Monte Carlo Planning for Stochastic Control on Constrained Markov Decision Processes

In the world of stochastic control, especially in economics and engineering, Markov Decision Processes (MDPs) can effectively model various stochastic decision processes, from asset management to transportation optimization. These underlying MDPs, upon closer examination, often reveal a specifically constrained causal structure concerning the transition and reward dynamics. By exploiting this structure, we can obtain a reduction in the causal representation of the problem setting, allowing us to solve of the optimal value function more efficiently. This work defines an MDP framework, the SD-MDP, where we disentangle the causal structure of MDPs' transition and reward dynamics, providing distinct partitions on the temporal causal graph. With this stochastic reduction, the SD-MDP reflects a general class of resource allocation problems. This disentanglement further enables us to derive theoretical guarantees on the estimation error of the value function under an optimal policy by allowing independent value estimation from Monte Carlo sampling. Subsequently, by integrating this estimator into well-known Monte Carlo planning algorithms, such as Monte Carlo Tree Search (MCTS), we derive bounds on the simple regret of the algorithm. Finally, we quantify the policy improvement of MCTS under the SD-MDP framework by demonstrating that the MCTS planning algorithm achieves higher expected reward (lower costs) under a constant simulation budget, on a tangible economic example based on maritime refuelling.