# Distributed Interference-Aware Power Optimization for Multi-Task Over-the-Air Federated Learning
## 4 Proposed Method
### Distributed Scheme  
This subsection investigates the decentralized power control strategy for parallel multi-task systems under IT constraints to characterize the optimality gap and the Pareto boundary. We define \(\Gamma_{j,m}\) as the IT value representing the maximum allowable interference power from devices in cell \(m\) to AP \(j\). This constraint ensures that the interference from cell \(m\) does not exceed a tolerable threshold at AP \(j\), thereby enabling independent local optimization while maintaining global coordination. Let \(\mathbf{\Gamma}\) be a vector of size \(M(M-1)\times 1\) containing all \(\Gamma_{j,m}\), and \(\mathbf{\Gamma}_{m}\) be a vector of size \(2(M-1)\times 1\) containing both \(\Gamma_{j,m}\) and \(\Gamma_{m,j}\). For AP \(m\in\mathcal{M}\), replace the interference term \(\sum\limits_{i\in\mathcal{K}_{j}}\frac{p_{i}\Re\{h_{j,m}(h_{j})^{H}\}^{2}}{|h_{ j,i}|^{2}}\) with \(\Gamma_{m,j}\). Define the interference channel from device \(k,\forall k\in\mathcal{K}_{m}\) to neighboring cells as \(\left|g_{jk}\right|^{2}=\frac{\Re\{h_{j,k}(h_{m,k})^{H}\}^{2}}{|h_{m,k}|^{2}}\). Consequently, the optimality gap minimization problem can be independently addressed at AP \(m\in\mathcal{M}\), formulated as  
\[(\mathrm{P6.}m): \min_{\{p_{k}\delta_{m}\geq 0\}}E_{\mathrm{m}}\left(\sum_{k \in\mathcal{K}_{m}}\left(\frac{\sqrt{p_{k}}|h_{m,k}|}{\sqrt{\theta_{m}}}-v_{k} \right)^{2}+\sum_{j\in\mathcal{M}\setminus\{m\}}\frac{\Gamma_{m,j}}{\theta_{m} }+\frac{\sigma_{m}^{2}}{2\theta_{m}}\right)\] (43) s.t. \[\sum_{k\in\mathcal{K}_{m}}p_{k}|g_{jk}|^{2}\leq\Gamma_{j,m},j\in \mathcal{M}\setminus\{m\}, \tag{44}\] \[0\leq p_{k}\leq P_{k}^{max},\forall k\in\mathcal{K}_{m},m\in \mathcal{M}. \tag{45}\]  
Due to the complex coupling relationship between the transmit power variables \(p_{k}\) and the denoising factor \(\theta_{m}\) in the objective function of problem \((\mathrm{P6. }m)\), the problem exhibits a non-convex structure and is challenging to solve. To address this, we introduce the inverse denoising factor \(\gamma_{m}=1/\theta_{m}\) and an auxiliary variable \(Q_{k}=\sqrt{p_{k}\gamma_{m}}\), thereby transforming problem \((\mathrm{P6. }m)\) as  
\[(\mathrm{P7}): \min_{\{Q_{k},\gamma_{m}\geq 0\}}E_{m}\left(\sum_{k\in \mathcal{K}_{m}}\left(Q_{k}|h_{m,k}|-v_{k}\right)^{2}+\gamma_{m}\left(\sum_{j \in\mathcal{M}\setminus\{m\}}\Gamma_{m,j}+\frac{\sigma_{m}^{2}}{2}\right)\right)\] (46) s.t. \[\sum_{k\in\mathcal{K}_{m}}|Q_{k}g_{jk}|^{2}\leq\Gamma_{j,m} \gamma_{m},j\in\mathcal{M}\setminus\{m\}, \tag{47}\] \[Q_{k}^{2}\leq P_{k}^{max}\gamma_{m},\forall k\in\mathcal{K}_{m}, m\in\mathcal{M}. \tag{48}\]  
Since problem \((\mathrm{P7})\) is a standard convex optimization problem, it can be directly solved using common convex optimization tools (e.g., CVX [31]). To further derive a closed-form expression and enhance theoretical understanding of the problem structure, we employ the Lagrangian dual method for analysis. Let \(\left\{\lambda_{j,m}\right\}_{j\in\mathcal{M}\setminus\{m\}}\geq 0\) represent the Lagrange multipliers corresponding to the \(j\)-th IT constraint, and \(\varphi_{k}\) denote the Lagrange multiplier associated with the \(k\)-th edge device's power constraint.