What is the Bellman Equation?
The Bellman equation is a fundamental recursive equation in Reinforcement Learning (RL) that defines the relationship between the value of a state and the values of its successor states. It is crucial for solving Markov Decision Processes (MDPs), where an agent learns to make decisions by maximizing cumulative rewards.
Formally, the Bellman equation expresses the value of a state \( V(s) \) as the expected reward from that state plus the discounted value of future states, mathematically represented as:
\( V(s) = R(s) + \gamma \sum_{s'} P(s'|s,a)V(s') \)
Here, \( R(s) \) denotes the immediate reward received upon reaching state \( s \), \( \gamma \) represents the discount factor (0 ≤ γ < 1), capturing the importance of future rewards, and \( P(s'|s,a) \) indicates the transition probability to the next state \( s' \) given the current state \( s \) and action \( a \).
The Bellman equation lays the groundwork for various RL algorithms, enabling the estimation of value functions through iterative methods such as Dynamic Programming and Q-learning. By solving the Bellman equation, an agent can derive optimal policies that maximize rewards in uncertain environments.
Understanding the Bellman equation is essential for anyone involved in the development of intelligent systems, as it encapsulates the core principles of decision-making under uncertainty.