Lower Bounds for Policy Iteration on Multi-action MDPs

TL;DR

This paper establishes new lower bounds on the number of iterations policy iteration algorithms require to solve multi-action Markov Decision Processes, extending previous results from two-action cases to more actions.

Contribution

It introduces lower bounds for policy iteration on MDPs with three or more actions, showing that the iteration count can grow exponentially with the number of states.

Findings

Policy iteration can take (k^{n/2}) iterations for certain multi-action MDPs.

Lower bounds are generalized from 2-action to k-action MDPs.

Randomized variants of policy iteration have lower bounds scaled by ( log(k)).

Abstract

Policy Iteration (PI) is a classical family of algorithms to compute an optimal policy for any given Markov Decision Problem (MDP). The basic idea in PI is to begin with some initial policy and to repeatedly update the policy to one from an improving set, until an optimal policy is reached. Different variants of PI result from the (switching) rule used for improvement. An important theoretical question is how many iterations a specified PI variant will take to terminate as a function of the number of states $n$ and the number of actions $k$ in the input MDP. While there has been considerable progress towards upper-bounding this number, there are fewer results on lower bounds. In particular, existing lower bounds primarily focus on the special case of $k = 2$ actions. We devise lower bounds for $k \geq 3$. Our main result is that a particular variant of PI can take $\Omega(k^{n/2})$ iterations to terminate. We also generalise existing constructions on $2$-action MDPs to scale lower bounds by a factor of $k$ for some common deterministic variants of PI, and by $\log(k)$ for corresponding randomised variants.