Reinforcement Learning with Depreciating Assets

A basic assumption of traditional reinforcement learning is that the value of a reward does not change once it is received by an agent. The present work forgoes this assumption and considers the situation where the value of a reward decays proportionally to the time elapsed since it was obtained. Emphasizing the inflection point occurring at the time of payment, we use the term asset to refer to a reward that is currently in the possession of an agent. Adopting this language, we initiate the study of depreciating assets within the framework of infinite-horizon quantitative optimization. In particular, we propose a notion of asset depreciation, inspired by classical exponential discounting, where the value of an asset is scaled by a fixed discount factor at each time step after it is obtained by the agent. We formulate an equational characterization of optimality in this context, establish that optimal values and policies can be computed efficiently, and develop a model-free reinforcement learning approach to obtain optimal policies.