research-article

An invariance principle for polytopes

Authors:

Prahladh Harsha,

Raghu MekaAuthors Info & Claims

STOC '10: Proceedings of the forty-second ACM symposium on Theory of computing

Pages 543 - 552

https://doi.org/10.1145/1806689.1806764

Published: 05 June 2010 Publication History

Abstract

Let X be randomly chosen from {-1,1}ⁿ, and let Y be randomly chosen from the standard spherical Gaussian on Rⁿ. For any (possibly unbounded) polytope P formed by the intersection of k halfspaces, we prove that |Pr[X ∈ P] - Pr[Y ∈ P]| ≤ log^8/5k • Δ, where Δ is a parameter that is small for polytopes formed by the intersection of "regular" halfspaces (i.e., halfspaces with low influence). The novelty of our invariance principle is the polylogarithmic dependence on k. Previously, only bounds that were at least linear in k were known.

We give two important applications of our main result: A bound of log^O(1)k • ε^1/6 on the Boolean noise sensitivity of intersections of k "regular" halfspaces (previous work gave bounds linear in k). This gives a corresponding agnostic learning algorithm for intersections of regular halfspaces. A pseudorandom generator (PRG) with seed length O(log n, poly(log k,1/Δ)) that Δ-fools all polytopes with k faces with respect to the Gaussian distribution.

We also obtain PRGs with similar parameters that fool polytopes formed by intersection of regular halfspaces over the hypercube. Using our PRG constructions, we obtain the first deterministic quasi-polynomial time algorithms for approximately counting the number of solutions to a broad class of integer programs, including dense covering problems and contingency tables.

References

[1]

N. Ailon and B. Chazelle. Approximate nearest neighbors and the fast Johnson Lindenstrauss transform. In STOC, pages 557--563. 2006.

Digital Library

[2]

Vidmantas K. Bentkus. Smooth approximations of the norm and differentiable functions with bounded support in Banach space l_ınfty ^k. Lithuanian Mathematical Journal, 30(3):223--230, July 1990.

[3]

---. On the dependence of the Berryâ??-Esseen bound on dimension. Journal of Statistical Planning and Inference, 113(2):385--402, May 2003.

[4]

Itai Benjamini, Gil Kalai, and Oded Schramm. Noise sensitivity of Boolean functions and applications to percolation. Inst. Hautes Études Sci. Publ. Math., 90(1):5--43, 1999. http://arxiv.org/abs/math/9811157/arXiv:math/9811157.

[5]

Matthias Beck and Sinai Robins. http://math.sfsu.edu/beck/ccd.html Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra. Undergraduate Texts in Mathematics. Springer, 1st edition, 2007.

[6]

Alexander Barvinok and Ellen Veomett. http://www.ams.org/bookstore-getitem/item=CONM-453 The computational complexity of convex bodies. In Jacob E. Goodman, János Pach, and Richard Pollack, eds., Surveys on Discrete and Computational Geometry: Twenty Years Later, volume 453 of Contemporary Mathematics, pages 117--137. AMS, 2008. http://arxiv.org/abs/math/0610325/arXiv:math/0610325.

[7]

Mary Cryan and Martin E. Dyer. A polynomial-time algorithm to approximately count contingency tables when the number of rows is constant. J. Comp. Sys. Sci., 67(2):291--310, 2003.

Digital Library

[8]

I. Diakonikolas, P. Harsha, A. Klivans, R. Meka, P. Raghavendra, R. Servedio, and L. Tan. Bounding the average sensitivity and noise sensitivity of polynomial threshold functions. In STOC, to appear. 2010.

Digital Library

[9]

Ilias Diakonikolas, Daniel M. Kane, and Jelani Nelson. Bounded independence fools degree-2 threshold functions, 2009. http://arxiv.org/abs/0911.3389/arXiv:0911.3389.

[10]

M. E. Dyer. Approximate counting by dynamic programming. In STOC, pages 693--699. 2003.

Digital Library

[11]

William Feller. An Introduction to Probability Theory and Its Applications, Volume 2. Wiley, 2nd edition, 1971.

[12]

P. Gopalan, R. O'Donnell, Y. Wu, and D. Zuckerman. Fooling functions of halfspaces under product distributions. In CCC, to appear. 2010.

[13]

Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42(6):1115--1145, 1995. (Preliminary version in 26th STOC, 1994).

Digital Library

[14]

Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798--859, July 2001. (Preliminary Version in 29th STOC, 1997).

Digital Library

[15]

Russell Impagliazzo, Noam Nisan, and Avi Wigderson. Pseudorandomness for network algorithms. In STOC, pages 356--364. 1994.

Digital Library

[16]

Mark Jerrum and Alistair Sinclair. http://www.ieor.berkeley.edu/ hochbaum/html/book-aanp.html The Markov chain Monte Carlo method: An approach to approximate counting and integration. In Dorit S. Hochbaum, ed., Approximation Algorithms for NP-hard Problems. PWS Publishing Company, 1997.

Digital Library

[17]

Gil Kalai. http://www.ratio.huji.ac.il/dp_files/dp399.pdf Noise sensitivity and chaos in social choice theory. Technical Report 399, Center for Rationality and Interactive Decision Theory, Hebrew University of Jerusalem, 2005.

[18]

Jeff Kahn, Gil Kalai, and Nathan Linial. The influence of variables on Boolean functions (extended abstract). In FOCS, pages 68--80. 1988.

Digital Library

[19]

Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM J. Computing, 37(1):319--357, 2007.

Digital Library

[20]

Adam Tauman Kalai, Adam R. Klivans, Yishay Mansour, and Rocco A. Servedio. Agnostically learning halfspaces. SIAM J. Computing, 37(6):1777--1805, 2008. (Preliminary version in 46th FOCS, 2005).

Digital Library

[21]

Adam R. Klivans, Ryan O'Donnell, and Rocco A. Servedio. Learning intersections and thresholds of halfspaces. J. Comp. Sys. Sci., 68(4):808--840, 2004. (Preliminary version in 43rd FOCS, 2002).

Digital Library

[22]

______. Learning geometric concepts via Gaussian surface area. In FOCS, pages 541--550. 2008.

Digital Library

[23]

Nathan Linial, Yishay Mansour, and Noam Nisan. Constant depth circuits, Fourier transform, and learnability. J. ACM, 40(3):607--620, 1993. (Preliminary version in 30th FOCS, 1989).

Digital Library

[24]

Yishay Mansour. http://www.springer.com/physics/complexity/book/978-0-7923-9478-5 Learning Boolean functions via the Fourier transform. In Vwani P. Roychowdhury, Kai-Yeung Siu, and Alon Orlitsky, eds., Theoretical Advances in Neural Computation and Learning, pages 391--424. Kluwer Academic Publishers, 1994.

[25]

Sanjeev Mahajan and Ramesh Hariharan. Derandomizing approximation algorithms based on semidefinite programming. SIAM J. Computing, 28(5):1641--1663, 1999. (Preliminary version in 36th FOCS, 1995).

Digital Library

[26]

Elchanan Mossel, Ryan O'Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions with low in.uences invariance and optimality. In FOCS, pages 21--30. 2005. http://arxiv.org/abs/math/0503503\patharXiv:math/0503503.

Digital Library

[27]

Elchanan Mossel. Gaussian bounds for noise correlation of functions and tight analysis of long codes. In FOCS, pages 156--165. 2008. http://arxiv.org/abs/math/0703683/arXiv:math/0703683.

Digital Library

[28]

R. Meka and D. Zuckerman. Pseudorandom generators for polynomial threshold functions. In STOC,to appear. 2010. http://arxiv.org/abs/0910.4122/arXiv:0910.4122.

Digital Library

[29]

Fedor Nazarov. On the maximal perimeter of a convex set in Rⁿ with respect to a Gaussian measure. In Geometric Aspects of Functional Analysis (Israel Seminar 2001--2002), volume 1807/2003 of Lecture Notes in Mathematics, pages 169--187. Springer, 2003.

[30]

Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM J. Computing, 22(4):838--856, August 1993. (Preliminary Version in 22nd STOC, 1990).

Digital Library

[31]

Ryan O'Donnell. Hardness amplification within NP. J. Comp. Sys. Sci., 69(1):68--94, 2004. (Preliminary version in 34th STOC, 2002).

Digital Library

[32]

____. Some topics in analysis of Boolean functions. In STOC, pages 569--578. 2008. http://eccc.hpi-web.de/report/2008/055/eccc:TR08-055.

Digital Library

[33]

Yuval Peres. Noise stability of weighted majority, 2004. http://arxiv.org/abs/math/0412377/arXiv:math/0412377.

[34]

Vygantas Paulauskas and Alfredas Ra\vckauskas. Approximation Theory in the Central Limit Theorem: Exact Results in Banach Spaces. Kluwer Academic Publishers, 1989. (Translated from Russian).

[35]

Yaoyun Shi. Lower bounds of quantum black-box complexity and degree of approximating polynomials by influence of Boolean variables. Inf. Process. Lett., 75(1-2):79--83, 2000. http://arxiv.org/abs/quant-ph/9904107/arXiv:quant-ph/9904107.

Digital Library

[36]

Pawel Wolff. Hypercontractivity of simple random variables. Studia Math, 180(3):219--236, 2007.

[37]

Günter M. Ziegler. http://www.springer.com/math/geometry/book/978-0-387-94365-7 Lectures on polytopes, volume 152 of Graduate texts in Mathematics. Springer, 1995.

Cited By

Rubinfeld RVasilyan ASaha BServedio R(2023)Testing Distributional Assumptions of Learning AlgorithmsProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585117(1643-1656)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585117
Ding NGu D(2021)New cryptographic hardness for learning intersections of halfspaces over boolean cubes with membership queriesInformation and Computation10.1016/j.ic.2021.104771(104771)Online publication date: Jun-2021
https://doi.org/10.1016/j.ic.2021.104771
Kane DShmoys D(2014)The average sensitivity of an intersection of half spacesProceedings of the forty-sixth annual ACM symposium on Theory of computing10.1145/2591796.2591798(437-440)Online publication date: 31-May-2014
https://dl.acm.org/doi/10.1145/2591796.2591798
Show More Cited By

Index Terms

An invariance principle for polytopes
1. Mathematics of computing
  1. Probability and statistics
    1. Probabilistic reasoning algorithms
      1. Random number generation

Recommendations

Fooling Polytopes
We give a pseudorandom generator that fools m-facet polytopes over {0, 1}ⁿ with seed length polylog(m) · log n. The previous best seed length had superlinear dependence on m.
An invariance principle for polytopes

Let X be randomly chosen from {-1,1}ⁿ, and let Y be randomly chosen from the standard spherical Gaussian on ℝⁿ. For any (possibly unbounded) polytope P formed by the intersection of k halfspaces, we prove that |Pr[X ∈ P] - Pr[Y ∈ P]| ≤ log^8/5k ⋅ Δ, ...
Fooling polytopes
STOC 2019: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

We give a pseudorandom generator that fools m-facet polytopes over {0,1}ⁿ with seed length polylog(m) · log(n). The previous best seed length had superlinear dependence on m. An immediate consequence is a deterministic quasipolynomial time algorithm for ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

STOC '10: Proceedings of the forty-second ACM symposium on Theory of computing

June 2010

812 pages

ISBN:9781450300506

DOI:10.1145/1806689

General Chair:
Michael Mitzenmacher
Harvard
,
Program Chair:
Leonard J. Schulman
Caltech

Copyright © 2010 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 05 June 2010

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

STOC'10

Sponsor:

SIGACT

STOC'10: Symposium on Theory of Computing

June 5 - 8, 2010

Massachusetts, Cambridge, USA

Acceptance Rates

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

Upcoming Conference

STOC '25

Sponsor:
sigact

57th Annual ACM Symposium on Theory of Computing (STOC 2025)

June 23 - 27, 2025

Prague , Czech Republic

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

8
Total Citations
View Citations
164
Total Downloads

Downloads (Last 12 months)2
Downloads (Last 6 weeks)0

Reflects downloads up to 23 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Rubinfeld RVasilyan ASaha BServedio R(2023)Testing Distributional Assumptions of Learning AlgorithmsProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585117(1643-1656)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585117
Ding NGu D(2021)New cryptographic hardness for learning intersections of halfspaces over boolean cubes with membership queriesInformation and Computation10.1016/j.ic.2021.104771(104771)Online publication date: Jun-2021
https://doi.org/10.1016/j.ic.2021.104771
Kane DShmoys D(2014)The average sensitivity of an intersection of half spacesProceedings of the forty-sixth annual ACM symposium on Theory of computing10.1145/2591796.2591798(437-440)Online publication date: 31-May-2014
https://dl.acm.org/doi/10.1145/2591796.2591798
Harsha PKlivans AMeka R(2013)An invariance principle for polytopesJournal of the ACM10.1145/2395116.239511859:6(1-25)Online publication date: 9-Jan-2013
https://dl.acm.org/doi/10.1145/2395116.2395118
Watson T(2013)Pseudorandom generators for combinatorial checkerboardsComputational Complexity10.1007/s00037-012-0036-622:4(727-769)Online publication date: 1-Dec-2013
https://dl.acm.org/doi/10.1007/s00037-012-0036-6
Meka RZuckerman DMitzenmacher MSchulman L(2010)Pseudorandom generators for polynomial threshold functionsProceedings of the forty-second ACM symposium on Theory of computing10.1145/1806689.1806749(427-436)Online publication date: 5-Jun-2010
https://dl.acm.org/doi/10.1145/1806689.1806749
Diakonikolas IKane DNelson J(2010)Bounded Independence Fools Degree-2 Threshold FunctionsProceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science10.1109/FOCS.2010.8(11-20)Online publication date: 23-Oct-2010
https://dl.acm.org/doi/10.1109/FOCS.2010.8
Gopalan PO'Donnell RWu YZuckerman D(2010)Fooling Functions of Halfspaces under Product DistributionsProceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity10.1109/CCC.2010.29(223-234)Online publication date: 9-Jun-2010
https://dl.acm.org/doi/10.1109/CCC.2010.29

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents