The Value Function Polytope in Reinforcement Learning
This provides a foundational geometric framework for analyzing reinforcement learning, which is incremental but clarifies theoretical properties for researchers.
The paper characterizes the shape of the value function space in finite Markov decision processes as a general polytope, and uses this geometric insight to introduce visualizations for understanding reinforcement learning algorithm dynamics.
We establish geometric and topological properties of the space of value functions in finite state-action Markov decision processes. Our main contribution is the characterization of the nature of its shape: a general polytope (Aigner et al., 2010). To demonstrate this result, we exhibit several properties of the structural relationship between policies and value functions including the line theorem, which shows that the value functions of policies constrained on all but one state describe a line segment. Finally, we use this novel perspective to introduce visualizations to enhance the understanding of the dynamics of reinforcement learning algorithms.