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Optimal Best Arm Identification in Two-Armed Bandits with a Fixed Budget under a Small Gap

Author

Listed:
  • Masahiro Kato
  • Kaito Ariu
  • Masaaki Imaizumi
  • Masahiro Nomura
  • Chao Qin

Abstract

We consider fixed-budget best-arm identification in two-armed Gaussian bandit problems. One of the longstanding open questions is the existence of an optimal strategy under which the probability of misidentification matches a lower bound. We show that a strategy following the Neyman allocation rule (Neyman, 1934) is asymptotically optimal when the gap between the expected rewards is small. First, we review a lower bound derived by Kaufmann et al. (2016). Then, we propose the "Neyman Allocation (NA)-Augmented Inverse Probability weighting (AIPW)" strategy, which consists of the sampling rule using the Neyman allocation with an estimated standard deviation and the recommendation rule using an AIPW estimator. Our proposed strategy is optimal because the upper bound matches the lower bound when the budget goes to infinity and the gap goes to zero.

Suggested Citation

  • Masahiro Kato & Kaito Ariu & Masaaki Imaizumi & Masahiro Nomura & Chao Qin, 2022. "Optimal Best Arm Identification in Two-Armed Bandits with a Fixed Budget under a Small Gap," Papers 2201.04469, arXiv.org, revised Dec 2022.
    • Handle: RePEc:arx:papers:2201.04469
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    File URL: http://arxiv.org/pdf/2201.04469
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    Cited by:

    1. Masahiro Kato & Masaaki Imaizumi & Takuya Ishihara & Toru Kitagawa, 2022. "Best Arm Identification with Contextual Information under a Small Gap," Papers 2209.07330, arXiv.org, revised Jan 2023.
    2. Karun Adusumilli, 2022. "Neyman allocation is minimax optimal for best arm identification with two arms," Papers 2204.05527, arXiv.org, revised Aug 2022.
    3. Jinglong Zhao, 2023. "Adaptive Neyman Allocation," Papers 2309.08808, arXiv.org, revised Sep 2023.

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