Reinforcement learning for long-run average cost

A large class of sequential decision-making problems under uncertainty can be modeled as Markov and semi-Markov decision problems (SMDPs), when their underlying probability structure has a Markov chain. They may be solved by using classical dynamic programming (DP) methods. However, DP methods suffer from the curse of dimensionality and break down rapidly in face of large state-spaces. In addition, DP methods require the exact computation of the so-called transition probabilities, which are often hard to obtain and are hence said to suffer from the curse of modeling as well. In recent years, a simulation-based method, called reinforcement learning (RL), has emerged in the literature. It can, to a great extent, alleviate stochastic DP of its curses by generating ‘near-optimal’ solutions to problems having large state-spaces and complex transition mechanisms. In this paper, a simulation-based algorithm that solves Markov and SMDPs is presented, along with its convergence analysis. The algorithm involves a step-size based transformation on two-time scales. Its convergence analysis is based on a recent result on asynchronous convergence of iterates on two-time scales. We present numerical results from the new algorithm on a classical preventive maintenance case study of a reasonable size, where results on the optimal policy are also available. In addition, we present a tutorial that explains the framework of RL in the context of long-run average cost SMDPs.