How does Bellman optimality differ from the regular Bellman equation?

Bellman optimality emphasizes finding the best policy that maximizes the expected cumulative reward over time, rather than just evaluating individual states as in the regular Bellman equation. In the context of reinforcement learning and dynamic programming, the Bellman equation provides a way to calculate the value of a state given a specific policy. In contrast, the Bellman optimality equation specifically looks for the highest values achievable across all possible policies, allowing one to derive the optimal policy ultimately.

By focusing on the principle of optimality, which states that any optimal policy should consist of optimal policies for subproblems, Bellman optimality sets the stage for determining a globally optimal policy rather than a locally optimal one. This means it provides a framework for improving policies iteratively, focusing on how to approach the best strategy instead of merely ranking the states based on their current values.

This distinction is significant in the design of algorithms that seek to optimize decision-making processes in environments with uncertainty and changing dynamics, such as Markov decision processes (MDPs).