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Explore First, Exploit Next: The True Shape of Regret in Bandit Problems

Author

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  • Aurélien Garivier

    (Institute de Mathématiques de Toulouse (IMT): Université Paul Sabatier—The French National Research Center (CNRS), 31062 Toulouse, France)

  • Pierre Ménard

    (Institute de Mathématiques de Toulouse (IMT): Université Paul Sabatier—The French National Research Center (CNRS), 31062 Toulouse, France)

  • Gilles Stoltz

    (HEC Paris Management Research Group (GREGHEC): HEC Paris—CNRS, 78351 Jouy-en-Josas, France)

Abstract

We revisit lower bounds on the regret in the case of multiarmed bandit problems. We obtain nonasymptotic, distribution-dependent bounds and provide simple proofs based only on well-known properties of Kullback–Leibler divergences. These bounds show in particular that in the initial phase the regret grows almost linearly, and that the well-known logarithmic growth of the regret only holds in a final phase. The proof techniques come to the essence of the information-theoretic arguments used and they involve no unnecessary complications.

Suggested Citation

  • Aurélien Garivier & Pierre Ménard & Gilles Stoltz, 2019. "Explore First, Exploit Next: The True Shape of Regret in Bandit Problems," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 377-399, May.
    • Handle: RePEc:inm:ormoor:v:44:y:2019:i:2:p:377-399
    DOI: 10.1287/moor.2017.0928
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    References listed on IDEAS

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    1. Sanjeev R. Kulkarni & Gábor Lugosi, 1997. "Minimax lower bounds for the two-armed bandit problem," Economics Working Papers 206, Department of Economics and Business, Universitat Pompeu Fabra.
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    Cited by:

    1. Apostolos Burnetas, 2022. "Learning and data-driven optimization in queues with strategic customers," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 517-519, April.
    2. Masahiro Kato & Kaito Ariu, 2021. "The Role of Contextual Information in Best Arm Identification," Papers 2106.14077, arXiv.org, revised Feb 2024.

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