Asymptotic learnability of reinforcement problems with arbitrary dependence

We address the problem of reinforcement learning in which observations may exhibit an arbitrary form of stochastic dependence on past observations and actions, i.e. environments more general than (PO) MDPs. The task for an agent is to attain the best possible asymptotic reward where the true generating environment is unknown but belongs to a known countable family of environments. We find some sufficient conditions on the class of environments under which an agent exists which attains the best asymptotic reward for any environment in the class. We analyze how tight these conditions are and how they relate to different probabilistic assumptions known in reinforcement learning and related fields, such as Markov Decision Processes and mixing conditions.