While standard DQN estimates the Q-value $Q(s, a)$ directly for each state-action pair, the Dueling Network Architecture proposes a different approach. It explicitly separates the representation of the state's value from the advantage of taking a specific action in that state.

Think about it intuitively: in some states, the value $V(s)$ (how good it is to be in state $s$ ) is high regardless of the action taken. In other states, the choice of action is highly significant, meaning the advantage $A(s, a)$ (how much better action $a$ is compared to other actions in state $s$ ) varies substantially. Dueling networks aim to disentangle these two components.

The core idea is based on the relationship:

Q(s, a) = V(s) + A(s, a)

Here, $V(s)$ represents the state value function, estimating the expected return from state $s$ . $A(s, a)$ is the advantage function, indicating the relative importance of action $a$ compared to the average action in state $s$ .

However, we cannot simply learn $V(s)$ and $A(s, a)$ independently and sum them up. Given a target $Q(s, a)$ value from our learning updates (like those used in DQN or Double DQN), there are infinitely many combinations of $V(s)$ and $A(s, a)$ that could produce it. This is an identifiability problem. For example, if we increase $V(s)$ by a constant $C$ and decrease all $A(s, a)$ values by $C$ , the resulting $Q(s, a)$ remains unchanged.

To address this, Dueling Networks introduce constraints when combining the value and advantage streams. A common approach is to force the advantage function estimate for the chosen action to be zero, or alternatively, to subtract the mean advantage across all actions:

Max Operator: $Q(s, a; \theta, \alpha, \beta) = V(s; \theta, \beta) + \left( A(s, a; \theta, \alpha) - \max_{a' \in |\mathcal{A}|} A(s, a'; \theta, \alpha) \right)$
Average Operator: $Q(s, a; \theta, \alpha, \beta) = V(s; \theta, \beta) + \left( A(s, a; \theta, \alpha) - \frac{1}{|\mathcal{A}|} \sum_{a' \in |\mathcal{A}|} A(s, a'; \theta, \alpha) \right)$

In these equations:

$\theta$ represents the parameters of shared network layers (e.g., convolutional layers processing input like images).
$\alpha$ represents the parameters of the network stream computing the advantages $A(s, a)$ .
$\beta$ represents the parameters of the network stream computing the state value $V(s)$ .
$|\mathcal{A}|$ is the number of discrete actions.

The average operator formulation is often preferred in practice as it increases the stability of the optimization. Subtracting the mean ensures that the average advantage is zero, making $V(s)$ a direct estimate of the state's value.

Network Structure

The network architecture typically involves initial layers (e.g., convolutional for image inputs, fully connected for vector inputs) shared between the two streams. After these shared layers, the network splits into two separate fully connected streams:

Value Stream: Outputs a single scalar value, $V(s; \theta, \beta)$ .
Advantage Stream: Outputs a vector with $|\mathcal{A}|$ values, one for each action, $A(s, a; \theta, \alpha)$ .

These two streams are then combined using one of the aggregation methods described above to produce the final Q-values for all actions in state $s$ .

Diagram illustrating the Dueling Network architecture. Input state features pass through shared layers before splitting into value and advantage streams. These streams are then recombined to produce the final Q-values.

Why Does It Help?

The primary benefit of the Dueling architecture stems from its ability to learn the state value $V(s)$ without needing to consider the effect of each action at every step. The value stream learns a representation of the state's overall usefulness. The advantage stream only needs to learn the relative desirability of actions when it actually matters.

This separation often leads to better policy evaluation in states where the actions don't significantly impact the outcome. The network can learn the state's value more efficiently, as updates to $V(s)$ affect the Q-values of all actions simultaneously through the aggregation layer. This often results in faster convergence and improved performance compared to standard DQN, particularly in environments with large action spaces or where many actions have similar effects.

Dueling networks can be readily combined with other DQN improvements like Double DQN (DDQN) and Prioritized Experience Replay (PER) for further performance gains. Integrating Dueling architecture into a DDQN agent simply involves replacing the standard Q-network and target network structures with their dueling counterparts, while keeping the DDQN update rule.