# DDPG-Based UAV-RIS Framework for Optimizing Mobility in Future Wireless Communication Networks
## 3 Problem Formulation
### Joint Optimization of UAV Location and RIS Phase Shifts
Additionally,  
Figure 2: Flow diagram of the proposed DDPG algorithm. actor and critic networks \(\mu(s|\theta^{\mu})\) and \(Q(s,a|\theta^{Q})\) are initialized with their respective weights \(\theta^{\mu}\) and \(\theta^{Q}\), respectively. To facilitate stable training, the target networks \(\mu^{\prime}\) and \(Q^{\prime}\) are initialized with weights \(\theta^{\mu^{\prime}}\leftarrow\theta^{\mu}\) and \(\theta^{Q^{\prime}}\leftarrow\theta^{Q}\). ```
Initialization:
* Replay buffer \(D\), discount factor \(\gamma\), soft update coefficient \(\tau\), and the minibatch size \(N_{B}\)
* Actor network \(\mu(s|\theta^{\mu})\) with weights \(\theta^{\mu}\), critic network \(Q(s,a|\theta^{Q})\) with weights \(\theta^{Q}\), target networks \(\mu^{\prime}\) and \(Q^{\prime}\) with weights \(\theta^{\mu^{\prime}}\leftarrow\theta^{\mu}\) and \(\theta^{Q^{\prime}}\leftarrow\theta^{Q}\)
1:for episode \(m,\dots,M\)do
2 Get the initial observed state \(s_{t}\) (18)
3 Initialize the random process \(\mathcal{N}\) for action exploration
4 for each time step \(t=1,\dots,T\)do
5 Select action \(a_{t}=\mu(s_{i}|\theta^{\mu})+\mathcal{N}_{t}\) based on the current policy and exploration noise
6 Execute the actions \(a_{t}\) and observe reward \(r_{t}\) and new state \(s_{t+1}\)
7 Store transition \((s_{t},a_{t},r_{t},s_{t+1})\) in replay buffer \(D\)
8 Sample the random \(N_{B}\) mini-batch transitions \((s_{t},a_{t},r_{t},s_{t+1})\) from \(D\)
9 Calculate target Q-value by \(z_{t}=r_{t}(s_{t},a_{t})+\gamma Q^{\prime}(s_{t+1},\mu^{\prime}(s_{t+1}|\theta ^{\mu^{\prime}})|\theta^{Q^{\prime}})\)
10 Update critic by minimizing the loss: \[L=\frac{1}{N_{B}}\frac{N_{B}}{\sum_{t}^{N_{B}}}(z_{t}-Q(s_{t},a_{t}|\theta^{Q} ))^{2}\]
11: Update the actor policy using the sampled policy gradient: \[\
abla_{\theta^{\mu}}M\approx\frac{1}{N_{B}}\sum\
abla_{a}Q(s,a|\theta^{Q})| _{s=s_{t},a=\mu(s_{t})}\
abla_{\theta^{\mu}}\mu(s|\theta^{\mu})|_{s_{t}}\]
12: Update the target networks: \[\theta^{\mathcal{N}} \leftarrow\tau_{t}\theta^{Q}+(1-\tau_{tc})\theta^{\mathcal{N}}\] \[\theta^{\mu^{\prime}} \leftarrow\tau_{ta}\theta^{\mu}+(1-\tau_{ta})\theta^{\mu^{\prime}}\]
13:\(s_{t}\gets s_{t+1}\)
14:endfor
15:endfor
```  
**Algorithm 2**Proposed DDPG Algorithm for Joint Optimization of UAV Location and RIS Phase Shifts  
In step 2, the algorithm obtains the initial observed state \(s_{t}\) of the environment by utilizing equation (18). In step 3, the DDPG algorithm initializes noise \(\mathcal{N}\) to facilitate the agent in achieving a better action and state exploration. In step 5, the agent selects an action \(a_{t}\) based on the current policy, represented by actor network \(\mu(s_{t}|\theta^{\mu})\), while adding exploration noise \(\mathcal{N}\) to encourage a broader exploration of the action space. This approach allows the agent to discover a wider range of actions during training, thereby improving its ability to learn an optimal policy. Next, in step 6, the agent performs the selected action \(a_{t}\) in the environment, observes the resulting reward \(r_{t}\), and transitions to the next state \(s_{t+1}\). This step is important for learning from environmental feedback and refining the agent's policy. In step 7, the algorithm stores the current state, action taken, reward received, and next state observed in the replay buffer \(D\) to retain experiences for learning. To train the neural networks, step 8 involves randomly selecting a mini-batch of \(N_{B}\) transitions from \(D\), based on the current state, action, reward, and next state. In step 9, the algorithm calculates the target Q-value \((z_{t})\) using the Bellman equation. Here, \(r_{t}(s_{t},a_{t})\) is the observed reward after taking \(a_{t}\) from \(s_{t}\), while \(Q^{\prime}(s_{t+1},\mu^{\prime}(s_{t+1}|\theta^{\mu^{\prime}})|\theta^{Q^{ \prime}})\) denotes the target Q-value for next state \(s_{t+1}\). Moving forward, step 10 computes the loss function \(L\), which measures the difference between the target Q-value \((z_{t})\) and training Q-value \(Q(s,a|\theta^{Q})\)[37].