We address the problem of identifying the optimal policy with a fixed confidence level in a multi-armed bandit setup, when \emph{the arms are subject to linear constraints}. Unlike the standard best-arm identification problem which is well studied, the optimal policy in this case may not be deterministic and could mix between several arms. This changes the geometry of the problem which we characterize via an information-theoretic lower bound. We introduce two asymptotically optimal algorithms for this setting, one based on the Track-and-Stop method and the other based on a game-theoretic approach. Both these algorithms try to track an optimal allocation based on the lower bound and computed by a weighted projection onto the boundary of a normal cone. Finally, we provide empirical results that validate our bounds and visualize how constraints change the hardness of the problem.

翻译：我们研究了在多臂赌博机框架下以固定置信度识别最优策略的问题，其中臂受到线性约束。与已被充分研究的标准最佳臂识别问题不同，此情形下的最优策略可能非确定性的，而会在多个臂之间进行混合。这改变了问题的几何结构，我们通过信息论下界对其进行了刻画。针对该设置，我们提出了两种渐近最优算法：一种基于跟踪-停止方法，另一种基于博弈论方法。这两种算法均试图基于下界并通过加权投影至法锥边界来计算最优分配。最后，我们提供了实证结果，验证了所提界并展示了约束如何改变问题的难度。