## 3 Proposed CNN-Based Power Allocation Method
### CNN Training and Optimization Objective  
To train the proposed CNN model, a supervised learning approach was adopted using data generated from traditional max-min power optimization. Specifically, the method introduced in [3] is followed, in which the downlink power control problem is solved under a max-min fairness criterion to maximize the minimum achievable SINR across all UEs under per-AP power constraints. This is achieved by iteratively transforming the problem into a sequence of second-order cone programs (SOCPs), which are globally optimal under convex feasibility. The resulting power allocation matrices provide fairness-centric ground truth for supervised training.  
To simulate practical network environments, the dataset included diverse network realizations with user distributions clustered around high-traffic regions such as urban centers or event areas. Each training sample consisted of a spatial layout of APs and UEs, large-scale fading coefficients, and corresponding input features. These features are used to form the input tensor to the CNN, while the optimal power values from the max-min solution form the output supervision targets. The CNN model was trained on 10,000 samples generated under Rayleigh fading channels with a path loss exponent of \(\alpha=3.76\). Each sample encodes spatial features such as clustered UE distributions, AP-UE distances, channel gain magnitudes, estimated interference levels, and initial SINR values. Training was conducted for 50 epochs using the Adam optimizer, with a batch size of 32 and a learning rate of \(10^{-3}\). A validation split of 20% was considered to apply early stopping if necessary.  
Figure 4 illustrates the training loss curve of the CNN model. The loss rapidly decreases within the first ten epochs and converges smoothly, remaining stable with minimal fluctuations throughout the remaining training process. This confirms that the model learns efficiently and is not prone to overfitting.  
The CNN was trained to minimize the discrepancy between its predicted power allocations and the max-min references while ensuring compliance with power constraints. By learning from optimal yet computationally expensive power control solutions, the CNN generalizes to new scenarios with far lower inference complexity and supports end-to-end hierarchical power control beyond the original optimization formulation. The loss function combines two mean squared error (MSE) terms, corresponding to the CPU-to-AP and AP-to-UE power variables, along with a regularization term used to ensure that the sum of AP-to-UE power does not exceed each AP's CPU-assigned power [11]:  
\[\mathcal{L}(\theta)=\frac{1}{N}\sum_{i=1}^{N}\left(\|P_{\text{CPU-AP},i}^{ \text{CNN}}-P_{\text{CPU-AP},i}^{\text{opt}}\|_{2}^{2}+\|P_{\text{AP-UE},i}^{ \text{CNN}}-P_{\text{AP-UE},i}^{\text{opt}}\|_{F}^{2}\right)+\lambda\sum_{i=1 }^{M}\max\left(0,\sum_{j=1}^{K}\hat{p}_{ij}-\hat{p}_{i}\right) \tag{12}\]  
where \(\|\cdot\|_{F}\) is the Frobenius norm and \(\lambda\) is a hyperparameter controlling the penalty for violating AP power constraints.  
Figure 3: Overall architecture of the proposed CNN model.