Reasoning and predicting POMDP planning complexity via covering numbers

Partially observable Markov decision processes (POMDPs) provide a rich mathematical framework for planning tasks in partially observable stochastic environments. The notion of the covering number, a metric of capturing the search space size of a POMDP planning problem, has been proposed as a complexity measure of approximate POMDP planning. Existing theoretical results are based on POMDPs with finite and discrete state spaces and measured in the l 1 -metric space. When considering heuristics, they are assumed to be always admissible. This paper extends the theoretical results on the covering numbers of different search spaces, including the newly defined space reachable under inadmissible heuristics, to the l n -metric spaces. We provide a simple but scalable algorithm for estimating covering numbers. Experimentally, we provide estimated covering numbers of the search spaces reachable by following different policies on several benchmark problems, and analyze their abilities to predict the runtime of POMDP planning algorithms.