On The Expressivity of Objective-Specification Formalisms in Reinforcement Learning
This work addresses the need for a cohesive understanding of objective-specification formalisms in RL, which is crucial for researchers and practitioners when specifying tasks, but it is incremental as it fills a gap in existing literature without introducing new methods.
The paper tackles the problem of comparing the expressivity of various objective-specification formalisms in reinforcement learning, such as Linear Temporal Logic and Multi-Objective Reinforcement Learning, by providing a comprehensive analysis of 17 formalisms and placing them in a preorder based on expressive power, revealing that no single formalism is both dominantly expressive and easy to optimize with current techniques.
Most algorithms in reinforcement learning (RL) require that the objective is formalised with a Markovian reward function. However, it is well-known that certain tasks cannot be expressed by means of an objective in the Markov rewards formalism, motivating the study of alternative objective-specification formalisms in RL such as Linear Temporal Logic and Multi-Objective Reinforcement Learning. To date, there has not yet been any thorough analysis of how these formalisms relate to each other in terms of their expressivity. We fill this gap in the existing literature by providing a comprehensive comparison of 17 salient objective-specification formalisms. We place these formalisms in a preorder based on their expressive power, and present this preorder as a Hasse diagram. We find a variety of limitations for the different formalisms, and argue that no formalism is both dominantly expressive and straightforward to optimise with current techniques. For example, we prove that each of Regularised RL, (Outer) Nonlinear Markov Rewards, Reward Machines, Linear Temporal Logic, and Limit Average Rewards can express a task that the others cannot. The significance of our results is twofold. First, we identify important expressivity limitations to consider when specifying objectives for policy optimization. Second, our results highlight the need for future research which adapts reward learning to work with a greater variety of formalisms, since many existing reward learning methods assume that the desired objective takes a Markovian form. Our work contributes towards a more cohesive understanding of the costs and benefits of different RL objective-specification formalisms.