Best of both worlds: Stochastic & adversarial best-arm identification.

We study a version of the bandit best-arm identification problem with potentially adversarial rewards. A simple random uniform strategy obtains the optimal rate of error in the adversarial scenario. However, this type of strategy is sub-optimal when the rewards are sampled stochastically. Can we design a learner that performs optimally in both the stochastic and adversarial problems while not being aware of the nature of the rewards? First, we show that designing such learner is impossible in general: to be robust to adversarial rewards, we can only guarantee optimal rates of error on a subset of the stochastic problems. We show a lower bound that characterizes the optimal rate in stochastic problems if the strategy is constrained to be robust to adversarial rewards. Finally, we design a simple parameter-free algorithm and show that its probability of error matches (up to log factors) the lower bound in stochastic problems, and it is also robust to adversarial problems.