Before revision:  In addition, the assumptions under which this regret guarantee is established are as weak as possible --- even a minimal weakening of them will imply non-existence of policies with regret increasing sub-linearly in the number of assignments. Furthermore, we provide an upper bound on the number of suboptimal assignments made by the FSA policy and show that every policy must make a number of suboptimal assignments at least of this order .
After revision:  In addition, the ETC type. Our results show, in particular, that the F-UCB policy (i) performs much better than any ETC policy, and (ii) is near minimax optimal. We also show that a suitably adapted F-UCB policy is near minimax optimal under minimal assumptions when covariate information is available .