On the Limitations of Markovian Rewards to Express Multi-Objective, Risk-Sensitive, and Modal Tasks

Joar Max Viktor Skalse, Alessandro Abate

Keywords: multi-objective reinforcement learning, morl, risk-sensitive reinforcement learning, risk-averse reinforcement learning, reward hypothesis

TL;DR: We study three classes of tasks (multi-objective tasks, risk-sensitive tasks, and modal tasks), and provide necessary and sufficient conditions for when these tasks can be expressed using scalar, Markovian reward functions.

Abstract: In this paper, we study the expressivity of scalar, Markovian reward functions in Reinforcement Learning (RL), and identify several limitations to what they can express. Specifically, we look at three classes of RL tasks; multi-objective RL, risk-sensitive RL, and modal RL. For each class, we derive necessary and sufficient conditions that describe when a problem in this class can be expressed using a scalar, Markovian reward. Moreover, we find that scalar, Markovian rewards are unable to express most of the instances in each of these three classes. We thereby contribute to a more complete understanding of what standard reward functions can and cannot express. In addition to this, we also call attention to modal problems as a new class of problems, since they have so far not been given any systematic treatment in the RL literature. We also briefly outline some approaches for solving some of the problems we discuss, by means of bespoke RL algorithms.