Continuous Experts and the Binning Algorithm

  • Jacob Abernethy
  • John Langford
  • Manfred K. Warmuth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4005)


We consider the design of online master algorithms for combining the predictions from a set of experts where the absolute loss of the master is to be close to the absolute loss of the best expert. For the case when the master must produce binary predictions, the Binomial Weighting algorithm is known to be optimal when the number of experts is large. It has remained an open problem how to design master algorithms based on binomial weights when the predictions of the master are allowed to be real valued. In this paper we provide such an algorithm and call it the Binning algorithm because it maintains experts in an array of bins. We show that this algorithm is optimal in a relaxed setting in which we consider experts as continuous quantities. The algorithm is efficient and near-optimal in the standard experts setting.


Successor State Exponential Weight Continuous Quantity Absolute Loss Good Expert 
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.


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  1. [ACBFS95]
    Auer, P., Cesa-Bianchi, N., Freund, Y., Schapire, R.E.: Gambling in a rigged casino: the adversarial multi-armed bandit problem. In: Proceedings of the 36th Annual Symposium on Foundations of Computer Science, pp. 322–331. IEEE Computer Society Press, Los Alamitos (1995)Google Scholar
  2. [CBFH+97]
    Cesa-Bianchi, N., Freund, Y., Haussler, D., Helmbold, D.P., Schapire, R.E., Warmuth, M.K.: How to use expert advice. Journal of the ACM 44(3), 427–485 (1997)CrossRefMathSciNetMATHGoogle Scholar
  3. [CBFHW96]
    Cesa-Bianchi, N., Freund, Y., Helmbold, D.P., Warmuth, M.K.: On-line prediction and conversion strategies. Machine Learning 25, 71–110 (1996)Google Scholar
  4. [FO02]
    Freund, Y., Opper, M.: Drifting games and Brownian motion. Journal of Computer and System Sciences 64, 113–132 (2002)CrossRefMathSciNetMATHGoogle Scholar
  5. [Fre95]
    Freund, Y.: Boosting a weak learning algorithm by majority. Information and Computation 121(2), 256–285 (1995)CrossRefMathSciNetMATHGoogle Scholar
  6. [FS97]
    Freund, Y., Schapire, R.E.: A decision-theoretic generalization of on-line learning and an application to boosting. Journal of Computer and System Sciences 55(1), 119–139 (1997)CrossRefMathSciNetMATHGoogle Scholar
  7. [HKW98]
    Haussler, D., Kivinen, J., Warmuth, M.K.: Sequential prediction of individual sequences under general loss functions. IEEE Transactions on Information Theory 44(2), 1906–1925 (1998)CrossRefMathSciNetMATHGoogle Scholar
  8. [HW98]
    Herbster, M., Warmuth, M.K.: Tracking the best expert. Journal of Machine Learning 32(2), 151–178 (1998)CrossRefMATHGoogle Scholar
  9. [KW97]
    Kivinen, J., Warmuth, M.K.: Additive versus exponentiated gradient updates for linear prediction. Information and Computation 132(1), 1–64 (1997)CrossRefMathSciNetMATHGoogle Scholar
  10. [KW99]
    Kivinen, J., Warmuth, M.K.: Averaging expert predictions. In: Fischer, P., Simon, H.U. (eds.) EuroCOLT 1999. LNCS, vol. 1572, pp. 153–167. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  11. [LW94]
    Littlestone, N., Warmuth, M.K.: The weighted majority algorithm. Information and Computation 108(2), 212–261 (1994); An early version appeared in FOCS 1989Google Scholar
  12. [Vov90]
    Vovk, V.: Aggregating strategies. In: Proc. 3rd Annu. Workshop on Comput. Learning Theory, pp. 371–383. Morgan Kaufmann, San Francisco (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jacob Abernethy
    • 1
  • John Langford
    • 1
  • Manfred K. Warmuth
    • 2
    • 3
  1. 1.Toyota Technological InstituteChicago
  2. 2.University of California at Santa Cruz 
  3. 3.Supported by NSF grant CCR CCR 9821087 

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