
  30. Badih Ghazi, Ravi Kumar, Pasin Manurangsi, and Thao Nguyen. Robust and private learning of halfspaces. In Arindam Banerjee and Kenji Fukumizu, editors, The 24th International Conference on Artificial Intelligence and Statistics, AISTATS 2021, April 13-15, 2021, Virtual Event, volume 130 of Proceedings of Machine Learning Research, pages 1603-1611. PMLR, 2021. URL: http://proceedings.mlr.press/v130/ghazi21a.html.
  31. Arijit Ghosh, Gopinath Mishra, Rahul Raychaudhury, and Sayantan Sen. Tolerant bipartiteness testing in dense graphs. In 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 69:1-69:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  32. Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. J. ACM, 45(4):653-750, 1998. Google Scholar
  33. Oded Goldreich and Dana Ron. On estimating the average degree of a graph. Electron. Colloquium Comput. Complex., 2004. Google Scholar
  34. Maoguo Gong, Yu Xie, Ke Pan, Kaiyuan Feng, and Alex Kai Qin. A survey on differentially private machine learning [review article]. IEEE Comput. Intell. Mag., 15(2):49-64, 2020. URL: https://doi.org/10.1109/MCI.2020.2976185.
  35. Anupam Gupta, Katrina Ligett, Frank McSherry, Aaron Roth, and Kunal Talwar. Differentially private combinatorial optimization. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 1106-1125, 2010. Google Scholar
  36. Moritz Hardt and Kunal Talwar. On the geometry of differential privacy. In Leonard J. Schulman, editor, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 705-714. ACM, 2010. URL: https://doi.org/10.1145/1806689.1806786.
  37. Rajesh Jayaram, David P. Woodruff, and Samson Zhou. Truly perfect samplers for data streams and sliding windows. In PODS: International Conference on Management of Data, pages 29-40, 2022. Google Scholar
  38. Daniel M. Kane, Jelani Nelson, Ely Porat, and David P. Woodruff. Fast moment estimation in data streams in optimal space. In Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC, pages 745-754, 2011. Google Scholar
  39. Daniel M. Kane, Jelani Nelson, and David P. Woodruff. An optimal algorithm for the distinct elements problem. In Proceedings of the Twenty-Ninth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS, pages 41-52, 2010. Google Scholar
  40. Darakhshan J. Mir, S. Muthukrishnan, Aleksandar Nikolov, and Rebecca N. Wright. Pan-private algorithms via statistics on sketches. In Proceedings of the 30th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS, pages 37-48, 2011. Google Scholar
  41. Kobbi Nissim, Sofya Raskhodnikova, and Adam D. Smith. Smooth sensitivity and sampling in private data analysis. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pages 75-84, 2007. Google Scholar
  42. Michal Parnas, Dana Ron, and Ronitt Rubinfeld. Tolerant property testing and distance approximation. J. Comput. Syst. Sci., 72(6):1012-1042, 2006. Google Scholar
  43. Adam D. Smith, Shuang Song, and Abhradeep Thakurta. The flajolet-martin sketch itself preserves differential privacy: Private counting with minimal space. In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems, NeurIPS, 2020. Google Scholar
  44. Xiaoming Sun and David P. Woodruff. The communication and streaming complexity of computing the longest common and increasing subsequences. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 336-345, 2007. Google Scholar
  45. Jakub Tetek. Additive noise mechanisms for making randomized approximation algorithms differentially private. CoRR, abs/2211.03695, 2022. URL: https://doi.org/10.48550/arXiv.2211.03695.
  46. Lun Wang, Iosif Pinelis, and Dawn Song. Differentially private fractional frequency moments estimation with polylogarithmic space. In The Tenth International Conference on Learning Representations, ICLR, 2022. Google Scholar
  47. David P. Woodruff and Samson Zhou. Tight bounds for adversarially robust streams and sliding windows via difference estimators. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 1183-1196. IEEE, 2021. Google Scholar
  48. Yuichi Yoshida, Masaki Yamamoto, and Hiro Ito. Improved constant-time approximation algorithms for maximum matchings and other optimization problems. SIAM J. Comput., 41(4):1074-1093, 2012. URL: https://doi.org/10.1137/110828691.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

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