Fast Direct Policy Evaluation Using Multiscale Analysis of Markov Diffusion Processes

Abstract

Policy evaluation is a critical step in the approximate solution of large Markov decision processes (MDPs), typically requiring O(|S|3) to directly solve the Bellman system of |S| linear equations (where |S| is the state space size in the discrete case, and the sample size in the continuous case). In this paper we apply a recently introduced multiscale framework for analysis on graphs to design a faster algorithm for policy evaluation. For a fixed policy π, this framework efficiently constructs a multiscale decomposition of the random walk Pπ associated with the policy π. This enables efficiently computing medium and long term state distributions, approximation of value functions, and the direct computation of the potential operator (I - γPπ)-1 needed to solve Bellman's equation. We show that even a preliminary non-optimized version of the solver competes with highly optimized iterative techniques, requiring in many cases a complexity of O(|S|).