Learning in Reactive Environments with Arbitrary Dependence

In reinforcement learning 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. This task generalises the sequence prediction problem, in which the environment does not react to the behaviour of the agent. Solomonoff induction solves the sequence prediction problem for any countable class of measures; however, it is easy to see that such result is impossible for reinforcement learning - not any countable class of environments can be learnt. 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