What is the Bellman equation used for in reinforcement learning?

The Bellman equation is a fundamental equation in reinforcement learning that is used to express the relationship between the value of a state and the values of its successor states. It provides a recursive definition of the optimal value function, which represents the maximum expected cumulative reward that an agent can achieve starting from a given state. There are two main forms of the Bellman equation: the Bellman equation for value functions and the Bellman equation for Q-functions. The Bellman equation for value functions expresses the value of a state 's' as the immediate reward received plus the discounted value of the best possible next state 's''. Mathematically, it can be written as V(s) = max_a [R(s, a) + γ Σ P(s'|s, a) V(s')], where V(s) is the value of state 's', R(s, a) is the immediate reward received for taking action 'a' in state 's', γ is the discount factor (between 0 and 1), P(s'|s, a) is the probability of transitioning to state 's'' after taking action 'a' in state 's', and the summation is over all possible next states 's''. The Bellman equation for Q-functions expresses the value of taking action 'a' in state 's' as the immediate reward received plus the discounted value of the best possible next action in the next state. Mathematically, it can be written as Q(s, a) = R(s, a) + γ Σ P(s'|s, a) max_a' Q(s', a'), where Q(s, a) is the Q-value of taking action 'a' in state 's', and max_a' Q(s', a') represents the maximum Q-value achievable in the next state 's''. The Bellman equation is used in various reinforcement learning algorithms, such as Value Iteration and Q-learning, to iteratively update the value function or Q-function until they converge to the optimal values. For instance, in Value Iteration, the Bellman equation is applied repeatedly to update the value of each state until the changes become negligible, at which point the optimal value function has been found. In Q-learning, the Bellman equation is used to update the Q-value of each state-action pair based on the observed rewards and transitions, leading to an estimate of the optimal Q-function.