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We address the problem of best arm identification (BAI) with a fixed budgetfor two-armed Gaussian bandits. In BAI, given multiple arms, we aim to find thebest arm, an arm with the highest expected reward, through an adaptiveexperiment. Kaufmann et al. (2016) develops a lower bound for the probabilityof misidentifying the best arm. They also propose a strategy, assuming that thevariances of rewards are known, and show that it is asymptotically optimal inthe sense that its probability of misidentification matches the lower bound asthe budget approaches infinity. However, an asymptotically optimal strategy isunknown when the variances are unknown. For this open issue, we propose astrategy that estimates variances during an adaptive experiment and draws armswith a ratio of the estimated standard deviations. We refer to this strategy asthe Neyman Allocation (NA)-Augmented Inverse Probability weighting (AIPW)strategy. We then demonstrate that this strategy is asymptotically optimal byshowing that its probability of misidentification matches the lower bound whenthe budget approaches infinity, and the gap between the expected rewards of twoarms approaches zero (small-gap regime). Our results suggest that under theworst-case scenario characterized by the small-gap regime, our strategy, whichemploys estimated variance, is asymptotically optimal even when the variancesare unknown.

Locally Optimal Fixed-Budget Best Arm Identification in Two-Armed Gaussian Bandits with Unknown Variances