Generalized Second-Order Value Iteration in Markov Decision Processes

Abstract

Value iteration is a fixed point iteration technique utilized to obtain the optimal value function and policy in a discounted reward Markov decision process (MDP). Here, a contraction operator is constructed and applied repeatedly to arrive at the optimal solution. Value iteration is a first-order method and, therefore, it may take a large number of iterations to converge to the optimal solution. Successive relaxation is a popular technique that can be applied to solve a fixed point equation. It has been shown in the literature that under a special structure of the MDP, successive overrelaxation technique computes the optimal value function faster than standard value iteration. In this article, we propose a second-order value iteration procedure that is obtained by applying the Newton–Raphson method to the successive relaxation value iteration scheme. We prove the global convergence of our algorithm to the optimal solution asymptotically and show the second-order convergence. Through experiments, we demonstrate the effectiveness of our proposed approach.