We study $(ε, δ)$-PAC best arm identification, where a decision-maker must identify an $ε$-optimal arm with probability at least $1 - δ$, while minimizing the number of arm pulls (samples). Most of the work on this topic is in the sequential setting, where there is no constraint on the time taken to identify such an arm; this allows the decision-maker to pull one arm at a time. In this work, the decision-maker is given a deadline of $T$ rounds, where, on each round, it can adaptively choose which arms to pull and how many times to pull them; this distinguishes the number of decisions made (i.e., time or number of rounds) from the number of samples acquired (cost). Such situations occur in clinical trials, where one may need to identify a promising treatment under a deadline while minimizing the number of test subjects, or in simulation-based studies run on the cloud, where we can elastically scale up or down the number of virtual machines to conduct as many experiments as we wish, but need to pay for the resource-time used. As the decision-maker can only make $T$ decisions, she may need to pull some arms excessively relative to a sequential algorithm in order to perform well on all possible problems. We formalize this added difficulty with two hardness results that indicate that unlike sequential settings, the ability to adapt to the problem difficulty is constrained by the finite deadline. We propose Elastic Batch Racing (EBR), a novel algorithm for this setting and bound its sample complexity, showing that EBR is optimal with respect to both hardness results. We present simulations evaluating EBR in this setting, where it outperforms baselines by several orders of magnitude.
PAC Identification of Many Good Arms in Stochastic Multi-Armed Bandits
Indian Institute of Technology Bombay（印度理工学院孟买分校）
研究了在$n$ -武装随机多臂强盗中从最佳$m$武装中识别任意$k$的问题.在PAC设置中，这个特定的问题概括了“最佳子集选择”和"从最佳m个臂中选择一个“的问题[arcsk 2017]。在众包和药物设计等应用中，确定单一的良好解决方案往往是不够的。此外，由于存在许多非常接近的解决方案，找到最佳子集可能很难。我们的概括识别出最好的$m$，其中$1 \leq k \leq m$，作为一个更有效的替代准确$k$武器。我们提出了一般$k$的最坏情况下的样本复杂度的下限，和一个完全顺序的PAC算法，\GLUCB，这是更简单的实例上的样本效率。此外，将我们的分析扩展到无限武装匪徒，我们提出了一种独立于$n$的PAC算法，该算法使用最多与下限相比的加性多对数样本数从最佳$ρ$部分的武器中识别出一个武器，从而改进了[arcsk 2017]和[Aziz+AKA：2018]。从最好的$ρ$分数识别$k > 1$不同的武器的问题并不总是定义良好的，对于这个问题的一个特殊的类，我们提出了下限和上限。最后，通过一个约化，我们建立了无限情况下的“one out of the best $ρ$”问题和有限情况下的“one out of the best $m$”问题的上界之间的关系。我们猜想，这是更有效地解决“小”有限的情况下，使用后者制定，而不是通过前者。
