# Distributed Interference-Aware Power Optimization for Multi-Task Over-the-Air Federated Learning
## 5 Numerical Results
### Simulation Setup and Benchmark Schemes  
We simulate a two-cell network, each hosting a distinct FL task. The first cell is centered at coordinates \((0,0)\), and the second at \((40,0)\). Devices within each cell are randomly distributed within a 20-meter radius around their respective APs. The wireless channelsbetween devices and APs follow a distance-dependent Rayleigh fading model. (In this study, we adopt Rayleigh fading to model the wireless channels between APs and devices. This choice captures a rich-scattering, non-line-of-sight (NLoS) environment and allows for tractable performance evaluation. However, we acknowledge that practical deployments may involve line-of-sight (LoS) components, in which case Rician or Nakagami-m fading models may provide more accurate characterizations. Incorporating such models to study the impact of LoS-dominant fading on Air-FL performance is an important direction for future work.) The channel gain between device \(k\) and AP \(m\) is modeled as \(h_{m,k}=\sqrt{\Omega_{0}d_{m,k}^{-\zeta}}h_{0}\), where \(\Omega_{0}\) denotes the path loss at a reference distance of 1 meter; \(\zeta\) is the path loss exponent; \(d_{m,k}\) is the distance between device \(k\) and AP \(m\); and \(h_{0}\sim\mathcal{CN}(0,I)\) is a complex Gaussian random variable with zero mean and unit variance. Interference channels between devices and non-associated APs are modeled similarly. Simulation parameters are detailed in Table 2.  
In terms of performance metrics, the optimality gap and prediction error are used to evaluate the ridge regression task on the synthetic dataset. To benchmark the proposed distributed scheme, we compare it against the following baseline methods:  
* **Benchmark with maximum power**: In this scheme, all devices transmit at their maximum power levels, i.e., \(p_{k}=p_{k}^{\text{max}}\). This scheme requires no CSI collection and represents the simplest power control strategy.
* **Benchmark without AirComp**: In this scheme, all devices transmit their local model updates to their respective APs, which perform aggregation without any interference. This scenario assumes an ideal communication environment, serving as an upper performance bound.
* **Benchmark without interference**: In this scheme, each AP optimizes the device transmit power and denoising factor based solely on intra-cell CSI, without coordinating with other APs. The optimization problem for AP \(m\) is formulated as \[\min_{\{0\leq p_{k}\leq p_{k}^{\text{max}},0\leq\theta_{m}\}}E_{m}\left(\sum_{k \in\mathcal{K}_{m}}\left(\frac{\sqrt{p_{k}}|h_{m,k}|}{\sqrt{\theta_{m}}}-v_{k} \right)^{2}+\frac{\sigma_{m}^{2}}{2\theta_{m}}\right).\] (72)  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(K\) & 10 \\ \(K_{\text{tot}}\) & 20 \\ \(\Omega_{0}\) & \(-60\) dB \\ \(\zeta\) & 3 \\ \(\sigma_{m}^{2}\) & \(10^{-7}\) W \\ \(p_{k}^{\text{max}}\) & \(1\) W \\ \(\iota\) & \(10^{-9}\) \\ \(\ell_{j,m}\) & \(0.5\) \\ \(\eta_{1}^{(n)}\) & \(2/(n+8)\) \\ \(\eta_{2}^{(n)}\) & \(2/(n+10)\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation parameters.