# Distributed Interference-Aware Power Optimization for Multi-Task Over-the-Air Federated Learning
## 4 Proposed Method
### Distributed Scheme
\[\sum_{k\in\mathcal{K}_{m}}p_{k}|g_{jk}|^{2}\leq\Gamma_{j,m},j\in \mathcal{M}\setminus\{m\}, \tag{44}\]8  
Furthermore, according to the stationarity condition (56) for \(Q_{k}\), we obtain  
\[\sum_{k\in\mathcal{K}_{m}}p_{k}|g_{jk}|^{2}\leq\Gamma_{j,m},j\in \mathcal{M}\setminus\{m\}, \tag{44}\]9 \[0\leq p_{k}\leq P_{k}^{max},\forall k\in\mathcal{K}_{m},m\in \mathcal{M}. \tag{45}\]0By rearranging the equations and considering the maximum device power constraint, the optimal solution for \(Q_{k}\) is derived as  
\[0\leq p_{k}\leq P_{k}^{max},\forall k\in\mathcal{K}_{m},m\in \mathcal{M}. \tag{45}\]1  
Similarly, to determine the optimal inverse denoising factor \(\gamma_{m}\), we consider the following optimization problem:  
\[0\leq p_{k}\leq P_{k}^{max},\forall k\in\mathcal{K}_{m},m\in \mathcal{M}. \tag{45}\]2  
Applying the stationarity condition (57) with respect to \(\gamma_{m}\), we can obtain  
\[0\leq p_{k}\leq P_{k}^{max},\forall k\in\mathcal{K}_{m},m\in \mathcal{M}. \tag{45}\]3 \[0\leq p_{k}\leq P_{k}^{max},\forall k\in\mathcal{K}_{m},m\in \mathcal{M}. \tag{45}\]4  
Therefore, the optimal solution for \(\gamma_{m}\) is derived as  
\[0\leq p_{k}\leq P_{k}^{max},\forall k\in\mathcal{K}_{m},m\in \mathcal{M}. \tag{45}\]5  
To find the optimal dual variables \(\left\{\lambda_{j,m}^{*}\right\}_{j\in\mathcal{M}\setminus\{m\}}\) and \(\varphi_{k}^{*}\), we can employ the ellipsoid method [33], which utilizes subgradients to iteratively refine the estimates of the dual variables. The subgradient for \(\left\{\lambda_{j,m}\right\}_{j\in\mathcal{M}\setminus\{m\}}\) is \(\sum\limits_{k\in\mathcal{K}_{m}}\left|Q_{k}^{*}\delta_{jk}\right|^{2}-\Gamma_ {j,m}\gamma_{m}^{*}\), and the subgradient for \(\varphi_{k}^{*}\) is \(\left(Q_{k}^{*}\right)^{2}-P_{k}^{\max}\gamma_{m}^{*}\). By iteratively updating \(\left\{\lambda_{j,m}\right\}_{j\in\mathcal{M}\setminus\{m\}}\) and \(\varphi_{k}^{*}\) using these subgradients, the ellipsoid method converges to the optimal dual variables. Once the optimal dual variables are obtained, these variables can be substituted back into the expressions (61) and (64) to determine the optimal primal variables. Finally, the optimal transmit power \(p_{k}^{*}\) and the denoising factor \(\theta_{m}^{*}\) are computed as  
\[0\leq p_{k}\leq P_{k}^{max},\forall k\in\mathcal{K}_{m},m\in \mathcal{M}. \tag{45}\]6  
\[0\leq p_{k}\leq P_{k}^{max},\forall k\in\mathcal{K}_{m},m\in \mathcal{M}. \tag{45}\]7  
These results indicate that when \(\varphi_{k}^{*}>0\), the device transmits at its maximum power \(P_{k}^{\max}\). Otherwise, the transmit power is adjusted based on the regularized inverse power transmission strategy, where the regularization term \(\sum\limits_{j\in\mathcal{M}\setminus\{m\}}\lambda_{j,m}^{*}\Big{|}\delta_{jk} \Big{|}^{2}\) accounts for inter-cell interference. The denoising factor \(\theta_{m}^{*}\) is influenced by both inter-cell interference and the power constraints of all devices. While these solutions satisfy the inter-cell interference constraints, they are limited to scenarios with fixed interference thresholds and may not achieve Pareto optimality. Therefore, an iterative algorithm for optimizing inter-cell interference thresholds is proposed to enhance overall system performance. To facilitate efficient distributed power control in parallel multi-task Air-FL systems, this paper introduces the concept of \(\Pi\) as a pivotal coordination variable to establish a local information-driven distributed optimization framework. (The proposed analysis and algorithms assume quasi-static channel conditions, where wireless links remain constant over each communication round. This assumption enables tractable optimization and reliable gradient aggregation using AirComp. In practical scenarios with mobility or fading, channel variations may affect aggregation accuracy. Nonetheless, our distributed scheme relies only on local CSI and permits iterative \(\Pi\) updates, offering inherent adaptability to slow channel variations.