Q-Function

The Q-Function, or state-action value function, is a fundamental concept in reinforcement learning that quantifies the expected return (cumulative future rewards) of taking a specific action in a given state and following a particular policy thereafter. Mathematically, the Q-Function is defined as Q(s, a) = E[R_t | S_t = s, A_t = a], where R_t is the total reward received after taking action A_t in state S_t. The Q-Function is central to algorithms such as Q-learning, which employs the Bellman equation to iteratively update Q-values based on observed rewards and estimated future values. The relationship between the Q-Function and the optimal value function is established through the Bellman optimality equation, which states that Q*(s, a) = R(s, a) + γ Σ P(s'|s, a) V*(s'), where γ is the discount factor, P(s'|s, a) is the transition probability, and V*(s') is the optimal value function for the next state. The Q-Function is crucial for deriving optimal policies in Markov Decision Processes (MDPs) and is foundational for various reinforcement learning techniques, including deep Q-networks (DQN).