Motivated by real-world applications that necessitate responsible experimentation, we introduce the problem of best arm identification (BAI) with minimal regret. This variant of the multi-armed bandit problem elegantly amalgamates two of its most ubiquitous objectives: regret minimization and BAI. More precisely, the agent's goal is to identify the best arm with a prescribed confidence level $δ$, while minimizing the cumulative regret up to the stopping time. Focusing on single-parameter exponential families of distributions, we leverage information-theoretic techniques to establish an instance-dependent lower bound on the expected cumulative regret. Moreover, we present an impossibility result that underscores the tension between cumulative regret and sample complexity in fixed-confidence BAI. Complementarily, we design and analyze the Double KL-UCB algorithm, which achieves asymptotic optimality as the confidence level tends to zero. Notably, this algorithm employs two distinct confidence bounds to guide arm selection in a randomized manner. Our findings elucidate a fresh perspective on the inherent connections between regret minimization and BAI.

翻译：受现实世界中需要负责任实验的应用启发，我们提出了最小遗憾的最佳臂识别（BAI）问题。这一多臂老虎机问题的变体优雅地融合了其两个最广泛的目标：遗憾最小化和BAI。具体而言，智能体的目标是以指定的置信水平δ识别最佳臂，同时最小化停止时刻前的累积遗憾。聚焦于单参数指数族分布，我们利用信息论技术建立了期望累积遗憾的实例相关下界。此外，我们展示了一个不可能性结果，揭示了固定置信度BAI中累积遗憾与样本复杂度之间的张力。互补地，我们设计并分析了双KL-UCB算法，该算法在置信水平趋于零时实现渐近最优性。值得注意的是，该算法采用两种不同的置信界以随机方式引导臂选择。我们的发现阐明了遗憾最小化与BAI之间内在联系的新视角。