A Note on Performance Limitations in Bandit Problems With Side Information

We consider a sequential adaptive allocation problem which is formulated as a traditional two armed bandit problem but with one important modification: at each time step t, before selecting which arm to pull, the decision maker has access to a random variable X_t which provides information on the reward in each arm. Performance is measured as the fraction of time an inferior arm (generating lower mean reward) is pulled. We derive a minimax lower bound that proves that in the absence of sufficient statistical "diversity" in the distribution of the covariate X, a property that we shall refer to as lack of persistent excitation, no policy can improve on the best achievable performance in the traditional bandit problem without side information.