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