Lower Bounds for Policy Iteration on Multi-action MDPs
This provides theoretical lower bounds for multi-action MDPs, addressing a gap in existing literature that focused on two-action cases, which is incremental but important for algorithm analysis in reinforcement learning.
The paper tackles the problem of bounding the number of iterations for Policy Iteration algorithms on Markov Decision Processes with multiple actions, showing that a specific variant can require Ω(k^{n/2}) iterations to terminate.
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 $Ω(k^{n/2})$ 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.