A Generalized Reduced Linear Program for Markov Decision Processes
For researchers in approximate dynamic programming and reinforcement learning, this work provides a theoretical foundation for a practical method (RLP) that was previously lacking rigorous guarantees.
The paper addresses the computational challenge of solving large Markov decision processes (MDPs) by proposing a generalized reduced linear program (GRLP) that uses positive linear combinations of constraints. It provides theoretical error bounds for any GRLP, justifying linear constraint approximation, and demonstrates consistency with experiments in controlled queues.
Markov decision processes (MDPs) with large number of states are of high practical interest. However, conventional algorithms to solve MDP are computationally infeasible in this scenario. Approximate dynamic programming (ADP) methods tackle this issue by computing approximate solutions. A widely applied ADP method is approximate linear program (ALP) which makes use of linear function approximation and offers theoretical performance guarantees. Nevertheless, the ALP is difficult to solve due to the presence of a large number of constraints and in practice, a reduced linear program (RLP) is solved instead. The RLP has a tractable number of constraints sampled from the original constraints of the ALP. Though the RLP is known to perform well in experiments the theoretical guarantees are available only for a specific RLP obtained under idealized assumptions. In this paper, we generalize the RLP to define a generalized reduced linear program (GRLP) which has a tractable number of constraints that are obtained as positive linear combinations of the original constraints of the ALP. The main contribution of this paper is the novel theoretical framework developed to obtain error bounds for any given GRLP. Central to our framework are two $\max$-norm contraction operators. Our result solves theoretically justifies linear approximation of constraints. We discuss the implication of our results in the contexts of ADP and reinforcement learning. We also demonstrate via an example in the domain of controlled queues that the experiments conform to the theory.