Markov Decision Processes under External Temporal Processes

Reinforcement Learning Algorithms are predominantly developed for stationary environments, and the limited literature that considers nonstationary environments often involves specific assumptions about changes that can occur in transition probability matrices and reward functions. Considering that real-world applications involve environments that continuously evolve due to various external events, and humans make decisions by discerning patterns in historical events, this study investigates Markov Decision Processes under the influence of an external temporal process. We establish the conditions under which the problem becomes tractable, allowing it to be addressed by considering only a finite history of events, based on the properties of the perturbations introduced by the exogenous process. We propose and theoretically analyze a policy iteration algorithm to tackle this problem, which learns policies contingent upon the current state of the environment, as well as a finite history of prior events of the exogenous process. We show that such an algorithm is not guaranteed to converge. However, we provide a guarantee for policy improvement in regions of the state space determined by the approximation error induced by considering tractable policies and value functions. We also establish the sample complexity of least-squares policy evaluation and policy improvement algorithms that consider approximations due to the incorporation of only a finite history of temporal events. While our results are applicable to general discrete-time processes satisfying certain conditions on the rate of decay of the influence of their events, we further analyze the case of discrete-time Hawkes processes with Gaussian marks. We performed experiments to demonstrate our findings for policy evaluation and deployment in traditional control environments.