General Discounting versus Average Reward

Abstract

Consider an agent interacting with an environment in cycles. In every interaction cycle the agent is rewarded for its performance. We compare the average reward U from cycle 1 to m (average value) with the future discounted reward V from cycle k to infinity (discounted value). We consider essentially arbitrary (non-geometric) discount sequences and arbitrary reward sequences (non-MDP environments). We show that asymptotically U for m->infinity and V for k->infinity are equal, provided both limits exist. Further, if the effective horizon grows linearly with k or faster, then existence of the limit of U implies that the limit of V exists. Conversely, if the effective horizon grows linearly with k or slower, then existence of the limit of V implies that the limit of U exists.