Causal Best Intervention Identification via Importance Sampling

Motivated by applications in computational advertising and systems biology, we consider the problem of identifying the best out of several possible soft interventions at a source node $V$ in a causal DAG, to maximize the expected value of a target node $Y$ (downstream of $V$). There is a fixed total budget for sampling under various interventions. Also, there are cost constraints on different types of interventions. We pose this as a best arm identification problem with $K$ arms, where each arm is a soft intervention at $V$. The key difference from the classical setting is that there is information leakage among the arms. Each soft intervention is a distinct known conditional probability distribution between $V$ and its parents $pa(V)$. We propose an efficient algorithm that uses importance sampling to adaptively sample using different interventions and leverage information leakage to choose the best. We provide the first gap dependent simple regret and best arm mis-identification error bounds for this problem. This generalizes the prior bounds available for the simpler case of no information leakage. In the case of no leakage, the number of samples required for identification is (upto polylog factors) $\tilde{O} (\max_i \frac{i}{\Delta_i^2})$ where $\Delta_i$ is the gap between the optimal and the $i$-th best arm. We generalize the previous result for the causal setting and show that $\tilde{O}(\max_i \frac{\sigma_i^2}{\Delta_i^2})$ is sufficient where $\sigma_i^2$ is the effective variance of an importance sampling estimator that eliminates the $i$-th best arm out of a set of arms with gaps roughly at most twice $\Delta_i$. We also show that $\sigma_i^2 << i$ in many cases. Our result also recovers (up to constants) prior gap independent bounds for this setting. We demonstrate that our algorithm empirically outperforms the state of the art, through synthetic experiments.