- Value function approximation
- Value iteration
- Value-based reinforcement learning
- Vapnik-Chervonenkis dimension
- Variance minimization
- Variance reduction
- Variance-based sensitivity analysis
- Variance-stabilizing transformation
- Variational autoencoder
- Variational dropout
- Variational generative adversarial network
- Variational inference
- Variational message passing
- Variational optimization
- Variational policy gradient
- Variational recurrent neural network
- Vector autoregression
- Vector quantization
- Vector space models
- VGGNet
- Video classification
- Video summarization
- Video understanding
- Visual attention
- Visual question answering
- Viterbi algorithm
- Voice cloning
- Voice recognition
- Voxel-based modeling

# What is Value-based reinforcement learning

##### Value-based Reinforcement Learning

**Reinforcement learning** is a type of machine learning that encourages agents to learn from the environment by trial and error. It's a powerful way to train agents to perform tasks that are difficult to specify algorithmically. One of the important approaches in reinforcement learning is value-based reinforcement learning. It is an approach to reinforcement learning where the agent learns the value of taking different actions in different states of the environment.

This approach is different from policy-based reinforcement learning where the agent learns the policy directly. In value-based reinforcement learning, the agent learns to estimate the value function, which provides a measure of how good each state or action is. The value function tells the agent how good it is to be in a particular state at a particular time, under a particular policy. It's an estimate of the expected reward the agent will receive by following a given policy.

The value function is usually represented as a function from the state or state-action space to the expected return. The return is the total discounted reward collected by the agent from the current time-step until the end of the episode. The discounting factor is a value between 0 and 1 that determines the importance of future rewards. If the discounting factor is close to 0, the agent will focus on immediate rewards. If it's close to 1, the agent will consider future rewards as being equally important as the immediate reward.

The value function estimates the expected return given that the agent starts in a particular state and follows a particular policy. The policy is a function that maps states to actions. The agent selects the action that maximizes the value function for the current state. The process repeats until the end of the episode.

One of the advantages of value-based reinforcement learning is that it works well in domains with large state spaces. It's not easy to learn a policy in such domains because there are so many possible states and actions. Value-based methods can handle this by learning to estimate the value function directly. Once the agent learns the value function, it can easily determine the best action to take in any given state by selecting the action with the highest value.

##### The Bellman Equation

The most important equation in value-based reinforcement learning is the Bellman equation. The equation relates the value function for a state or action in the current time-step to the value function for states or actions at later time-steps. It allows the agent to update its estimate of the value function through experience.

The Bellman equation has two forms, one for the state-value function V and one for the action-value function Q. They are given below:

- Bellman equation for state-value function V:

**V _{t}(s) = E_{π} [G_{t} | S_{t} = s]**

Here, V_{t}(s) is the value function at time-step t for state s under policy π. G_{t} is the return starting from time-step t, and S_{t} is the state at time-step t. The expectation is taken over all possible future rewards starting from the current time-step, given that the agent follows the policy π.

- Bellman equation for action-value function Q:

**Q _{t}(s, a) = E_{π} [G_{t} | S_{t} = s, A_{t} = a]**

Here, Q_{t}(s, a) is the value function at time-step t for taking action a in state s under policy π. A_{t} is the action taken at time-step t. The expectation is taken over all possible future rewards starting from the current time-step, given that the agent follows the policy π.

##### Q-Learning

Q-learning is a model-free, off-policy algorithm that estimates the optimal action-value function by bootstrapping from the current estimate of the Q-value for the next state or action. The Q-value is the expected discounted return when taking a given action a from a state s and then following the optimal policy. The optimal policy selects the action that leads to the highest Q-value for the next state or action.

The algorithm updates the Q-value using the Bellman equation given below:

Q(s,a) = Q(s,a) + α[r + γ max_{a'}Q(s',a') - Q(s,a)]

Here, α is the learning rate that determines how much the new estimate replaces the old estimate, r is the reward received for taking action a in state s, s' is the next state, γ is the discount factor, and max_{a'}Q(s',a') is the maximum Q-value for the next state s'.

Q-learning is an off-policy algorithm in the sense that it learns the optimal Q-value function regardless of the policy being followed. It selects actions based on the ε-greedy policy, which selects the optimal action with probability 1- ε and a random action with probability ε.

One of the drawbacks of Q-learning is that it can take a long time to converge. It can also be unstable in some cases. There are several extensions to Q-learning that address these issues.

##### Deep Q-Networks

The deep Q-network (DQN) is an extension of Q-learning that uses a deep neural network to estimate the Q-value function. The neural network takes the state as input and outputs the Q-value for each action. The Q-value is computed using a separate target network that has the same architecture as the Q-network but with frozen parameters. The target network is used to compute the target Q-value for the Bellman update.

The loss function for training the Q-network is given below:

Loss = (r + γ max_{a'}Q_{θ'}(s_{t+1},a_{t+1}) - Q_{θ}(s_{t},a_{t}))^{2}

Here, Q_{θ} is the Q-value function parameterized by θ, and Q_{θ'} is the target Q-value function. The target Q-value is computed using the target network. The input to the network is the state s_{t+1} and the output is the Q-value for each action.

The DQN algorithm works well in environments with large state spaces and discrete actions. It has been used to play games such as Go and Atari. However, it can be slow to train and requires a large amount of data.

##### Conclusion

Value-based reinforcement learning is an important approach to reinforcement learning that learns the value function of different states or actions. The value function estimates the expected return given that the agent starts in a particular state and follows a particular policy. The Bellman equation is the most important equation in value-based reinforcement learning. It relates the value function for a state or action in the current time-step to the value function for states or actions at later time-steps.

Q-learning is a popular value-based reinforcement learning algorithm that estimates the optimal action-value function by bootstrapping. It updates the Q-value using the Bellman equation. One of the drawbacks of Q-learning is that it can take a long time to converge. The DQN algorithm is an extension of Q-learning that uses a deep neural network to estimate the Q-value function.