Interactive Value Iteration for Markov Decision Processes with Unknown Rewards

Abstract

To tackle the potentially hard task of defining the reward function in a Markov Decision Process, we propose a new approach, based on Value Iteration, which interweaves the elicitation and optimization phases. We assume that rewards whose numeric values are unknown can only be ordered, and that a tutor is present to help comparing sequences of re- wards. We first show how the set of possible reward functions for a given preference relation can be rep- resented as a polytope. Then our algorithm, called Interactive Value Iteration, searches for an optimal policy while refining its knowledge about the pos- sible reward functions, by querying a tutor when necessary. We prove that the number of queries needed before finding an optimal policy is upper- bounded by a polynomial in the size of the problem, and we present experimental results which demon- strate that our approach is efficient in practice.