Combinatorial Pure Exploration of Causal Bandits

The combinatorial pure exploration of causal bandits is the following online learning task: given a causal graph with unknown causal inference distributions, in each round we choose a subset of variables to intervene or do no intervention, and observe the random outcomes of all random variables, with the goal that using as few rounds as possible, we can output an intervention that gives the best (or almost best) expected outcome on the reward variable $Y$ with probability at least $1-\delta$, where $\delta$ is a given confidence level. We provide the first gap-dependent and fully adaptive pure exploration algorithms on two types of causal models --- the binary generalized linear model (BGLM) and general graphs. For BGLM, our algorithm is the first to be designed specifically for this setting and achieves polynomial sample complexity, while all existing algorithms for general graphs have either sample complexity exponential to the graph size or some unreasonable assumptions. For general graphs, our algorithm provides a significant improvement on sample complexity, and it nearly matches the lower bound we prove. Our algorithms achieve such improvement by a novel integration of prior causal bandit algorithms and prior adaptive pure exploration algorithms, the former of which utilize the rich observational feedback in causal bandits but are not adaptive to reward gaps, while the latter of which have the issue in reverse.