Analysis of Lower Bounds for Simple Policy Iteration

Policy iteration is a family of algorithms that are used to find an optimal policy for a given Markov Decision Problem (MDP). Simple Policy iteration (SPI) is a type of policy iteration where the strategy is to change the policy at exactly one improvable state at every step. Melekopoglou and Condon [1990] showed an exponential lower bound on the number of iterations taken by SPI for a 2 action MDP. The results have not been generalized to $k-$action MDP since. In this paper, we revisit the algorithm and the analysis done by Melekopoglou and Condon. We generalize the previous result and prove a novel exponential lower bound on the number of iterations taken by policy iteration for $N-$state, $k-$action MDPs. We construct a family of MDPs and give an index-based switching rule that yields a strong lower bound of $\mathcal{O}\big((3+k)2^{N/2-3}\big)$.