A Characterization of Strong Learnability in the Statistical Query Model

  • Hans Ulrich Simon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4393)


In this paper, we consider Kearns’ [4] Statistical Query Model of learning. It is well known [3] that the number of statistical queries, needed for “weakly learning” an unknown target concept (i.e. for gaining significant advantage over random guessing) is polynomially related to the so-called Statistical Query dimension of the concept class. In this paper, we provide a similar characterization for “strong learning” where the learners final hypothesis is required to approximate the unknown target concept up to a small rate of misclassification. The quantity that characterizes strong learnability in the Statistical Query model is a surprisingly close relative of (though not identical to) the Statistical Query dimension. For the purpose of proving the main result, we provide other characterizations of strong learnability which are given in terms of covering numbers and related notions. These results might find some interest in their own right. All characterizations are purely information-theoretical and ignore computational issues.


Version Space Function Class Target Function Concept Class Parameterized Class 
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  1. 1.
    Balcázar, J.L., Castro, J., Guijarro, D.: A new abstract combinatorial dimension for exact learning via queries. Journal of Computer and System Sciences 64(1), 2–21 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Ben-David, S., Itai, A., Kushilevitz, E.: Learning by distances. Information and Computation 117(2), 240–250 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Blum, A., et al.: Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In: Proceedings of the 26th Annual Symposium on Theory of Computing, pp. 253–263 (1994)Google Scholar
  4. 4.
    Kearns, M.: Efficient noise-tolerant learning from statistical queries. Journal of the Association on Computing Machinery 45(6), 983–1006 (1998)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Köbler, J., Lindner, W.: A general dimension for approxmately learning boolean functions. In: Cesa-Bianchi, N., Numao, M., Reischuk, R. (eds.) ALT 2002. LNCS (LNAI), vol. 2533, pp. 139–148. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Simon, H.U.: Spectral norm in learning theory: some selected topics (Invited Talk) (Invited Talk). In: Balcázar, J.L., Long, P.M., Stephan, F. (eds.) ALT 2006. LNCS (LNAI), vol. 4264, pp. 13–27. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Valiant, L.G.: A theory of the learnable. Communications of the ACM 27(11), 1134–1142 (1984)CrossRefzbMATHGoogle Scholar
  8. 8.
    Yang, K.: New lower bounds for statistical query learning. Journal of Computer and System Sciences 70(4), 485–509 (2005)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Hans Ulrich Simon
    • 1
  1. 1.Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 BochumGermany

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