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Solving Linear Programs with Differential Privacy

Authors Alina Ene , Huy Le Nguyen , Ta Duy Nguyen, Adrian Vladu

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Security and privacy
Keywords
  • Differential Privacy
  • Linear Programming

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Abstract

We study the problem of solving linear programs of the form Ax ≤ b, x ≥ 0 with differential privacy. For homogeneous LPs Ax ≥ 0, we give an efficient (ε,δ)-differentially private algorithm which with probability at least 1-β finds in polynomial time a solution that satisfies all but O(d²/ε log²(d/(δβ))√{log 1/ρ₀}) constraints, for problems with margin ρ₀ > 0. This improves the bound of O(d⁵/ε log^{1.5} 1/ρ₀ polylog(d,1/δ,1/β)) by [Kaplan-Mansour-Moran-Stemmer-Tur, STOC '25]. For general LPs Ax ≤ b, x ≥ 0 with potentially zero margin, we give an efficient (ε,δ)-differentially private algorithm that w.h.p drops O(d⁴/ε log^{2.5} d/δ √{log dU}) constraints, where U is an upper bound for the entries of A and b in absolute value. This improves the result by Kaplan et al. by at least a factor of d⁵. Our techniques build upon privatizing a rescaling perceptron algorithm by [Hoberg-Rothvoss, IPCO '17] and a more refined iterative procedure for identifying equality constraints by Kaplan et al.

Cite As Get BibTex

Alina Ene, Huy Le Nguyen, Ta Duy Nguyen, and Adrian Vladu. Solving Linear Programs with Differential Privacy. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 65:1-65:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.65

Author Details

Alina Ene
  • Department of Computer Science, Boston University, MA, USA
Huy Le Nguyen
  • Khoury College of Computer Sciences, Northeastern University, Boston, MA, USA
Ta Duy Nguyen
  • Department of Computer Science, Boston University, MA, USA
Adrian Vladu
  • CNRS & IRIF, Université Paris Cité, France

Funding

  • Ene, Alina: AE was supported in part by an Alfred P. Sloan Research Fellowship.
  • Le Nguyen, Huy: HN was supported in part by NSF CCF 2311649 and a gift from Apple Inc.
  • Vladu, Adrian: AV was supported by the French Agence Nationale de la Recherche (ANR) under grant ANR-21-CE48-0016 (project COMCOPT).

References

  1. Amos Beimel, Shay Moran, Kobbi Nissim, and Uri Stemmer. Private center points and learning of halfspaces. In COLT, volume 99 of Proceedings of Machine Learning Research, pages 269-282. PMLR, 2019. URL: http://proceedings.mlr.press/v99/beimel19a.html.
  2. Omri Ben-Eliezer, Dan Mikulincer, and Ilias Zadik. Archimedes meets privacy: On privately estimating quantiles in high dimensions under minimal assumptions. In NeurIPS, 2022. Google Scholar
  3. Mark Bun, Kobbi Nissim, Uri Stemmer, and Salil P. Vadhan. Differentially private release and learning of threshold functions. In FOCS, pages 634-649. IEEE Computer Society, 2015. URL: https://doi.org/10.1109/FOCS.2015.45.
  4. John Dunagan and Santosh S. Vempala. A simple polynomial-time rescaling algorithm for solving linear programs. Math. Program., 114(1):101-114, 2008. URL: https://doi.org/10.1007/S10107-007-0095-7.
  5. Cynthia Dwork, Krishnaram Kenthapadi, Frank McSherry, Ilya Mironov, and Moni Naor. Our data, ourselves: Privacy via distributed noise generation. In EUROCRYPT, volume 4004 of Lecture Notes in Computer Science, pages 486-503. Springer, 2006. URL: https://doi.org/10.1007/11761679_29.
  6. 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. URL: https://doi.org/10.29012/JPC.V7I3.405.
  7. Yue Gao and Or Sheffet. Differentially private approximations of a convex hull in low dimensions. In ITC, volume 199 of LIPIcs, pages 18:1-18:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ITC.2021.18.
  8. Rebecca Hoberg and Thomas Rothvoss. An improved deterministic rescaling for linear programming algorithms. In IPCO, volume 10328 of Lecture Notes in Computer Science, pages 267-278. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-59250-3_22.
  9. Justin Hsu, Aaron Roth, Tim Roughgarden, and Jonathan R. Ullman. Privately solving linear programs. In ICALP (1), volume 8572 of Lecture Notes in Computer Science, pages 612-624. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-43948-7_51.
  10. Haim Kaplan, Yishay Mansour, Shay Moran, Uri Stemmer, and Nitzan Tur. On differentially private linear algebra. CoRR, abs/2411.03087, 2024. URL: https://doi.org/10.48550/arXiv.2411.03087.
  11. Haim Kaplan, Yishay Mansour, Uri Stemmer, and Eliad Tsfadia. Private learning of halfspaces: Simplifying the construction and reducing the sample complexity. In NeurIPS, 2020. Google Scholar
  12. Andrés Muñoz Medina, Umar Syed, Sergei Vassilvitskii, and Ellen Vitercik. Private optimization without constraint violations. In AISTATS, volume 130 of Proceedings of Machine Learning Research, pages 2557-2565. PMLR, 2021. URL: http://proceedings.mlr.press/v130/munoz21a.html.
  13. Kobbi Nissim, Uri Stemmer, and Salil P. Vadhan. Locating a small cluster privately. In PODS, pages 413-427. ACM, 2016. URL: https://doi.org/10.1145/2902251.2902296.
  14. Javier Peña and Negar Soheili. A deterministic rescaled perceptron algorithm. Math. Program., 155(1-2):497-510, 2016. URL: https://doi.org/10.1007/S10107-015-0860-Y.
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