Skip to main content
SpringerLink
Log in

New degree bounds for polynomial threshold functions

  • Published:
Combinatorica Aims and scope Submit manuscript

Abstract

A real multivariate polynomial p(x 1, …, x n ) is said to sign-represent a Boolean function f: {0,1}n→{−1,1} if the sign of p(x) equals f(x) for all inputs x∈{0,1}n. We give new upper and lower bounds on the degree of polynomials which sign-represent Boolean functions. Our upper bounds for Boolean formulas yield the first known subexponential time learning algorithms for formulas of superconstant depth. Our lower bounds for constant-depth circuits and intersections of halfspaces are the first new degree lower bounds since 1968, improving results of Minsky and Papert. The lower bounds are proved constructively; we give explicit dual solutions to the necessary linear programs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Ambainis, A. Childs, B. Reichardt, R. Spalek and S. Zhang: Any ANDOR formula of size n can be evaluated in time n1/2+o(1) on a quantum computer, in: Proc. 48th IEEE Symposium on Foundations of Computer Science (FOCS), pages 363–372, 2007.

  2. D. Angluin: Queries and concept learning, Machine Learning2 (1988), 319–342.

    Google Scholar 

  3. J. Aspnes, R. Beigel, M. Furst and S. Rudich: The expressive power of voting polynomials, Combinatorica14(2) (1994), 1–14.

    Article  MathSciNet  Google Scholar 

  4. R. Beigel: The polynomial method in circuit complexity, in: Proceedings of the Eigth Conference on Structure in Complexity Theory, pages 82–95, 1993.

  5. R. Beigel: Perceptrons, PP, and the Polynomial Hierarchy, Computational Complexity4 (1994), 339–349.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Beigel, N. Reingold and D. Spielman: PP is closed under intersection, Journal of Computer & System Sciences50(2) (1995), 191–202.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Bruck: Harmonic analysis of polynomial threshold functions, SIAM Journal on Discrete Mathematics3(2) (1990), 168–177.

    Article  MATH  MathSciNet  Google Scholar 

  8. H. Buhrman, R. Cleve and A. Wigderson: Quantum vs. classical communication and computation, in: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, pages 63–68. ACM Press, 1998.

  9. H. Buhrman and R. de Wolf: Complexity measures and decision tree complexity: a survey; Theoretical Computer Science288(1) (2002), 21–43.

    Article  MATH  MathSciNet  Google Scholar 

  10. E. Cheney: Introduction to approximation theory, McGraw-Hill, New York, New York, 1966.

    MATH  Google Scholar 

  11. L. Comtet: Advanced Combinatorics: The Art of Finite and Infinite Expansions; Reidel, Dordrecht, Netherlands, 1974.

    MATH  Google Scholar 

  12. Y. Freund: Boosting a weak learning algorithm by majority, Information and Computation121(2) (1995), 256–285.

    Article  MATH  MathSciNet  Google Scholar 

  13. M. Goldmann: On the power of a threshold gate at the top, Information Processing Letters63(6) (1997), 287–293.

    Article  MathSciNet  Google Scholar 

  14. A. Hajnal, W. Maass, P. Pudlák, M. Szegedy and Gy. Turán: Threshold circuits of bounded depth, Journal of Computer and System Sciences46 (1993), 129–154.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Kearns and U. Vazirani: An introduction to computational learning theory. MIT Press, Cambridge, MA, 1994.

    Google Scholar 

  16. A. Klivans, R. O’Donnell and R. Servedio: Learning intersections and thresholds of halfspaces, in: Proceedings of the 43rd Annual Symposium on Foundations of Computer Science, pages 177–186, 2002.

  17. A. Klivans and R. Servedio: Learning DNF in time 2Õ(n1/3), in: Proceedings of the Thirty-Third Annual Symposium on Theory of Computing, pages 258–265, 2001.

  18. M. Krause and P. Pudlák: Computing boolean functions by polynomials and threshold circuits, Computational Complexity7(4) (1998), 346–370.

    Article  MATH  MathSciNet  Google Scholar 

  19. N. Linial, Y. Mansour and N. Nisan: Constant depth circuits, Fourier transform and learnability; Journal of the ACM40(3) (1993), 607–620.

    Article  MATH  MathSciNet  Google Scholar 

  20. N. Littlestone: Learning quickly when irrelevant attributes abound: a new linearthreshold algorithm; Machine Learning2 (1988), 285–318.

    Google Scholar 

  21. M. Minsky and S. Papert: Perceptrons: an introduction to computational geometry (expanded edition); MIT Press, Cambridge, MA, 1988.

    MATH  Google Scholar 

  22. D. J. Newman: Rational approximation to |x|, Michigan Mathematical Journal11 (1964), 11–14.

    Article  MATH  MathSciNet  Google Scholar 

  23. N. Nisan and M. Szegedy: On the degree of Boolean functions as real polynomials, in: Proceedings of the Twenty-Fourth Annual Symposium on Theory of Computing, pages 462–467, 1992.

  24. R. O’Donnell and R. Servedio: New degree bounds for polynomial threshold functions, in: Proceedings of the 35th ACM Symposium on Theory of Computing, pages 325–334, 2003.

  25. R. Paturi and M. Saks: Approximating threshold circuits by rational functions, Information and Computation112(2) (1994), 257–272.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Saks: Slicing the hypercube, in: Surveys in Combinatorics 1993 (Keith Walker, ed.), London Mathematical Society Lecture Note Series 187, pages 211–257, 1993.

  27. D. Sieling: Minimization of decision trees is hard to approximate, Journal of Computer and System Sciences74(3) (2008), 394–403.

    Article  MATH  MathSciNet  Google Scholar 

  28. L. Valiant: A theory of the learnable, Communications of the ACM27(11) (1984), 1134–1142.

    Article  MATH  Google Scholar 

  29. N. Vereshchagin: Lower bounds for perceptrons solving some separation problems and oracle separation of AM from PP, in: Proceedings of the Third Annual Israel Symposium on Theory of Computing and Systems, pages 46–51, 1995.

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA, 15213-3891, USA

    Ryan O’Donnell

  2. Department of Computer Science, Columbia University, 1214 Amsterdam Avenue, Mailcode 0401, New York, NY, 10027-7003, USA

    Rocco A. Servedio

Authors
  1. Ryan O’Donnell
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rocco A. Servedio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rocco A. Servedio.

Additional information

A preliminary version of these results appeared as [24].

This work was done while at the Department of Mathematics, MIT, Cambridge, MA, and while supported by NSF grant 99-12342.

Supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship and by NSF grant CCR-98-77049. This work was done while at the Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA.

Rights and permissions

Reprints and permissions

About this article

Cite this article

O’Donnell, R., Servedio, R.A. New degree bounds for polynomial threshold functions. Combinatorica 30, 327–358 (2010). https://doi.org/10.1007/s00493-010-2173-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-010-2173-3

Mathematics Subject Classification (2000)

Search

Navigation