Combining DQN Improvements

Having explored Double DQN (DDQN) to mitigate value overestimation and Dueling Network Architectures to better estimate state values and action advantages, a natural question arises: can we use these improvements together? Fortunately, the answer is yes. These techniques address different aspects of the DQN training process and are largely complementary. Combining them often leads to significantly improved performance and stability compared to using any single enhancement alone.

Integrating Double DQN and Dueling Networks

The core idea is to use the Dueling architecture for both the online network (used for action selection and gradient calculation) and the target network (used for calculating the target Q-value), while employing the Double DQN update rule for calculating that target value.

Recall the Dueling architecture computes the Q-value as:

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

Here, $\theta$ represents parameters shared between the value ($V$) and advantage ($A$) streams, while $\beta$ and $\alpha$ are parameters specific to the value and advantage streams, respectively. Let's denote the parameters of the online network as $(\theta, \alpha, \beta)$ and the target network as $(\theta^-, \alpha^-, \beta^-)$.

Now, let's incorporate the Double DQN principle. The DDQN target $Y_t$ is calculated as:

$$Y_t^{DDQN} = R_{t+1} + \gamma Q_{\text{target}}(S_{t+1}, \arg\max_{a'} Q_{\text{online}}(S_{t+1}, a'; \theta, \alpha, \beta); \theta^-, \alpha^-, \beta^-)$$

To combine these:

1. Action Selection: Use the online network with its Dueling architecture to determine the best action $a^*$ for the next state $S_{t+1}$:

   $$a^* = \arg\max_{a'} Q_{\text{online}}(S_{t+1}, a'; \theta, \alpha, \beta)$$

   Specifically, you'd compute the advantages $A(S_{t+1}, a'; \theta, \alpha)$ for all actions $a'$ using the online network's advantage stream and select the action $a^*$ corresponding to the maximum advantage (since the $V(S_{t+1})$ term is constant across actions for a given state).

2. Action Evaluation: Use the target network, also with its Dueling architecture, to evaluate the Q-value of taking action $a^*$ in state $S_{t+1}$:

   $$Q_{\text{target}}(S_{t+1}, a^*; \theta^-, \alpha^-, \beta^-) = V_{\text{target}}(S_{t+1}; \theta^-, \beta^-) + \left( A_{\text{target}}(S_{t+1}, a^*; \theta^-, \alpha^-) - \frac{1}{|\mathcal{A}|} \sum_{a''} A_{\text{target}}(S_{t+1}, a''; \theta^-, \alpha^-) \right)$$

3. Target Calculation: Form the final target value $Y_t$ using the reward $R_{t+1}$ and the evaluated Q-value from the target network:

   $$Y_t = R_{t+1} + \gamma Q_{\text{target}}(S_{t+1}, a^*; \theta^-, \alpha^-, \beta^-)$$

4. Loss Calculation: Calculate the loss (e.g., Mean Squared Error or Huber loss) between the target $Y_t$ and the Q-value predicted by the online network for the original state-action pair $(S_t, A_t)$:

   $$\text{Loss} = \mathcal{L}(Y_t - Q_{\text{online}}(S_t, A_t; \theta, \alpha, \beta))$$

5. Gradient Update: Update the parameters $(\theta, \alpha, \beta)$ of the online network using gradients derived from this loss. Periodically update the target network parameters $(\theta^-, \alpha^-, \beta^-)$ by copying the online network parameters.

This combined approach, often referred to as Dueling DDQN, benefits from both the reduced overestimation bias of Double DQN and the improved feature learning capabilities of the Dueling architecture.

Diagram illustrating the data flow for calculating the target value ($Y_t$) in a Dueling Double DQN. Action selection uses the online network, while action evaluation uses the target network, both featuring the Dueling architecture.

Further Enhancements: Prioritized Experience Replay (PER)

As briefly mentioned before, Prioritized Experience Replay (PER) is another significant improvement that can be layered on top of Dueling DDQN. Instead of sampling transitions uniformly from the replay buffer, PER samples transitions based on their TD error. Transitions where the agent's prediction was highly inaccurate (large TD error) are considered more "surprising" or informative and are replayed more frequently.

Integrating PER involves:

1. Storing TD Errors: When a transition $(S_t, A_t, R_{t+1}, S_{t+1})$ is added to the replay buffer, calculate its initial TD error and store it alongside the transition.
2. Prioritized Sampling: When sampling a minibatch, assign probabilities based on these stored TD errors (or ranks derived from them). Higher error means higher probability of being sampled.
3. Importance Sampling Weights: To correct for the bias introduced by non-uniform sampling, compute importance sampling (IS) weights for each transition in the minibatch. These weights adjust the gradient update, down-weighting updates from frequently sampled transitions.
4. Updating Priorities: After performing a gradient update for a sampled transition, update its priority in the replay buffer using the newly calculated TD error.

Combining Dueling DDQN with PER often results in state-of-the-art performance for many discrete action space tasks, such as those found in the Arcade Learning Environment (ALE). The implementation complexity increases, but the potential gains in sample efficiency and final performance can be substantial.

In summary, the modular nature of these DQN improvements allows them to be combined effectively. Starting with the base DQN, adding Double DQN addresses overestimation, Dueling networks improve value function representation, and Prioritized Experience Replay focuses learning on the most informative transitions. Implementing these combinations provides a powerful toolkit for tackling complex reinforcement learning problems.