Document

How to Find a Point in the Convex Hull Privately

File

LIPIcs.SoCG.2020.52.pdf
• Filesize: 0.52 MB
• 15 pages

Acknowledgements

We thank Santosh Vempala for many helpful discussions.

Cite As

Haim Kaplan, Micha Sharir, and Uri Stemmer. How to Find a Point in the Convex Hull Privately. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SoCG.2020.52

Abstract

We study the question of how to compute a point in the convex hull of an input set S of n points in ℝ^d in a differentially private manner. This question, which is trivial without privacy requirements, turns out to be quite deep when imposing differential privacy. In particular, it is known that the input points must reside on a fixed finite subset G ⊆ ℝ^d, and furthermore, the size of S must grow with the size of G. Previous works [Amos Beimel et al., 2010; Amos Beimel et al., 2019; Amos Beimel et al., 2013; Mark Bun et al., 2018; Mark Bun et al., 2015; Haim Kaplan et al., 2019] focused on understanding how n needs to grow with |G|, and showed that n=O(d^2.5 ⋅ 8^(log^*|G|)) suffices (so n does not have to grow significantly with |G|). However, the available constructions exhibit running time at least |G|^d², where typically |G|=X^d for some (large) discretization parameter X, so the running time is in fact Ω(X^d³). In this paper we give a differentially private algorithm that runs in O(n^d) time, assuming that n=Ω(d⁴ log X). To get this result we study and exploit some structural properties of the Tukey levels (the regions D_{≥ k} consisting of points whose Tukey depth is at least k, for k=0,1,…). In particular, we derive lower bounds on their volumes for point sets S in general position, and develop a rather subtle mechanism for handling point sets S in degenerate position (where the deep Tukey regions have zero volume). A naive approach to the construction of the Tukey regions requires n^O(d²) time. To reduce the cost to O(n^d), we use an approximation scheme for estimating the volumes of the Tukey regions (within their affine spans in case of degeneracy), and for sampling a point from such a region, a scheme that is based on the volume estimation framework of Lovász and Vempala [László Lovász and Santosh S. Vempala, 2006] and of Cousins and Vempala [Ben Cousins and Santosh S. Vempala, 2018]. Making this framework differentially private raises a set of technical challenges that we address.

Subject Classification

ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
• Theory of computation → Randomness, geometry and discrete structures
• Security and privacy → Formal methods and theory of security
Keywords
• Differential privacy
• Tukey depth
• Convex hull

Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

References

1. Amos Beimel, Shiva Prasad Kasiviswanathan, and Kobbi Nissim. Bounds on the sample complexity for private learning and private data release. In TCC, volume 5978 of LNCS, pages 437-454. Springer, 2010.
2. Amos Beimel, Shay Moran, Kobbi Nissim, and Uri Stemmer. Private center points and learning of halfspaces. In Conference on Learning Theory (COLT), pages 269-282, 2019.
3. Amos Beimel, Kobbi Nissim, and Uri Stemmer. Private learning and sanitization: Pure vs. approximate differential privacy. In APPROX-RANDOM, volume 8096 of LNCS, pages 363-378. Springer, 2013.
4. Mark Bun, Cynthia Dwork, Guy N. Rothblum, and Thomas Steinke. Composable and versatile privacy via truncated cdp. In 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 74-86, 2018.
5. Mark Bun, Kobbi Nissim, Uri Stemmer, and Salil P. Vadhan. Differentially private release and learning of threshold functions. In IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), pages 634-649, 2015.
6. Bernard Chazelle. An optimal convex hull algorithm in any fixed dimension. Discrete Comput. Geom., 10:377-409, 1993.
7. Ben Cousins and Santosh S. Vempala. Gaussian cooling and O^*(n³) algorithms for volume and gaussian volume. SIAM J. Comput., 47(3):1237-1273, 2018.
8. Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. In TCC, volume 3876 of LNCS, pages 265-284. Springer, 2006.
9. Cynthia Dwork and Aaron Roth. The algorithmic foundations of differential privacy. Found. Trends Theor. Comput. Sci., 9(3-4), 2014.
10. Herbert Edelsbrunner, Raimund Seidel, and Micha Sharir. On the zone theorem for hyperplane arrangements. SIAM J. Comput., 22(2):418-429, 1993.
11. Haim Kaplan, Katrina Ligett, Yishay Mansour, Moni Naor, and Uri Stemmer. Privately learning thresholds: Closing the exponential gap. CoRR, 2019. URL: http://arxiv.org/abs/1911.10137.
12. Haim Kaplan, Micha Sharir, and Uri Stemmer. How to find a point in the convex hull privately, 2020. URL: http://arxiv.org/abs/2003.13192.
13. Xiaohui Liu, Karl Mosler, and Pavlo Mozharovskyi. Fast computation of Tukey trimmed regions and median in dimension p > 2. J. of Comput. and Graph. Stat., 28(3):682-697, 2019.
14. László Lovász and Santosh S. Vempala. Simulated annealing in convex bodies and an O^*(n⁴) volume algorithm. J. Comput. Syst. Sci., 72(2):392-417, 2006.
15. Jiří Matousek. Lectures on Discrete Geometry. Springer-Verlag, Berlin, Heidelberg, 2002.
16. Frank McSherry and Kunal Talwar. Mechanism design via differential privacy. In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 94-103, 2007.
17. Peter J. Rousseeuw and Ida Ruts. Constructing the bivariate Tukey median. Statistica Sinica, 8(3):827-839, 1998.
18. John W. Tukey. Mathematics and the picturing of data. In Proc. of the International Congress of Mathematicians, volume 2, page 523–531, 1975.
19. Salil Vadhan. The complexity of differential privacy. In Yehuda Lindell, editor, Tutorials on the Foundations of Cryptography: Dedicated to Oded Goldreich, pages 347-450. Springer, 2017.