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On Martingale Extensions of Vapnik–Chervonenkis Theory with Applications to Online Learning

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Abstract

We review recent advances on uniform martingale laws of large numbers and the associated sequential complexity measures. These results may be considered as forming a non-i.i.d. generalization of Vapnik–Chervonenkis theory. We discuss applications to online learning, provide a recipe for designing online learning algorithms, and illustrate the techniques on the problem of online node classification. We outline connections to statistical learning theory and discuss inductive principles of stochastic approximation and empirical risk minimization.

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Notes

  1. 1.

    We may also consider the absolute value of the average without any complications.

  2. 2.

    Issues of measurability can be addressed with the techniques in [15].

  3. 3.

    It is also possible to study an intermediate setting, where some knowledge about the sequence is available (see, e.g., [27]).

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Acknowledgments

We gratefully acknowledge the support of NSF under grants CAREER DMS-0954737 and CCF-1116928.

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Authors and Affiliations

  1. University of Pennsylvania, Philadelphia, USA

    Alexander Rakhlin & Karthik Sridharan

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  1. Alexander Rakhlin
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  2. Karthik Sridharan
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Correspondence to Alexander Rakhlin .

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Editors and Affiliations

  1. Dept. of Computer Science, Royal Holloway, Univ of London, Egham, Surrey, United Kingdom

    Vladimir Vovk

  2. Frederick University, Nicosia, Cyprus

    Harris Papadopoulos

  3. Dept. of Computer Science, University of London, Egham, Surrey, United Kingdom

    Alexander Gammerman

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Rakhlin, A., Sridharan, K. (2015). On Martingale Extensions of Vapnik–Chervonenkis Theory with Applications to Online Learning. In: Vovk, V., Papadopoulos, H., Gammerman, A. (eds) Measures of Complexity. Springer, Cham. https://doi.org/10.1007/978-3-319-21852-6_15

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  • DOI: https://doi.org/10.1007/978-3-319-21852-6_15

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