The value function in reinforcement learning and the importance of its Bellman equation

Reinforcement learning is a branch of machine learning that aims to learn optimal actions in a given environment through trial and error. Among them, the value function and Bellman equation are key concepts in reinforcement learning and help us understand the basic principles of this field.

The value function is the expected value of the long-term return expected to be obtained in a given state. In reinforcement learning, we often use rewards to evaluate the merits of an action. Rewards can be immediate or delayed, with effects occurring in future time steps. Therefore, we can divide value functions into two categories: state value functions and action value functions. State value functions evaluate the value of taking an action in a certain state, while action value functions evaluate the value of taking a specific action in a given state. By computing and updating a value function, reinforcement learning algorithms can find optimal strategies to maximize long-term returns.

The state value function is the expected return that can be obtained by adopting the optimal strategy in a specific state. We can estimate the state value function by calculating the expected return from executing a certain strategy in the current state. The Monte Carlo method and the time difference learning method are commonly used methods to estimate the state value function.

The action value function refers to the expected return that may be obtained after taking an action in a specific state. Q-learning algorithm and SARSA algorithm can be used to estimate the action value function. These algorithms make estimates by calculating the expected return from taking a certain action in the current state.

The Bellman equation is an important concept in reinforcement learning and is used to recursively calculate the value function of the state. The Bellman equation can be divided into two types: the Bellman equation for the state value function and the Bellman equation for the action value function. The former is calculated through the value function of the subsequent state and the immediate reward, while the latter needs to consider the impact of the action taken on the value. These equations play a key role in reinforcement learning algorithms, helping agents learn and make optimal decisions.

The Bellman equation of the state value function states that the value function of a state can be calculated recursively through the value function of the next state and the immediate reward of the state. The mathematical formula is:

V(s)=E[R γV(s')]

where V(s) represents the state The value function of s; R represents the immediate reward after taking an action in state s; γ represents the discount factor, used to measure the importance of future returns; E represents the expected value; s' represents the next state.

The Bellman equation of the action value function expresses that the value function of taking an action in one state can be calculated recursively through the value function of the next state of the action and the immediate reward. The mathematical formula is:

Q(s,a)=E[R γQ(s',a')]

where, Q (s,a) represents the value function of taking action a in state s; R represents the immediate reward after taking action a in state s; γ represents the discount factor; E represents the expected value; s' represents the next step after taking action a. A state; a' represents the optimal action to take in the next state s'.

The Bellman equation is a very important equation in reinforcement learning. It provides an effective recursive calculation method for estimating the state value function and action value function. The Bellman equation can be calculated recursively using value function-based reinforcement learning algorithms, such as value iteration algorithms, policy iteration algorithms, and Q-learning algorithms.

In short, the value function and the Bellman equation are two important concepts in reinforcement learning, and they are the basis for understanding reinforcement learning. By estimating the value function and recursively calculating the Bellman equation, we can find the optimal strategy to take the optimal action in a specific environment and maximize the long-term return.

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