Improved and Generalized Upper Bounds on the Complexity of Policy Iteration

TL;DR

This paper provides improved upper bounds on the number of iterations for policy iteration algorithms in Markov Decision Processes, considering both discount factors and structural properties, demonstrating strong polynomiality under certain conditions.

Contribution

It introduces tighter bounds for Howard's and Simplex-PI, including discount-independent bounds based on structural properties, and extends results to broader classes of MDPs.

Findings

Howard's PI terminates after at most O(m/(1-γ) log(1/(1-γ))) iterations.

Simplex-PI terminates after at most O(nm/(1-γ) log(1/(1-γ))) iterations.

Under structural assumptions, Simplex-PI is strongly polynomial, and bounds are provided for both algorithms.

Abstract

Given a Markov Decision Process (MDP) with $n$ states and a totalnumber $m$ of actions, we study the number of iterations needed byPolicy Iteration (PI) algorithms to converge to the optimal$\gamma$-discounted policy. We consider two variations of PI: Howard'sPI that changes the actions in all states with a positive advantage,and Simplex-PI that only changes the action in the state with maximaladvantage. We show that Howard's PI terminates after at most $O\left(\frac{m}{1-\gamma}\log\left(\frac{1}{1-\gamma}\right)\right)$iterations, improving by a factor $O(\log n)$ a result by Hansen etal., while Simplex-PI terminates after at most $O\left(\frac{nm}{1-\gamma}\log\left(\frac{1}{1-\gamma}\right)\right)$iterations, improving by a factor $O(\log n)$ a result by Ye. Undersome structural properties of the MDP, we then consider bounds thatare independent of the discount factor~$\gamma$: quantities ofinterest are bounds $\tau\_t$ and $\tau\_r$---uniform on all states andpolicies---respectively on the \emph{expected time spent in transientstates} and \emph{the inverse of the frequency of visits in recurrentstates} given that the process starts from the uniform distribution.Indeed, we show that Simplex-PI terminates after at most $\tilde O\left(n^3 m^2 \tau\_t \tau\_r \right)$ iterations. This extends arecent result for deterministic MDPs by Post & Ye, in which $\tau\_t\le 1$ and $\tau\_r \le n$, in particular it shows that Simplex-PI isstrongly polynomial for a much larger class of MDPs. We explain whysimilar results seem hard to derive for Howard's PI. Finally, underthe additional (restrictive) assumption that the state space ispartitioned in two sets, respectively states that are transient andrecurrent for all policies, we show that both Howard's PI andSimplex-PI terminate after at most $\tilde O(m(n^2\tau\_t+n\tau\_r))$iterations.