# Distributed Interference-Aware Power Optimization for Multi-Task Over-the-Air Federated Learning
## 4 Proposed Method
### Centralized Scheme
#### 4.1.2 Device Transmit Power Optimization  
Substituting the solution for \(\theta_{m}\) in Equation (30) back into problem \((\mathrm{P1})\), problem \((\mathrm{P1})\) is rewritten as  
\[(\mathrm{P3}):\ \ \min_{\{p_{k},0<\xi\}}\xi \tag{31}\] \[\mathrm{s.t.}\ \left(\sum\limits_{k\in\mathcal{K}_{m}}v_{k}^{2}- \frac{\kappa_{m}\xi}{E_{m}}\right)\left(\sum\limits_{k\in\mathcal{K}_{m}}p_{k} |h_{m,k}|^{2}+\sum\limits_{j\in\mathcal{M}\backslash\{m\}}\sum\limits_{i\in \mathcal{K}_{j}}\frac{p_{i}\Re\{h_{m,i}(h_{j,i})^{H}\}^{2}}{|h_{j,i}|^{2}}+ \frac{\sigma_{m}^{2}}{2}\right)\] \[\ \ \ \ \ \ \leq\left(\sum\limits_{k\in\mathcal{K}_{m}}\sqrt{p_{k}}|h_ {m,k}|v_{k}\right)^{2},\forall m\in\mathcal{M},\] (32) \[\ \ \ \ \ 0\leq p_{k}\leq P_{k}^{max},\forall k\in\mathcal{K}_{m},m \in\mathcal{M}. \tag{33}\]  
For any given \(\xi\), problem \((\mathrm{P3})\) can be reformulated as  
\[(\mathrm{P4}):\ \ \mathrm{Find}\{p_{k}\} \tag{34}\] \[\mathrm{s.t.}\ \psi_{m}\left(\sum\limits_{k\in\mathcal{K}_{m}}p_{k}|h_{m,k}|^{2} +\sum\limits_{j\in\mathcal{M}\backslash\{m\}}\sum\limits_{i\in\mathcal{K}_{j}} \frac{p_{i}\Re\{h_{m,i}(h_{j,i})^{H}\}^{2}}{|h_{j,i}|^{2}}+\frac{\sigma_{m}^{2 }}{2}\right)\] \[\ \ \ \ \ \ \leq\left(\sum\limits_{k\in\mathcal{K}_{m}}\sqrt{p_{k}}|h_ {m,k}|v_{k}\right)^{2},\forall m\in\mathcal{M},\] (35) \[\ \ \ \ \ 0\leq p_{k}\leq P_{k}^{max},\forall k\in\mathcal{K}_{m},m \in\mathcal{M}, \tag{36}\]  
where \(\psi_{m}=\sum\limits_{k\in\mathcal{K}_{m}}v_{k}^{2}-\frac{\kappa_{m}\xi}{E_{m}}\). Introduce the interference matrix \(\mathbf{\Lambda}_{m}\triangleq[\Lambda_{m,1},\Lambda_{m,2},\ldots,\Lambda_{m,K_ {\mathrm{tot}}}]\) with \(K_{\mathrm{tot}}=KM\), defined as  
\[\Lambda_{m,k}=\left\{ \begin{array}{ll}|h_{m,k}|,&k\in\mathcal{K}_{m},\\ \frac{\Re\{h_{m,j}(h_{j,i})^{H}\}}{|h_{j,i}|},&k\in\mathcal{K}_{j},\ j\in \mathcal{M}\setminus\{m\}.\end{array} \right. \tag{37}\]Constraint (35) can be equivalently expressed as  
\[\sqrt{\psi_{m}\left(\sum_{m\in\mathcal{M}}\sum_{k\in\mathcal{K}_{m}}\Lambda_{m,k} ^{2}p_{k}+\frac{\sigma_{m}^{2}}{2}\right)}\leq\sum_{k\in\mathcal{K}_{m}}\sqrt{p _{k}}|h_{m,k}|v_{k}. \tag{38}\]  
Define \(\mathbf{q}=\big{[}\sqrt{p_{1}},\sqrt{p_{2}},\ldots,\sqrt{p_{K_{\text{tot}}}}\big{]} ^{\mathsf{T}},\mathbf{\kappa}_{m}=\text{diag}\big{(}[\sigma_{m}^{2}/2,\mathbf{\Lambda} _{m}]\big{)},\mathbf{\beta}_{m}=[\beta_{m,1},\beta_{m,2},\ldots,\beta_{m,K_{\text{ tot}}}]^{\mathsf{T}}\) with \(\beta_{m,k}\) satisfying:  
\[\beta_{m,k}=\left\{ \begin{array}{ll}|h_{m,k}|v_{k},&k\in K_{m},\\ 0,&k\in K_{j},j\in\mathcal{M}\setminus\{m\}.\end{array} \right. \tag{39}\]  
Therefore, problem (P4) can be transformed into a standard second-order cone program (SOCP) problem, given as  
\[\text{(P5)}:\text{Find }\{\mathbf{q}\} \tag{40}\] \[\text{s.t. }\sqrt{\psi_{m}}\big{\|}[1;\mathbf{q}]^{\mathsf{T}}\mathbf{ \kappa}_{m}\big{\|}\leq\mathbf{q}^{\mathsf{T}}\beta_{m},\forall m\in\mathcal{M},\] (41) \[0\leq q_{k}\leq\sqrt{p_{k}^{\text{max}}},\forall k\in\mathcal{K} _{m},m\in\mathcal{M}. \tag{42}\]  
Problem (P5) can be efficiently solved using standard convex optimization solvers (e.g., CVX [31]). Once the optimal solution \(\mathbf{q}^{*}\) is obtained, the transmit powers follow as \(p_{k}^{*}=(q_{k}^{*})^{2}\). To determine the optimal \(\varsigma^{*}\), we employ a bisection search over feasible \(\varsigma\) values and solve the corresponding SOCP at each step. The complete process is summarized in Algorithm 1. ```
1:Input: Local gradient variance \(v_{k}\), profile vector \(\mathbf{\kappa}\), convergence threshold \(\iota\), and maximum power budget \(P_{k}^{\text{max}}\). 2:Set \(\varsigma^{\text{low}}\gets 0,\varsigma^{\text{up}}\leftarrow\min_{m\in \mathcal{M}}\big{(}E_{m}\sum_{k\in\mathcal{K}_{m}}v_{k}^{2}/\kappa_{m}\big{)}\). 3:while\(\big{|}\varsigma^{\text{up}}-\varsigma^{\text{low}}\big{|}<\iota\)do
4:\(\varsigma\leftarrow\frac{\varsigma^{\text{low}}+\varsigma^{\text{up}}}{2}\). 5: Solve problem (P5) to obtain \(p_{k}^{*}\). 6:ifproblem (P5) is feasiblethen
7:\(\varsigma^{\text{up}}\leftarrow\varsigma\).