What is the value function in reinforcement learning?

The value function in reinforcement learning (RL) quantifies the expected long-term reward an agent can accumulate starting from a specific state (or state-action pair) while following a given policy. It serves as a guide for the agent to evaluate which states or actions are more beneficial over time, beyond immediate rewards. There are two primary types: the state-value function (V(s)), which estimates the expected return from a state under a policy, and the action-value function (Q(s,a)), which estimates the return from taking a specific action in a state and then following the policy. These functions help the agent prioritize actions that lead to higher cumulative rewards, even if they involve short-term trade-offs.

For example, consider a robot navigating a grid to reach a goal. The immediate reward for moving into a wall might be -1, while reaching the goal gives +10. The value function accounts for not just these immediate rewards but also future outcomes. If a state has a high value, it means the robot can reliably reach the goal from there. Suppose the robot has two paths: a shorter route with risky terrain (e.g., slippery tiles) and a longer, safer path. The value function would assign higher values to states along the safer path if the risk of penalties (e.g., falling into a pit) outweighs the shorter path’s benefits. This is calculated using the Bellman equation, which recursively breaks down the value of a state into its immediate reward plus the discounted (scaled-down) value of future states.

In practice, value functions are central to algorithms like Q-Learning and Deep Q-Networks (DQN). Q-Learning, for instance, iteratively updates Q(s,a) estimates using observed rewards and the maximum Q-value of the next state, fostering a balance between exploration and exploitation. For example, in a game-playing agent, the Q-function might learn that sacrificing a pawn in chess (immediate loss) leads to a stronger board position (higher long-term value). Value functions also enable techniques like temporal difference (TD) learning, where the agent adjusts its estimates incrementally as it interacts with the environment. By grounding decisions in long-term outcomes, value functions provide a structured way for RL agents to optimize behavior in complex, uncertain environments.