Value Function

The value function in reinforcement learning and Markov Decision Processes (MDPs) quantifies the expected cumulative reward that an agent can obtain from a given state or state-action pair. It is denoted as V(s) for the state value function, which estimates the expected return from state s when following a particular policy, or Q(s, a) for the action value function, which estimates the expected return from taking action a in state s. The value function is computed using the Bellman equation, which establishes a recursive relationship between the value of a state and the values of its successor states. The formulation can be expressed as V(s) = R(s) + γ * Σ P(s'|s, a)V(s'), where R(s) is the immediate reward, γ is the discount factor, and P(s'|s, a) represents the transition probabilities to subsequent states. The value function is fundamental for evaluating and improving policies, as it provides a basis for algorithms such as Q-learning and policy gradient methods, enabling agents to learn optimal strategies over time.