Optimal Fixed-Budget Best Arm Identification using the Augmented Inverse Probability Weighting Estimator in Two-Armed Gaussian Bandits with Unknown Variances

Masahiro Kato, Kaito Ariu, Masaaki Imaizumi, Masatoshi Uehara, Masahiro Nomura, Chao Qin

Abstract: We consider fixed-budget best arm identification in two-armed bandit problems. One of the longstanding open questions is a tight lower bound on the probability of misidentifying the best arm and a strategy whose upper bound matches the lower bound when the optimal target allocation ratio of arm draws is unknown. We address this problem when the gap between the expected rewards is small. First, we introduce a distribution-dependent lower bound. Then, we propose the ``RS-AIPW'' strategy, which consists of the random sampling (RS) rule using the estimated optimal target allocation ratio and the recommendation rule using the augmented inverse probability weighting (AIPW) estimator. Our proposed strategy is optimal in the sense that the upper bound achieves the lower bound when the budget goes to infinity and the gap goes to zero. In the course of the analysis, we present a novel large deviation bound for martingales.