Distributed Stochastic Bandits with Corrupted and Defective Input Commands

We analyze a distributed stochastic bandit model in which an agent controls multiple independent stochastic bandit machines. At each time step, the agent selects several machines for parallel exploitation but the arm pulled by each machine may differ from the command received either randomly (defective command) or adversarially (corrupted command). Machines that faithfully execute commands are called honest. We study situations in which the number of honest machines is either known or unknown and define appropriate notions of regret. With at least one honest machine and a known number of honest bandits, we provide a simple algorithm that achieves $\tilde O\left( {{n^{1/2}}} \right)$ regrets when commands are corrupted. Lower bounds on regret established by drawing connections to the problem of "low probability of detection," show the near optimality of the regret achieved by the algorithms.