Improved and Generalized Upper Bounds on the Complexity of Policy Iteration

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 optimalγ-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 Oleft(fracm1-γlogleft(frac11-γright)right)iterations, improving by a factor O(log n) a result by Hansen etal., while Simplex-PI terminates after at most Oleft(fracnm1-γlogleft(frac11-γright)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~γ: quantities ofinterest are bounds τt and τr---uniform on all states andpolicies---respectively on the expected time spent in transientstates and 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 Oleft(n3 m2 τt τr right) iterations. This extends arecent result for deterministic MDPs by Post & Ye, in which τt≤ 1 and τr ≤ 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(n2τt+nτr))iterations.