Loss Functions, Complexities, and the Legendre Transformation

  • Yuri Kalnishkan
  • Michael V. Vyugin
  • Volodya Vovk
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2225)

Abstract

The paper introduces a way of re-constructing a loss function from predictive complexity. We show that a loss function and expectations of the corresponding predictive complexity w.r.t. the Bernoulli distribution are related through the Legendre transformation. It is shown that if two loss functions specify the same complexity then they are equivalent in a strong sense.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin. Optimal Control. Plenum, New York, 1987.Google Scholar
  2. [2]
    Y. Kalnishkan. General linear relations among different types of predictive complexity. In Proc. 10th International Conference on Algorithmic Learning Theory — ALT’ 99, volume 1720 of Lecture Notes in Artificial Intelligence, pages 323–334. Springer-Verlag, 1999.Google Scholar
  3. [3]
    M. Li and P. Vitányi. An Introduction to Kolmogorov Complexity and Its Applications. Springer, New York, 2nd edition, 1997.MATHGoogle Scholar
  4. [4]
    A. Wayne Roberts and Dale E. Varberg. Convex Functions. Academic Press, 1973.Google Scholar
  5. [5]
    R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, 1970.Google Scholar
  6. [6]
    V. Vovk. Aggregating strategies. In M. Fulk and J. Case, editors, Proceedings of the 3rd Annual Workshop on Computational Learning Theory, pages 371–383, San Mateo, CA, 1990. Morgan Kaufmann.Google Scholar
  7. [7]
    V. Vovk. A game of prediction with expert advice. Journal of Computer and System Sciences, 56:153–173, 1998.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    V. Vovk and C. J. H. C. Watkins. Universal portfolio selection. In Proceedings of the 11th Annual Conference on Computational Learning Theory, pages 12–23, 1998.Google Scholar
  9. [9]
    V. V. V'yugin. Algorithmic entropy (complexity) of finite objects and its applications to defining randomness and amount of information. Selecta Mathematica formerly Sovietica, 13:357–389, 1994.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Yuri Kalnishkan
    • 1
  • Michael V. Vyugin
    • 1
  • Volodya Vovk
    • 1
  1. 1.Department of Computer Science, Royal HollowayUniversity of LondonSurreyUK

Personalised recommendations