# Distributed Interference-Aware Power Optimization for Multi-Task Over-the-Air Federated Learning
## 4 Proposed Method
### Distributed Scheme
Thus, the partial Lagrangian function for AP \(m\in\mathcal{M}\) is expressed as  
\[\mathcal{L}_{m}\Big{(}Q_{k},\gamma_{m}\left\{\lambda_{j,m}\right\} _{j\in\mathcal{M}\setminus\{m\}},\varphi_{k}\Big{)} =E_{\mathrm{m}}\Bigg{(}\sum_{k\in\mathcal{K}_{m}}\left(Q_{k}|h_{ m,k}|-v_{k}\right)^{2}\] \[+\gamma_{m}\left(\sum_{j\in\mathcal{M}\setminus\{m\}}\Gamma_{m,j }+\frac{\sigma_{m}^{2}}{2}\right)\Bigg{)}+\sum_{k\in\mathcal{K}_{m}}\varphi_{ k}\Big{(}Q_{k}^{2}-P_{k}^{\max}\gamma_{m}\Big{)}\] \[+\sum_{j\in\mathcal{M}\setminus\{m\}}\lambda_{j,m}\left(\sum_{k \in\mathcal{K}_{m}}\left|Q_{k}g_{jk}\right|^{2}-\Gamma_{j,m}\gamma_{m}\right). \tag{49}\]This leads to the dual function  
\[\mathcal{R}_{m}\Big{(}\big{\{}\lambda_{j,m}\big{\}}_{j\in\mathcal{M} \setminus\{m\}}\Big{)} = \min_{\{Q_{k}\geq 0,\gamma_{m}\geq 0\}}\mathcal{L}_{m}\Big{(}Q_{k}, \gamma_{m},\big{\{}\lambda_{j,m}\big{\}}_{j\in\mathcal{M}\setminus\{m\}}, \varphi_{k}\Big{)}\] (50) 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  
Therefore, the dual problem is formulated as  
\[\sum_{k\in\mathcal{K}_{m}}p_{k}|g_{jk}|^{2}\leq\Gamma_{j,m},j\in \mathcal{M}\setminus\{m\}, \tag{44}\]1  
Since problem \((\mathrm{P}7)\) satisfies [PERSON]'s condition, strong duality holds between its dual problem and the primal problem, which allows us to apply the Karush-Kuhn-Tucker (KKT) optimality conditions [32] to solve the problem. Define the optimal solutions of the dual problem as \(\big{\{}\lambda_{j,m}^{*}\big{\}}_{j\in\mathcal{M}\setminus\{m\}}\) and \(\varphi_{k}^{*}\). According to the KKT optimality conditions, the variables \(\left\{Q_{k}^{*},\gamma_{m^{*}}^{*},\left\{\lambda_{j,m}^{*}\right\}_{j\in \mathcal{M}\setminus\{m\}},\varphi_{k}^{*}\right\}\) satisfy the following complementary slackness conditions, given as  
\[\sum_{k\in\mathcal{K}_{m}}p_{k}|g_{jk}|^{2}\leq\Gamma_{j,m},j\in \mathcal{M}\setminus\{m\}, \tag{44}\]2 \[\sum_{k\in\mathcal{K}_{m}}p_{k}|g_{jk}|^{2}\leq\Gamma_{j,m},j\in \mathcal{M}\setminus\{m\}, \tag{44}\]3 (54) \[\sum_{k\in\mathcal{K}_{m}}p_{k}|g_{jk}|^{2}\leq\Gamma_{j,m},j\in \mathcal{M}\setminus\{m\}, \tag{44}\]4 (55) \[\sum_{k\in\mathcal{K}_{m}}p_{k}|g_{jk}|^{2}\leq\Gamma_{j,m},j\in \mathcal{M}\setminus\{m\}, \tag{44}\]5 (56) \[\sum_{k\in\mathcal{K}_{m}}p_{k}|g_{jk}|^{2}\leq\Gamma_{j,m},j\in \mathcal{M}\setminus\{m\}, \tag{44}\]6  
To determine the optimal auxiliary variable \(Q_{k}\) of each device, we begin by decomposing the Lagrangian dual function into \(K_{m}\) independent subproblems, each corresponding to a device's power optimization task. This decomposition allows for parallel optimization across devices. The complementary slackness conditions indicate that when \(\varphi_{k}^{*}>0\), the power constraint is strictly tight, and the device transmits at its maximum power \(P_{k}^{max}\). Conversely, if \(\varphi_{k}^{*}=0\), the device transmit power satisfies \(Q_{k}^{2}\leq P_{k}^{max}\gamma_{m}\). Accordingly, each subproblem is equivalently formulated as  
\[\sum_{k\in\mathcal{K}_{m}}p_{k}|g_{jk}|^{2}\leq\Gamma_{j,m},j\in \mathcal{M}\setminus\{m\}, \tag{44}\]7 (58) s.t.