Encyclopedia of Machine Learning

2010 Edition
| Editors: Claude Sammut, Geoffrey I. Webb

VC Dimension

  • Thomas Zeugmann
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30164-8_875
  • 274 Downloads

Motivation and Background

We define an important combinatorial parameter that measures the combinatorial complexity of a family of subsets taken from a given universe (learning domain) X. This parameter was originally defined by Vapnik and Chervonenkis (1971) and is thus commonly referred to as Vapnik–Chervonenkis dimension, abbreviated as VC dimension. Subsequently, Dudley (19781979) generalized Vapnik and Chervonenkis (1971) results. The reader is also referred to Vapnik’s (2000) book in which he greatly extends the original ideas. This results in a theory which is called  structural risk minimization.

The importance of the VC dimension for  PAC Learning was discovered by Blumer, Ehrenfeucht, Haussler, & Warmuth (1989), who introduced the notion to computational learning theory.

As Anthony and Biggs (1992, p. 71) have put it, “The development of this notion is probably the most significant contribution that mathematics has made to Computational Learning Theory.”

Recall that we use...

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

Recommended Reading

  1. Anthony, M., & Bartlett, P. L. (1999). Neural network learning: Theoretical foundations. Cambridge: Cambridge University Press.zbMATHCrossRefGoogle Scholar
  2. Anthony, M., & Biggs, N. (1992). Computational learning theory. Cambridge tracts in theoretical computer science (No. 30). Cambridge: Cambridge University Press.Google Scholar
  3. Blumer, A., Ehrenfeucht, A., Haussler, D., & Warmuth, M. K. (1989). Learnability and the Vapnik–Chervonenkis dimension. Journal of the ACM, 36(4), 929–965.zbMATHCrossRefMathSciNetGoogle Scholar
  4. Dudley, R. M. (1978). Central limit theorems for empirical measures. Annals of Probability, 6(6), 899–929.zbMATHCrossRefMathSciNetGoogle Scholar
  5. Dudley, R. M. (1979). Corrections to “Central limit theorems for empirical measures”. Annals of Probability, 7(5), 909–911.zbMATHCrossRefMathSciNetGoogle Scholar
  6. Goldberg, P. W., & Jerrum, M. R. (1995). Bounding the Vapnik–Chervonenkis dimension of concept classes parameterized by real numbers. Machine Learning, 18(2–3), 131–148.zbMATHGoogle Scholar
  7. Gurvits, L. (1997). Linear algebraic proofs of VC-dimension based inequalities. In S. Ben-David (Ed.), Computational learning theory, third European conference, EuroCOLT ’97, Jerusalem, Israel, March 1997, Proceedings, Lecture notes in artificial intelligence (Vol. 1208, pp. 238–250). Springer.Google Scholar
  8. Haussler, D., & Welz, E. (1987). Epsilon nets and simplex range queries. Discrete & Computational Geometry, 2, 127–151.zbMATHCrossRefMathSciNetGoogle Scholar
  9. Haussler, D., & Littlestone, N., & Warmuth, M. K. (1994). Predicting f0; 1g functions on randomly drawn points. Information and Computation, 115(2), 248–292.zbMATHCrossRefMathSciNetGoogle Scholar
  10. Karpinski, M., & Macintyre, A. (1995). Polynomial bounds for VC dimension of sigmoidal neural networks. In Proceedings of twenty-seventh annual ACM symposium on theory of computing (pp. 200–208). New York: ACM Press.CrossRefGoogle Scholar
  11. Karpinski, M., & Werther, T. (1994). VC dimension and sampling complexity of learning sparse polynomials and rational functions. In S. J. Hanson, G. A. Drastal, and R. L. Rivest (Eds.), Computational learning theory and natural learning systems, Vol. I: Constraints and prospects (Chap. 11, pp. 331–354). Cambridge, MA: MIT Press.Google Scholar
  12. Kearns, M. J., & Vazirani, U. V. (1994). An introduction to computational learning theory. Cambridge, MA: MIT Press.Google Scholar
  13. Littlestone, N. (1988). Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning, 2(4), 285–318.Google Scholar
  14. Maass, W., & Turan, G. (1990). On the complexity of learning from counterexamples and membership queries. In Proceedings of the thirty-first annual symposium on Foundations of Computer Science (FOCS 1990), St. Louis, Missouri, October 22–24, 1990 (pp. 203–210). Los Alamitos, CA: IEEE Computer Society Press.Google Scholar
  15. Mitchell, A., Scheffer, T., Sharma, A., & Stephan, F. (1999). The VC-dimension of subclasses of pattern languages. In O. Watanabe & T. Yokomori (Eds.), Algorithmic learning theory, tenth international conference, ALT’99, Tokyo, Japan, December 1999, Proceedings, Lecture notes in artificial intelligence (Vol. 1720, pp. 93–105). Springer.Google Scholar
  16. Natschläger, T., & Schmitt, M. (1996). Exact VC-dimension of Boolean monomials. Information Processing Letters, 59(1), 19–20.CrossRefMathSciNetGoogle Scholar
  17. Sakurai, A. (1995). On the VC-dimension of depth four threshold circuits and the complexity of Boolean-valued functions. Theoretical Computer Science, 137(1), 109–127. Special issue for ALT ’93Google Scholar
  18. Sauer, N. (1972). On the density of families of sets. Journal of Combinatorial Theory (A), 13(1), 145–147.zbMATHCrossRefMathSciNetGoogle Scholar
  19. Schaefer, M. (1999). Deciding the Vapnik–Červonenkis dimension is Σ 3 p-complete. Journal of Computer System Sciences, 58(1), 177–182.zbMATHCrossRefMathSciNetGoogle Scholar
  20. Vapnik, V. N. (2000). The nature of statistical learning theory, (2nd ed.). Berlin: Springer.zbMATHGoogle Scholar
  21. Vapnik, V. N., & Chervonenkis, A. Y. (1971). On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probabability and Its Applications, 16(2), 264–280.zbMATHCrossRefMathSciNetGoogle Scholar
  22. Vapnik, V. N., & Chervonenkis, A. Y. (1974). Theory of pattern recognition. Moskwa: Nauka (in Russian).Google Scholar
  23. Wenocur, R. S., & Dudley, R. M. (1981). Some special Vapnik–Chervonenkis classes. Discrete Mathematics, 33, 313–318.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Thomas Zeugmann

There are no affiliations available