On the Noise Sensitivity of Monotone Functions

  • Elchanan Mossel
  • Ryan O’Donnell
Conference paper
Part of the Trends in Mathematics book series (TM)


It is known that for all monotone functions f:{0, 1}n→ {0, 1}, if x∈ {0, 1}nis chosen uniformly at random and is obtained from by flipping each of the bits of independently with probability ∈, then \(P[{f_n}(x) \ne {f_n}(y)] < c \in \sqrt n\) for some c > 0. Previously, the best construction of monotone functions satisfying \(P[{f_n}(x) \ne {f_n}(y)] \geqslant \delta\) where 0< δ < 1/2, required ∈ ≥ c(δ)n−α where α = 1- ln 2/ln 3 = 0.36907…, and c(δ) > 0. We improve this result by achieving for every \(0 < \delta < 1/2, P[{f_n}(x) \ne {f_n}(y)] \geqslant \delta\) with:
  • ∈ = c(δ)n−α for any α < 1/2, using the recursive majority function with arity k = k(α);

  • \(\in = c(\delta ){n^{ - 1/2}}{\log ^t}n\) for \(t = {\log _2}\sqrt {\pi /2} = .3257 \ldots\) using an explicit recursive majority function with increasing arities; and

  • \(\in = c(\delta ){n^{ - 1/2}}\), non-constructively following a probabilistic CNF construction due to Talagrand.


Boolean Function Monotone Function Balance Function Noise Sensitivity Noise Rate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AN72]
    K. Athreya, P. Ney. Branching processes. Springer-Verlag, New York-Heidelberg, 1972.zbMATHCrossRefGoogle Scholar
  2. [BDG88]
    J. Balcázar, J. Díaz, J. Gabarró. Structural Complexity I, II. Springer-Verlag, Heidelberg, 1988.CrossRefGoogle Scholar
  3. [BJT99]
    N. Bshouty, J. Jackson, T. Tamon. Uniform-distribution attribute noise learnability. Workshop on Computational Learning Theory 1999.Google Scholar
  4. [BKS98]
    I. Benjamini, G. Kalai, O. Schramm. Noise sensitivity of boolean functions and applications to percolation. Preprint.Google Scholar
  5. [BL90]
    M. Ben-Or, N. Linial. Collective coin flipping. In Randomness and Computation S. Micali ed. Academic Press, New York, 1990.Google Scholar
  6. [BT96]
    N. Bshouty, C. Tamon. On the Fourier spectrum of monotone functions. Journal of the ACM 43(4), 1996.MathSciNetGoogle Scholar
  7. [DKOO]
    D.-Z. Du, K.-I Ko. Theory of Computational Complexity. Wiley Inter-science, New York, 2000.zbMATHGoogle Scholar
  8. [F98]
    E. Friedgut. Boolean functions with low average sensitivity depend on few coordinates. Combinatorica 18(1), 1998, 27–36.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [FK96]
    E. Friedgut, G. Kalai. Every monotone graph property has a sharp threshold. Proc. Amer. Math. Soc. 124, 1996, 2993–3002.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [H99]
    S. Haykin. Neural Networks, 2nd Edition. Prentice Hall, 1999.Google Scholar
  11. [J97]
    J. Jackson. An efficient membership-query algorithm for learning DNF with respect to the uniform distribution. Journal of Computer and System Sciences 55(3), 1997.Google Scholar
  12. [KKL88]
    J. Kahn, G. Kalai, N. Linial. The influence of variables on boolean functions. Foundations of Computer Science 1988.Google Scholar
  13. [KOS02]
    A. Klivans, R. O’Donnell, R. Servedio. Learning intersections and thresholds of halfspaces. To appear.Google Scholar
  14. [LMN93]
    N. Linial, Y. Mansour, N. Nisan. Constant depth circuits, Fourier transform, and learnability. J. Assoc. Comput. Mach. 40, 1993, 607–620.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [M94]
    Y. Mansour. Learning boolean functions via the Fourier transform. Theo-retical Advances in Neural Computing and Learning Kluwer Acad. Publ., Dordrecht (1994), 391–424.CrossRefGoogle Scholar
  16. [M98]
    E. Mossel. Recursive reconstruction on periodic trees. Random Structures Algorithms 13, 1998, no. 1, 81–97.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [002]
    R. O’Donnell. Hardness amplification within NP. Symposium on the Theory Of Computation, 2002.Google Scholar
  18. [Pa93]
    C. Papadimitriou. Computational Complexity. Addison Wesley, Reading, MA, 1993.Google Scholar
  19. [Pe98]
    Y. Peres. Personal communication, 1998.Google Scholar
  20. [T96]
    M. Talagrand. How much are increasing sets positively correlated Combinatorica 16, 1996, no. 2, 243–258.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [V84]
    L. Valiant. A theory of the learnable. Communications of the ACM, 40,1994, no. 2, 445–474.Google Scholar

Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Elchanan Mossel
    • 1
  • Ryan O’Donnell
    • 2
  1. 1.Hebrew Universit of Jerusalem and Microsoft ResearchHebrewIsrael
  2. 2.MIT Mathematics DepartmentUSA

Personalised recommendations