Causal Bandits with Propagating Inference

Bandit is a framework for designing sequential experiments, where a learner selects an arm $A \in \mathcal{A}$ and obtains an observation corresponding to $A$ in each experiment. Theoretically, the tight regret lower-bound for the general bandit is polynomial with respect to the number of arms $|\mathcal{A}|$, and thus, to overcome this bound, the bandit problem with side-information is often considered. Recently, a bandit framework over a causal graph was introduced, where the structure of the causal graph is available as side-information and the arms are identified with interventions on the causal graph. Existing algorithms for causal bandit overcame the $\Omega(\sqrt{|\mathcal{A}|/T})$ simple-regret lower-bound; however, their algorithms work only when the interventions $\mathcal{A}$ are localized around a single node (i.e., an intervention propagates only to its neighbors). We then propose a novel causal bandit algorithm for an arbitrary set of interventions, which can propagate throughout the causal graph. We also show that it achieves $O(\sqrt{ \gamma^*\log(|\mathcal{A}|T) / T})$ regret bound, where $\gamma^*$ is determined by using a causal graph structure. In particular, if the maximum in-degree of the causal graph is a constant, then $\gamma^* = O(N^2)$, where $N$ is the number of nodes.