Advertisement

Abstract

We give an algorithm that with high probability properly learns random monotone DNF with t(n) terms of length ≈ logt(n) under the uniform distribution on the Boolean cube {0,1} n . For any function t(n) ≤ poly(n) the algorithm runs in time poly(n,1/ε) and with high probability outputs an ε-accurate monotone DNF hypothesis. This is the first algorithm that can learn monotone DNF of arbitrary polynomial size in a reasonable average-case model of learning from random examples only.

Keywords

Boolean Function Full Version Random Draw Term Length Boolean Cube 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AM02]
    Amano, K., Maruoka, A.: On learning monotone boolean functions under the uniform distribution. In: Proc. 13th ALT, pp. 57–68 (2002)Google Scholar
  2. [AP95]
    Aizenstein, H., Pitt, L.: On the learnability of disjunctive normal form formulas. Machine Learning 19, 183–208 (1995)zbMATHGoogle Scholar
  3. [BBL98]
    Blum, A., Burch, C., Langford, J.: On learning monotone boolean functions. In: Proc. 39th FOCS, pp. 408–415 (1998)Google Scholar
  4. [BFJ+94]
    Blum, A., Furst, M., Jackson, J., Kearns, M., Mansour, Y., Rudich, S.: Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In: Proc. 26th STOC, pp. 253–262 (1994)Google Scholar
  5. [Blu03a]
    Blum, A.: Learning a function of r relevant variables (open problem). In: Proc. 16th COLT, pp. 731–733 (2003)Google Scholar
  6. [Blu03b]
    Blum, A.: Machine learning: a tour through some favorite results, directions, and open problems. In: FOCS 2003 tutorial slides (2003)Google Scholar
  7. [BT96]
    Bshouty, N., Tamon, C.: On the Fourier spectrum of monotone functions. Journal of the ACM 43(4), 747–770 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [HM91]
    Hancock, T., Mansour, Y.: Learning monotone k-μ DNF formulas on product distributions. In: Proc. 4th COLT, pp. 179–193 (1991)Google Scholar
  9. [Jac97]
    Jackson, J.: An efficient membership-query algorithm for learning DNF with respect to the uniform distribution. JCSS 55, 414–440 (1997)zbMATHGoogle Scholar
  10. [JS05]
    Jackson, J., Servedio, R.: Learning random log-depth decision trees under the uniform distribution. SICOMP 34(5), 1107–1128 (2005)zbMATHMathSciNetGoogle Scholar
  11. [JS06]
    Jackson, J., Servedio, R.: On learning random DNF formulas under the uniform distribution. Theory of Computing 2(8), 147–172 (2006)MathSciNetCrossRefGoogle Scholar
  12. [JT97]
    Jackson, J., Tamon, C.: Fourier analysis in machine learning. In: ICML/COLT 1997 tutorial slides (1997)Google Scholar
  13. [KLV94]
    Kearns, M., Li, M., Valiant, L.: Learning Boolean formulas. Journal of the ACM 41(6), 1298–1328 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  14. [KMSP94]
    Kučera, L., Marchetti-Spaccamela, A., Protassi, M.: On learning monotone DNF formulae under uniform distributions. Information and Computation 110, 84–95 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  15. [KV94]
    Kearns, M., Vazirani, U.: An introduction to computational learning theory. MIT Press, Cambridge (1994)Google Scholar
  16. [Man95]
    Mansour, Y.: An O(n loglogn) learning algorithm for DNF under the uniform distribution. JCSS 50, 543–550 (1995)zbMATHMathSciNetGoogle Scholar
  17. [MO03]
    Mossel, E., O’Donnell, R.: On the noise sensitivity of monotone functions. Random Structures and Algorithms 23(3), 333–350 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  18. [OS06]
    O’Donnell, R., Servedio, R.: Learning monotone decision trees in polynomial time. In: Proc. 21st CCC, pp. 213–225 (2006)Google Scholar
  19. [Sel08]
    Sellie, L.: Learning Random Monotone DNF Under the Uniform Distribution. In: Proc. 21st COLT (to appear, 2008)Google Scholar
  20. [Ser04]
    Servedio, R.: On learning monotone DNF under product distributions. Information and Computation 193(1), 57–74 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  21. [SM00]
    Sakai, Y., Maruoka, A.: Learning monotone log-term DNF formulas under the uniform distribution. Theory of Computing Systems 33, 17–33 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  22. [Val84]
    Valiant, L.: A theory of the learnable. CACM 27(11), 1134–1142 (1984)zbMATHGoogle Scholar
  23. [Ver90]
    Verbeurgt, K.: Learning DNF under the uniform distribution in quasi-polynomial time. In: Proc. 3rd COLT, pp. 314–326 (1990)Google Scholar
  24. [Ver98]
    Verbeurgt, K.: Learning sub-classes of monotone DNF on the uniform distribution. In: Proc. 9th ALT, pp. 385–399 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Jeffrey C. Jackson
    • 1
  • Homin K. Lee
    • 2
  • Rocco A. Servedio
    • 2
  • Andrew Wan
    • 2
  1. 1.Duquesne UniversityPittsburgh 
  2. 2.Columbia UniversityNew York 

Personalised recommendations