Asymptotically Minimax Optimal Fixed-Budget Best Arm Identification for Expected Simple Regret Minimization

We investigate fixed-budget best arm identification (BAI) for expected simple regret minimization. In each round of an adaptive experiment, a decision maker draws one of multiple treatment arms based on past observations and subsequently observes the outcomes of the chosen arm. After the experiment, the decision maker recommends a treatment arm with the highest projected outcome. We evaluate this decision in terms of the expected simple regret, a difference between the expected outcomes of the best and recommended treatment arms. Due to the inherent uncertainty, we evaluate the regret using the minimax criterion. For distributions with fixed variances (location-shift models), such as Gaussian distributions, we derive asymptotic lower bounds for the worst-case expected simple regret. Then, we show that the Random Sampling (RS)-Augmented Inverse Probability Weighting (AIPW) strategy proposed by Kato et al. (2022) is asymptotically minimax optimal in the sense that the leading factor of its worst-case expected simple regret asymptotically matches our derived worst-case lower bound. Our result indicates that, for location-shift models, the optimal RS-AIPW strategy draws treatment arms with varying probabilities based on their variances. This result contrasts with the results of Bubeck et al. (2011), which shows that drawing each treatment arm with an equal ratio is minimax optimal in a bounded outcome setting.