Locally Optimal Fixed-Budget Best Arm Identification in Two-Armed Gaussian Bandits with Unknown Variances
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未知方差双臂高斯老虎机中固定预算最优臂识别的局部最优性
- 研究问题
- 在双臂高斯老虎机中，奖励方差未知且实验预算固定时，如何自适应分配抽样以降低选错最高期望奖励臂的概率？
- 方法与思路
- 作者在实验中估计奖励方差，按估计标准差之比分配抽样，并结合增广逆概率加权，构造 Neyman 分配与 AIPW 策略。
- 材料陈述的贡献
- 摘要称，当预算趋于无穷且两臂期望奖励差趋于零时，该策略的误识别概率匹配既有下界，从而获得小差距情形下的渐近最优性。
- 阅读关注点
- 应核对预算与奖励差距趋近的联合条件、方差估计和抽样初始化、概率下界的适用范围，以及局部结果能否支持有限预算表现。
- 阅读边界
- 这是 arXiv 预印本，同行评审或期刊接收未获核实；最优性限定于摘要所述高斯双臂与小差距渐近情形，有限预算保证未说明。
原文摘要
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.