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# A Tighter Analysis of Randomised Policy Iteration

*Proceedings of The 35th Uncertainty in Artificial Intelligence Conference*, PMLR 115:519-529, 2020.

#### Abstract

Policy Iteration (PI) is a popular family of algorithms to compute an optimal policy for a givenMarkov Decision Problem (MDP). Starting from an arbitrary initial policy, PI repeatedly performs locally-improving switches until an optimal policy is found. The exact form of the switching rule gives rise to different variants of PI. Two decades ago, Mansour and Singh [1999] provided the first non-trivial “strong” upper bound on the number of iterations taken by “Howard’s PI” (HPI), a widelyused variant of PI (strong bounds depend only on the number of states and actions in the MDP). They also proposed a randomised variant (RPI) and showed an even tighter strong upper bound. Their bounds for HPI and RPI have not been improved subsequently.\\{We} revisit the algorithms and analysis of Mansour and Singh [1999]. We prove a novel result on the structure of the policy space for k-action MDPs, $k\geq 2$, which generalises a result known for k = 2. Also proposing a new counting argument, we obtain a strong bound of (O$(\sqrt{k \log k}))^n$ iterations for an algorithm akin to RPI, improving significantly upon Mansour and Singh’s original bound of roughly O($(k/2)^n$). Similar analysis of a randomised variant of HPI also yields a strong upper bound of (O($\sqrt{k \log k}))^n$ iterations, registering the first exponential improvement for HPI over the trivial bound of $k^n$. Our other contributions include a lower bound of $\Omega(n)$ iterations for RPI and an upper bound of $1.6001^n$ iterations for a randomised variant of “Batch-Switching PI” [Kalyanakrishnan et al., 2016a] on 2-action MDPs—the tightest strong upper bound shown yet for the PI family.