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On prediction of individual sequences

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Abstract

Sequential randomized prediction of an arbitrary binary sequence is investigated. No assumption is made on the mechanism of generating the bit sequence. The goal of the predictor is to minimize its relative loss, i.e., to make (almost) as few mistakes as the best ``expert'' in a fixed, possibly infinite, set of experts. We point out a surprising connection between this prediction problem and empirical process theory. First, in the special case of static (memoryless) experts, we completely characterize the minimax relative loss in terms of the maximum of an associated Rademacher process. Then we show general upper and lower bounds on the minimax relative loss in terms of the geometry of the class of experts. As main examples, we determine the exact order of magnitude of the minimax relative loss for the class of autoregressive linear predictors and for the class of Markov experts.

Suggested Citation

  • Nicolo Cesa Bianchi & Gábor Lugosi, 1998. "On prediction of individual sequences," Economics Working Papers 324, Department of Economics and Business, Universitat Pompeu Fabra.
    • Handle: RePEc:upf:upfgen:324
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    Cited by:

    1. Carl Remlinger & Bri`ere Marie & Alasseur Cl'emence & Joseph Mikael, 2021. "Expert Aggregation for Financial Forecasting," Papers 2111.15365, arXiv.org, revised Jul 2023.
    2. Davide Viviano & Jelena Bradic, 2019. "Synthetic learner: model-free inference on treatments over time," Papers 1904.01490, arXiv.org, revised Aug 2022.
    3. Gábor Lugosi & Shie Mannor & Gilles Stoltz, 2008. "Strategies for Prediction Under Imperfect Monitoring," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 513-528, August.
    4. A. Borodin & R. El-Yaniv & V. Gogan, 2011. "Can We Learn to Beat the Best Stock," Papers 1107.0036, arXiv.org.
    5. Sancetta, A., 2005. "Forecasting Distributions with Experts Advice," Cambridge Working Papers in Economics 0517, Faculty of Economics, University of Cambridge.
    6. Nicolò Cesa-Bianchi & Gábor Lugosi & Gilles Stoltz, 2006. "Regret Minimization Under Partial Monitoring," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 562-580, August.
    7. Sancetta, Alessio, 2007. "Online forecast combinations of distributions: Worst case bounds," Journal of Econometrics, Elsevier, vol. 141(2), pages 621-651, December.

    More about this item

    Keywords

    Universal prediction; prediction with experts; absolute loss; empirical processes; covering numbers; finite-state machines;
    All these keywords.

    JEL classification:

    • C20 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - General

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