Robust control of the multi-armed bandit problem

We study a robust model of the multi-armed bandit (MAB) problem in which the transition probabilities are ambiguous and belong to subsets of the probability simplex. We first show that for each arm there exists a robust counterpart of the Gittins index that is the solution to a robust optimal stopping-time problem and can be computed effectively with an equivalent restart problem. We then characterize the optimal policy of the robust MAB as a project-by-project retirement policy but we show that arms become dependent so the policy based on the robust Gittins index is not optimal. For a project selection problem, we show that the robust Gittins index policy is near optimal but its implementation requires more computational effort than solving a non-robust MAB problem. Hence, we propose a Lagrangian index policy that requires the same computational effort as evaluating the indices of a non-robust MAB and is within 1 % of the optimum in the robust project selection problem.