Practice: Implementing Double DQN

Double DQN (DDQN) mitigates the overestimation bias inherent in standard Q-learning and DQN. Implementing DDQN from a standard DQN setup involves a small, yet significant, modification to the calculation of target Q-values used in the loss function.

Recall the standard DQN target calculation for a transition $(s, a, r, s', d)$ (where $d$ indicates if $s'$ is a terminal state):

The target $y_i$ for the $i$-th sample from the experience replay buffer is:

$$y_i = r_i + \gamma \max_{a'} Q_{target}(s'_i, a') \quad \text{(if } s'_i \text{ is not terminal)}$$ $$y_i = r_i \quad \text{(if } s'_i \text{ is terminal)}$$

Here, the target network $Q_{target}$ is used for both selecting the best next action ($a'$) and evaluating the value of that action. This coupling is where the overestimation can arise.

The Double DQN Modification

Double DQN decouples this process. It uses the online network ($Q_{online}$) to select the best action in the next state $s'_i$, and then uses the target network ($Q_{target}$) to evaluate the value of that chosen action.

The Double DQN target $y_i$ calculation becomes:

$$a'_{max} = \arg\max_{a'} Q_{online}(s'_i, a')$$ $$y_i = r_i + \gamma Q_{target}(s'_i, a'_{max}) \quad \text{(if } s'_i \text{ is not terminal)}$$ $$y_i = r_i \quad \text{(if } s'_i \text{ is terminal)}$$

Notice the change: we first find the action $a'_{max}$ that maximizes the Q-value according to the online network in state $s'_i$. Then, we plug this specific action $a'_{max}$ into the target network to get the Q-value estimate for the target calculation.

Implementing the Change

Let's assume you have a standard DQN implementation, likely with a learn or compute_loss method that processes a batch of experiences sampled from the replay buffer. You'll need to modify the part where you calculate the target Q-values.

Here's a breakdown comparing the target calculation snippets (assuming online_net and target_net are your network models, and next_states, rewards, dones are tensors/arrays from the sampled batch):

Standard DQN Target Calculation (Snippet):

# Assuming next_states is a batch of next states from the replay buffer
# Get Q-values for next states from the target network
next_q_values_target = target_net(next_states)

# Select the maximum Q-value for each next state
max_next_q_values = next_q_values_target.max(dim=1)[0] # Or axis=1 in NumPy/TF

# Calculate the target y_i (handle terminal states where dones=True)
target_q_values = rewards + gamma * max_next_q_values * (1 - dones)

Double DQN Target Calculation (Snippet):

# Assuming next_states, rewards, dones are batches from the replay buffer
# 1. Select the best actions in the next states using the *online* network
next_q_values_online = online_net(next_states)
best_next_actions = next_q_values_online.argmax(dim=1) # Or axis=1

# 2. Evaluate these selected actions using the *target* network
# Get all Q-values for next states from the target network
next_q_values_target = target_net(next_states)
# Select the Q-values corresponding to the best_next_actions
# Needs careful indexing (e.g., gather in PyTorch/TF)
q_values_of_best_actions = next_q_values_target.gather(1, best_next_actions.unsqueeze(-1)).squeeze(-1)

# 3. Calculate the target y_i (handle terminal states where dones=True)
target_q_values = rewards + gamma * q_values_of_best_actions * (1 - dones)

The core change involves these steps:

1. Perform a forward pass on next_states using the online_net to find the argmax action ($a'_{max}$).
2. Perform a forward pass on next_states using the target_net to get the Q-values for all possible next actions.
3. Select the Q-values from step 2 that correspond to the actions chosen in step 1. This requires careful tensor indexing (like gather in PyTorch or tf.gather_nd in TensorFlow).
4. Use these selected Q-values to compute the final target $y_i$.

The rest of your DQN code, including the experience replay mechanism, the periodic updates of the target network weights from the online network weights, the optimizer step, and the epsilon-greedy action selection during environment interaction, typically remains the same.

Diagram illustrating the calculation of the target Q-value component ($\gamma Q_{target}(s', \arg\max_{a'} Q_{online}(s', a'))$) in Double DQN. The online network selects the best action, and the target network evaluates that specific action's value.

Modify your existing DQN agent code based on the snippets and the diagram above. Test your implementation, perhaps again on the CartPole environment or a more complex Atari environment if you are using those. Observe if the training appears more stable or if the agent achieves better performance compared to your standard DQN implementation, keeping in mind that results can vary based on hyperparameters and environment specifics. This hands-on modification provides direct experience with enhancing DQN algorithms.