Mixability and the Existence of Weak Complexities

  • Yuri Kalnishkan
  • Michael V. Vyugin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2375)


This paper investigates the behaviour of the constant c(β) from the Aggregating Algorithm. Some conditions for mixability are derived and it is shown that for many non-mixable games c(β) still converges to 1. The condition c(β) → 1 is shown to imply the existence of weak predictive complexity and it is proved that many games specify complexity up to √n.


Loss Function Expert Advice Canonical Representation Weak Complexity Computational Learn Theory 
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. [CBFH+97]
    N. Cesa-Bianchi, Y. Freund, D. Haussler, D. P. Helmbold, R. E. Schapire, and M. K. Warmuth. How to use expert advice. Journal of the ACM, 44(3):427–485, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [HKW98]
    D. Haussler, J. Kivinen, and M. K. Warmuth. Sequential prediction of individual sequences under general loss functions. IEEE Transactions on Information Theory, 44(5):1906–1925, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [KVV01]
    Y. Kalnishkan, M. Vyugin, and V. Vovk. Losses, complexities and the Legendre transformation. In Proceedings of the 12th International Conference on Algorithmic Learning Theory, ALT 2001, volume 2225 of Lecture Notes in Artificial Intelligence. Springer-Verlag, 2001.Google Scholar
  4. [LW94]
    N. Littlestone and M. K. Warmuth. The weighted majority algorithm. Information and Computation, 108:212–261, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [Vov90]
    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
  6. [Vov98]
    V. Vovk. A game of prediction with expert advice. Journal of Computer and System Sciences, 56:153–173, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [VW98]
    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
  8. [V’y00]
    V. V. V’yugin. Sub-optimal measures of predictive complexity for absolute loss function. Technical Report CLRC TR-00-05, Computer Learning Research Centre, Royal Holloway College, University of London, 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Yuri Kalnishkan
    • 1
  • Michael V. Vyugin
    • 1
  1. 1.Department of Computer Science, Royal HollowayUniversity of LondonEghamUK

Personalised recommendations