We study the causal bandit problem when the causal graph is unknown and develop an efficient algorithm for finding the parent node of the reward node using atomic interventions. We derive the exact equation for the expected number of interventions performed by the algorithm and show that under certain graphical conditions it could perform either logarithmically fast or, under more general assumptions, slower but still sublinearly in the number of variables. We formally show that our algorithm is optimal as it meets the universal lower bound we establish for any algorithm that performs atomic interventions. Finally, we extend our algorithm to the case when the reward node has multiple parents. Using this algorithm together with a standard algorithm from bandit literature leads to improved regret bounds.

翻译：我们研究了因果图未知情况下的因果赌博机问题，并开发了一种高效算法，通过原子干预寻找奖励节点的父节点。我们推导了该算法执行干预次数的精确期望方程，并证明在特定图结构条件下，干预次数可呈对数级增长；而在更一般假设下，虽速度较慢但仍保持次线性于变量数量。我们严格证明了该算法的最优性，因其达到了为所有原子干预算法建立的通用下界。进一步地，我们将算法扩展到奖励节点具有多个父节点的情形。将该算法与赌博机文献中的标准算法相结合，可实现更优的遗憾界。