Theoretical remarks on feudal hierarchies and reinforcement learning

Diogo S. Carvalho, Francisco S. Melo, Pedro A. Santos

Abstract: Hierarchical reinforcement learning is an increasingly demanded resource for learning to make sequential decisions towards long term goals. Feudal hierarchies are among the most deployed frameworks. However, there are few theoretical results for hierarchical structures. In this work, we formalize the common two-level feudal hierarchy as two Markov decision processes, with the one on the high level being dependent on the policy executed at the low level. Despite the non-stationarity raised by the dependency, we show that each of the processes presents stable behavior. We then build on the first result to show that, regardless of the convergent learning algorithm used for the low level, convergence of both prediction and control algorithms at the high-level is guaranteed. Our results contribute with theoretical support for the use of feudal hierarchies in combination with standard reinforcement learning methods at each level.