Polynomial Value Iteration Algorithms for Detrerminstic MDPs

TL;DR

This paper proves that value iteration for deterministic MDPs converges in polynomial time and introduces extensions that further improve efficiency, supported by empirical results showing fast convergence on sparse graphs.

Contribution

It establishes a polynomial bound for value iteration on deterministic MDPs and proposes two extensions that achieve near-linear time complexity.

Findings

Value iteration converges in heta(n^2) iterations for DMDPs.

Two extensions reduce convergence time to heta(mn).

Empirical results show faster convergence on sparse graphs.

Abstract

Value iteration is a commonly used and empirically competitive method in solving many Markov decision process problems. However, it is known that value iteration has only pseudo-polynomial complexity in general. We establish a somewhat surprising polynomial bound for value iteration on deterministic Markov decision (DMDP) problems. We show that the basic value iteration procedure converges to the highest average reward cycle on a DMDP problem in heta(n^2) iterations, or heta(mn^2) total time, where n denotes the number of states, and m the number of edges. We give two extensions of value iteration that solve the DMDP in heta(mn) time. We explore the analysis of policy iteration algorithms and report on an empirical study of value iteration showing that its convergence is much faster on random sparse graphs.