What is the significance of the Bellman equation in Reinforcement Learning?
Instruction: Explain the Bellman equation and its role in Reinforcement Learning.
Updated
Example Answer
A Bellman equation decomposes a long-term value into an immediate reward and the discounted value of what follows. That recursive structure lets us estimate returns without waiting for every full trajectory.
I distinguish two versions. A Bellman expectation equation evaluates a particular policy by averaging over its action choices and possible transitions. A Bellman optimality equation chooses the best available action. Solving the first gives the value of the specified policy; it does not by itself make that policy optimal.
This distinction appears directly in algorithms. Policy evaluation uses the current policy's expected continuation value, while a Q-learning target uses the maximum estimated next-action value. The estimates and transition samples can be noisy, so a Bellman equation is a relationship to satisfy, not evidence that a learned neural value function is already correct.
Worked Example
Suppose action A gives reward 2 and then ends. Action B gives reward 1 and reaches a state worth 4. With γ = 0.9, their one-step backed-up values are 2 and 1 + 0.9 × 4 = 4.6. A policy choosing each half the time has expected value 3.3; the optimal choice has value 4.6.