What is a value function in RL?

A value function in reinforcement learning (RL) is a mathematical tool that estimates the expected cumulative reward an agent can achieve from a given state or state-action pair. It serves as a guide for the agent to evaluate the long-term desirability of states or actions, helping it make decisions that maximize total rewards. There are two primary types of value functions: the state-value function (V(s)), which estimates the expected return from a state, and the action-value function (Q(s,a)), which estimates the return from taking a specific action in a state. For example, in a grid-world game where an agent navigates to a goal, V(s) might assign higher values to states closer to the goal, while Q(s,a) would rank moving “up” or “right” as better actions in specific cells.

Value functions are foundational to many RL algorithms. They are often computed using the Bellman equations, which recursively break down the problem into immediate rewards and future discounted rewards. For instance, in Q-learning, the Q-value for a state-action pair is updated iteratively using the formula: Q(s,a) = Q(s,a) + α [r + γ * max Q(s',a') - Q(s,a)], where α is the learning rate, γ is the discount factor, and max Q(s',a') represents the best future value from the next state (s’). This approach allows agents to balance immediate rewards (like picking up a coin) against long-term goals (like reaching the end of a level). Algorithms like policy iteration and value iteration use these equations to refine value estimates until they converge to optimal values.

In practice, computing exact value functions becomes infeasible in large or continuous state spaces (e.g., video games with complex environments). To address this, developers often use function approximators like neural networks, as seen in Deep Q-Networks (DQN). For example, a DQN trained to play Atari games uses a neural network to approximate Q-values for each possible action (e.g., moving a paddle left or right) based on pixel inputs. Challenges include balancing exploration (trying new actions) and exploitation (using known high-value actions) and ensuring stable training. Despite these hurdles, value functions remain a core component of RL, enabling applications from robotics (e.g., optimizing movement paths) to recommendation systems (e.g., predicting user engagement over time).