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Best Arm Identification with Contextual Information under a Small Gap

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  • Masahiro Kato
  • Masaaki Imaizumi
  • Takuya Ishihara
  • Toru Kitagawa

Abstract

We study the best-arm identification (BAI) problem with a fixed budget and contextual (covariate) information. In each round of an adaptive experiment, after observing contextual information, we choose a treatment arm using past observations and current context. Our goal is to identify the best treatment arm, which is a treatment arm with the maximal expected reward marginalized over the contextual distribution, with a minimal probability of misidentification. In this study, we consider a class of nonparametric bandit models that converge to location-shift models when the gaps go to zero. First, we derive lower bounds of the misidentification probability for a certain class of strategies and bandit models (probabilistic models of potential outcomes) under a small-gap regime. A small-gap regime is a situation where gaps of the expected rewards between the best and suboptimal treatment arms go to zero, which corresponds to one of the worst cases in identifying the best treatment arm. We then develop the ``Random Sampling (RS)-Augmented Inverse Probability weighting (AIPW) strategy,'' which is asymptotically optimal in the sense that the probability of misidentification under the strategy matches the lower bound when the budget goes to infinity in the small-gap regime. The RS-AIPW strategy consists of the RS rule tracking a target sample allocation ratio and the recommendation rule using the AIPW estimator.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:2209.07330
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    References listed on IDEAS

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    1. Jinyong Hahn & Keisuke Hirano & Dean Karlan, 2011. "Adaptive Experimental Design Using the Propensity Score," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 29(1), pages 96-108, January.
    2. Karlan, Dean & Wood, Daniel H., 2017. "The effect of effectiveness: Donor response to aid effectiveness in a direct mail fundraising experiment," Journal of Behavioral and Experimental Economics (formerly The Journal of Socio-Economics), Elsevier, vol. 66(C), pages 1-8.
    3. Hidehiko Ichimura & Whitney K. Newey, 2022. "The influence function of semiparametric estimators," Quantitative Economics, Econometric Society, vol. 13(1), pages 29-61, January.
    4. Keisuke Hirano & Jack R. Porter, 2009. "Asymptotics for Statistical Treatment Rules," Econometrica, Econometric Society, vol. 77(5), pages 1683-1701, September.
    5. Dehejia, Rajeev H., 2005. "Program evaluation as a decision problem," Journal of Econometrics, Elsevier, vol. 125(1-2), pages 141-173.
    6. Shantanu Gupta & Zachary C. Lipton & David Childers, 2021. "Efficient Online Estimation of Causal Effects by Deciding What to Observe," Papers 2108.09265, arXiv.org, revised Oct 2021.
    7. Maximilian Kasy & Anja Sautmann, 2021. "Adaptive Treatment Assignment in Experiments for Policy Choice," Econometrica, Econometric Society, vol. 89(1), pages 113-132, January.
    8. Toru Kitagawa & Aleksey Tetenov, 2018. "Who Should Be Treated? Empirical Welfare Maximization Methods for Treatment Choice," Econometrica, Econometric Society, vol. 86(2), pages 591-616, March.
    9. Manski, Charles F., 2000. "Identification problems and decisions under ambiguity: Empirical analysis of treatment response and normative analysis of treatment choice," Journal of Econometrics, Elsevier, vol. 95(2), pages 415-442, April.
    10. Maria Dimakopoulou & Zhimei Ren & Zhengyuan Zhou, 2021. "Online Multi-Armed Bandits with Adaptive Inference," Papers 2102.13202, arXiv.org, revised Jun 2021.
    11. Annie Liang & Xiaosheng Mu & Vasilis Syrgkanis, 2019. "Dynamically Aggregating Diverse Information," PIER Working Paper Archive 19-005, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
    12. Masahiro Kato & Kaito Ariu, 2021. "The Role of Contextual Information in Best Arm Identification," Papers 2106.14077, arXiv.org, revised Feb 2024.
    13. 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.
    14. Jinyong Hahn, 1998. "On the Role of the Propensity Score in Efficient Semiparametric Estimation of Average Treatment Effects," Econometrica, Econometric Society, vol. 66(2), pages 315-332, March.
    15. Heejung Bang & James M. Robins, 2005. "Doubly Robust Estimation in Missing Data and Causal Inference Models," Biometrics, The International Biometric Society, vol. 61(4), pages 962-973, December.
    16. Athey, Susan & Wager, Stefan, 2017. "Efficient Policy Learning," Research Papers 3506, Stanford University, Graduate School of Business.
    17. Imbens,Guido W. & Rubin,Donald B., 2015. "Causal Inference for Statistics, Social, and Biomedical Sciences," Cambridge Books, Cambridge University Press, number 9780521885881.
    18. Kaito Ariu & Masahiro Kato & Junpei Komiyama & Kenichiro McAlinn & Chao Qin, 2021. "Policy Choice and Best Arm Identification: Asymptotic Analysis of Exploration Sampling," Papers 2109.08229, arXiv.org, revised Nov 2021.
    19. Masahiro Kato & Shota Yasui & Kenichiro McAlinn, 2020. "The Adaptive Doubly Robust Estimator for Policy Evaluation in Adaptive Experiments and a Paradox Concerning Logging Policy," Papers 2010.03792, arXiv.org, revised Jun 2021.
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