Approximate Allocation Matching for Structural Causal Bandits with Unobserved Confounders

Structural causal bandit provides a framework for online decision-making problems when causal information is available. It models the stochastic environment with a structural causal model (SCM) that governs the causal relations between random variables. In each round, an agent applies an intervention (or no intervention) by setting certain variables to some constants and receives a stochastic reward from a non-manipulable variable. Though the causal structure is given, the observational and interventional distributions of these random variables are unknown beforehand, and they can only be learned through interactions with the environment. Therefore, to maximize the expected cumulative reward, it is critical to balance the explore-versus-exploit tradeoff. We assume each random variable takes a finite number of distinct values, and consider a semi-Markovian setting, where random variables are affected by unobserved confounders. Using the canonical SCM formulation to discretize the domains of unobserved variables, we efficiently integrate samples to reduce model uncertainty. This gives the decision maker a natural advantage over those in a classical multi-armed bandit setup. We provide a logarithmic asymptotic regret lower bound for the structural causal bandit problem. Inspired by the lower bound, we design an algorithm that can utilize the causal structure to accelerate the learning process and take informative and rewarding interventions. We establish that our algorithm achieves a logarithmic regret and demonstrate that it outperforms the existing methods via simulations.