Constrained Best Arm Identification

In real-world decision-making problems, one needs to pick among multiple policies the one that performs best while respecting economic constraints. This motivates the problem of constrained best-arm identification for bandit problems where every arm is a joint distribution of reward and cost. We investigate the general case where reward and cost are dependent. The goal is to accurately identify the arm with the highest mean reward among all arms whose mean cost is below a given threshold. We prove information-theoretic lower bounds on the sample complexity for three models: Gaussian with fixed covariance, Gaussian with unknown covariance, and non-parametric distributions of rectangular support. We propose a combination of a sampling and a stopping rule that correctly identifies the constrained best arm and matches the optimal sample complexities for each of the three models. Simulations demonstrate the performance of our algorithms.