Efficient Reward Identification In Max Entropy Reinforcement Learning with Sparsity and Rank Priors

In this paper, we consider the problem of recovering time-varying reward functions from either optimal policies or demonstrations coming from a max entropy reinforcement learning problem. This problem is highly ill-posed without additional assumptions on the underlying rewards. However, in many applications, the rewards are indeed parsimonious, and some prior information is available. We consider two such priors on the rewards: 1) rewards are mostly constant and they change infrequently, 2) rewards can be represented by a linear combination of a small number of feature functions. We first show that the reward identification problem with the former prior can be recast as a sparsification problem subject to linear constraints. Moreover, we give a polynomial-time algorithm that solves this sparsification problem exactly. Then, we show that identifying rewards representable with the minimum number of features can be recast as a rank minimization problem subject to linear constraints, for which convex relaxations of rank can be invoked. In both cases, these observations lead to efficient optimization-based reward identification algorithms. Several examples are given to demonstrate the accuracy of the recovered rewards as well as their generalizability.

Learning Reward Machines from Partially Observed Optimal Policies

Inverse reinforcement learning is the problem of inferring a reward function from an optimal policy. In this work, it is assumed that the reward is expressed as a reward machine whose transitions depend on atomic propositions associated with the state of a Markov Decision Process (MDP). Our goal is to identify the true reward machine using finite information. To this end, we first introduce the notion of a prefix tree policy which associates a distribution of actions to each state of the MDP and each attainable finite sequence of atomic propositions. Then, we characterize an equivalence class of reward machines that can be identified given the prefix tree policy. Finally, we propose a SAT-based algorithm that uses information extracted from the prefix tree policy to solve for a reward machine. It is proved that if the prefix tree policy is known up to a sufficient (but finite) depth, our algorithm recovers the exact reward machine up to the equivalence class. This sufficient depth is derived as a function of the number of MDP states and (an upper bound on) the number of states of the reward machine. Several examples are used to demonstrate the effectiveness of the approach.