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 (, 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.
The dual formulation consists of maximizing the expected return for a given initial state distribution, as a function of the discounted state action visit frequency distribution. Contrary to the primal form, any feasible discounted state action visit frequency distribution maps to an policy @cite_9 .
@cite_9
2149418961
We investigate the dual approach to dynamic programming and reinforcement learning, based on maintaining an explicit representation of stationary distributions as opposed to value functions. A significant advantage of the dual approach is that it allows one to exploit well developed techniques for representing, approximating and estimating probability distributions, without running the risks associated with divergent value function estimation. A second advantage is that some distinct algorithms for the average reward and discounted reward case in the primal become unified under the dual. In this paper, we present a modified dual of the standard linear program that guarantees a globally normalized state visit distribution is obtained. With this reformulation, we then derive novel dual forms of dynamic programming, including policy evaluation, policy iteration and value iteration. Moreover, we derive dual formulations of temporal difference learning to obtain new forms of Sarsa and Q-learning. Finally, we scale these techniques up to large domains by introducing approximation, and develop new approximate off-policy learning algorithms that avoid the divergence problems associated with the primal approach. We show that the dual view yields a viable alternative to standard value function based techniques and opens new avenues for solving dynamic programming and reinforcement learning problems