# Distributed Interference-Aware Power Optimization for Multi-Task Over-the-Air Federated Learning
## 5 Numerical Results
### Multi-Task Ridge Regression Performance  
We consider that each cell independently trains a ridge regression model on distinct datasets. Each cell's dataset follows a unique distribution, which reflects heterogeneous data environments. The sample loss function for cell \(m\in\mathcal{M}\) is defined as  
\[H_{m}(\mathbf{e}_{m},\mathbf{x}_{m},\tau_{m})=\frac{1}{2}\Big{\|}\mathbf{x}_{m}^{T}\mathbf{e}_{m }-\tau_{m}\Big{\|}^{2}+\rho R(\mathbf{e}_{m}), \tag{73}\]  
where \(\rho=5\times 10^{-5}\) is the regularization hyperparameter. The input sample vector \(\mathbf{x}_{m}\in\mathbb{R}^{C}\) for each cell is drawn from a standard normal distribution, i.e., \(\mathbf{x}_{m}\sim\mathcal{N}(\mathbf{0},\mathbf{I})\), with model dimension \(C=20\). The target labels for each cell are generated as \(\tau_{1}=x_{1}(2)+3x_{1}(5)+0.2z\) and \(\tau_{2}=x_{2}(3)+2x_{2}(8)+4x_{2}(10)+0.3z\), where \(z\) represents standard Gaussian noise. Each device within a cell holds \(D_{k}=500,\forall k\in\mathcal{K}\) data samples, resulting in a total of \(D_{\text{m}}=\sum_{k\in\mathcal{K}}D_{k}=5000\) samples per cell. To evaluate the convergence behavior, we compute the smoothness parameter \(L_{m}\) and the Polyak-Lojasiewicz parameter \(\mu_{m}\) for each cell as in Assumptions 1 and 3, based on the eigenvalues of the sample covariance matrix \(\mathbf{X}_{m}^{T}\mathbf{X}_{m}/D_{m}+10^{-4}\mathbf{I}\)[36]. The optimal model parameters \(\mathbf{e}_{m}^{*}\) are obtained as \(\mathbf{e}_{m}^{*}=(\mathbf{X}_{m}^{T}\mathbf{X}_{m}+\rho\mathbf{I})^{-1}\mathbf{X}_{m}^{T}\mathbf{ \tau}_{m}\) with \(\mathbf{\tau}_{m}=[\tau_{1},\cdots,\tau_{\text{D}_{m}}]^{T}\). The initial model for each cell is initialized as a zero vector. Figure 3 presents the evolution of the optimality gap and prediction error across communication rounds \(N\) for various power control strategies in both cells. The proposed distributed scheme demonstrates a consistent decline in both metrics as \(N\) increases, with the prediction error stabilizing in later rounds. This trend indicates that the local models of both cells progressively converge towards their respective optimal solutions on the training set and ultimately achieve strong generalization performance on the test set. Analyzing the performance disparity between the two cells, Cell 1 consistently exhibits superior outcomes in both optimality gap and prediction error. This advantage is primarily due to the simpler label structure and lower data noise in Cell 1's task, thereby enabling more efficient learning and faster convergence under identical training conditions. Comparative assessments reveal that the proposed distributed scheme significantly outperforms both the benchmark with maximum power and benchmark without interference in both cells. Specifically, the proposed scheme not only accelerates the reduction in training errors to bring the model closer to the optimal training solution but also enhances generalization capability as evidenced by the lower prediction errors on the test set. This performance underscores its effectiveness in balancing rapid convergence with robust generalization. Furthermore, Figure 3 also includes the benchmark without AirComp as an ideal performance upper bound. In later communication rounds, the prediction error achieved by the proposed distributed scheme closely approaches this ideal benchmark with only a marginal gap. This proximity illustrates the scheme's capacity to approximate optimal performance despite practical challenges such as communication losses and inter-cell interference. In contrast, other baseline schemes continue to exhibit significant error at high communication rounds, particularly the benchmark with maximum power, which highlights their limitations in achieving efficient convergence and generalization. Figure 4 illustrates the achievable regions of the optimality gap for two cells at communication rounds \(N=100\).