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Worst-case bounds for the logarithmic loss of predictors


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  • Nicolò Cesa Bianchi
  • Gábor Lugosi



We investigate on-line prediction of individual sequences. Given a class of predictors, the goal is to predict as well as the best predictor in the class, where the loss is measured by the self information (logarithmic) loss function. The excess loss (regret) is closely related to the redundancy of the associated lossless universal code. Using Shtarkov's theorem and tools from empirical process theory, we prove a general upper bound on the best possible (minimax) regret. The bound depends on certain metric properties of the class of predictors. We apply the bound to both parametric and nonparametric classes of predictors. Finally, we point out a suboptimal behavior of the popular Bayesian weighted average algorithm.

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Bibliographic Info

Paper provided by Department of Economics and Business, Universitat Pompeu Fabra in its series Economics Working Papers with number 418.

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Date of creation: Oct 1999
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Handle: RePEc:upf:upfgen:418

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Keywords: Universal prediction; universal coding; empirical processes; on-line learning; metric entropy;

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Cited by:
  1. Alessio Sancetta, 2010. "Bootstrap model selection for possibly dependent and heterogeneous data," Annals of the Institute of Statistical Mathematics, Springer, Springer, vol. 62(3), pages 515-546, June.


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