We study the real-valued combinatorial pure exploration of the multi-armed bandit in the fixed-budget setting. We first introduce the Combinatorial Successive Asign (CSA) algorithm, which is the first algorithm that can identify the best action even when the size of the action class is exponentially large with respect to the number of arms. We show that the upper bound of the probability of error of the CSA algorithm matches a lower bound up to a logarithmic factor in the exponent. Then, we introduce another algorithm named the Minimax Combinatorial Successive Accepts and Rejects (Minimax-CombSAR) algorithm for the case where the size of the action class is polynomial, and show that it is optimal, which matches a lower bound. Finally, we experimentally compare the algorithms with previous methods and show that our algorithm performs better.

翻译：我们研究了固定预算设置下多臂赌博机的实值组合纯探索问题。首先，我们提出了组合连续指派（CSA）算法，这是首个能够在动作类规模相对于臂数呈指数增长时识别最优动作的算法。我们证明CSA算法的错误概率上界与下界在指数上仅相差一个对数因子。接着，针对动作类规模为多项式的情况，我们提出了另一种名为极小化极大组合连续接受与拒绝（Minimax-CombSAR）的算法，并证明该算法是最优的，其性能与下界相匹配。最后，我们通过实验将所提算法与先前方法进行比较，结果表明我们的算法表现更优。