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In Deep QLearning (DQN), a wellknown algorithm that uses fitted value iteration, the update rule involves the loss function,
 [3668a]
where,
θ are the parameters of the Qnetwork.
θ^{} represents the parameters of a target network that is periodically updated.
The basic procedure of Qlearning with function approximation, specifically using Deep QNetworks (DQN), is:

Initialize Parameters:
Define the environment, state space, action space, and the reward function.
Set hyperparameters such as learning rate, discount factor (gamma), explorationexploitation tradeoff (epsilon), replay buffer size, etc.
Initialize the Qnetwork with random weights.
Initializing the Qvalues for all stateaction pairs to zero at the beginning of the learning process. This is a common initialization strategy in reinforcement learning, including Qlearning and its variants like Deep QNetworks (DQN). The idea behind this initialization is to start with a neutral stance, assuming no prior knowledge about the quality of actions in different states. As the agent interacts with the environment, it updates these Qvalues based on observed rewards and learns which actions are more favorable in different states.
Learning Process: As the agent takes actions and receives rewards, it updates the Qvalues using the Qlearning update rule, given by:
 [3668b]
where,
Q(s,a) is the Qvalue for stateaction pair (s, a).
α is the learning rate.
R is the observed reward.
γ is the discount factor, balancing immediate rewards against future rewards.
s′ is the next state after taking action a.
max_{a′}Q(s′, a′) is the maximum Qvalue for the next state s', estimating the expected future rewards.
is the temporal difference error, and updating Q(s,a) with this error combines the immediate reward and the estimated future rewards.
The learning process continues iteratively, with the Qvalues being updated based on the agent's experiences in the environment.
 Initialize Replay Buffer:
Create a replay buffer to store experiences (state, action, reward, next state) for replay during training.

Define QNetwork:
Design a neural network architecture to represent the Qfunction. It takes the state as input and outputs Qvalues for all possible actions.
The network is often a deep neural network with one or more hidden layers. 
Define Loss Function:
Define the loss function, typically the mean squared error between the predicted Qvalues and the target Qvalues. 
Explore and Exploit:
Use an epsilongreedy strategy to balance exploration and exploitation.
With probability epsilon, choose a random action. Otherwise, choose the action with the highest Qvalue from the current state. 
Interact with Environment:
Take an action in the environment based on the explorationexploitation strategy.
Observe the next state and the reward from the environment. 
Store Experience in Replay Buffer:
Store the experience (state, action, reward, next state) in the replay buffer. 
Sample MiniBatch:
Randomly sample a minibatch of experiences from the replay buffer for training. 
Calculate Target QValues:
Use the Qnetwork to calculate Qvalues for the next state.
Calculate the target Qvalues using the reward and the maximum Qvalue for the next state. 
Update QNetwork:
Minimize the loss between the predicted Qvalues and the target Qvalues.
Backpropagate the error through the network and update the weights using optimization algorithms like stochastic gradient descent (SGD) or variants like Adam.
Estimate the value of Q(s, a) based on current reward and expected future rewards.

Repeat:
Repeat steps 610 for a predetermined number of episodes or until convergence. 
Evaluation:
After training, evaluate the performance of the learned Qnetwork by letting it interact with the environment without exploration.
DQN incorporates experience replay and target networks to stabilize training and improve convergence. It is essential to carefully tune hyperparameters and monitor the learning process to achieve effective results.
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