We consider the best-k-arm identification problem for multi-armed bandits, where the objective is to select the exact set of k arms with the highest mean rewards by sequentially allocating measurement effort. We characterize the necessary and sufficient conditions for the optimal allocation using dual variables. Remarkably these optimality conditions lead to the extension of top-two algorithm design principle (Russo, 2020), initially proposed for best-arm identification. Furthermore, our optimality conditions induce a simple and effective selection rule dubbed information-directed selection (IDS) that selects one of the top-two candidates based on a measure of information gain. As a theoretical guarantee, we prove that integrated with IDS, top-two Thompson sampling is (asymptotically) optimal for Gaussian best-arm identification, solving a glaring open problem in the pure exploration literature (Russo, 2020). As a by-product, we show that for k > 1, top-two algorithms cannot achieve optimality even with an oracle tuning parameter. Numerical experiments show the superior performance of the proposed top-two algorithms with IDS and considerable improvement compared with algorithms without adaptive selection.

翻译：我们考虑多臂老虎机中的最佳k臂识别问题，其目标是通过顺序分配测量资源，精确选出具有最高平均奖励的k个臂的集合。我们利用对偶变量刻画了最优分配的必要和充分条件。值得注意的是，这些最优条件导致了最初为最佳臂识别提出的Top-Two算法设计原则（Russo, 2020）的扩展。此外，我们的最优条件催生了一种简单而有效的选择规则，称为信息导向选择（IDS），该规则基于信息增益度量从两个候选臂中选择一个。作为理论保证，我们证明集成IDS后，Top-Two汤普森采样在高斯最佳臂识别问题中是（渐近）最优的，从而解决了纯探索文献中一个显著未解决的问题（Russo, 2020）。作为副产品，我们表明当k > 1时，即使使用预言机调优参数，Top-Two算法也无法达到最优性。数值实验表明，所提出的带有IDS的Top-Two算法具有优越性能，并且与无自适应选择的算法相比有显著改进。