The Bellman equation in reinforcement learning (RL) is a fundamental recursive equation used to calculate the value function. It expresses the relationship between the value of a state and the values of its possible successor states, based on the reward function and the expected future rewards.
The Bellman equation allows the agent to break down the problem of estimating the value of a state into smaller subproblems, making it possible to compute the value of each state iteratively. For a given state 𝑠, the value 𝑉(𝑠) is calculated as the immediate reward plus the expected value of the next state, discounted by a factor that reflects the agent’s preference for short-term vs. long-term rewards. The equation is typically written as: 𝑉(𝑠)=𝑅(𝑠)+𝛾⋅max𝑎𝑉(𝑠′), where 𝑅(𝑠) is the immediate reward, 𝛾 is the discount factor, and 𝑠′ is the next state.
The Bellman equation is the foundation of many RL algorithms, including value iteration and Q-learning. It provides a way to iteratively improve the agent’s value estimates, helping it find the optimal policy for decision-making.