# Distributed Interference-Aware Power Optimization for Multi-Task Over-the-Air Federated Learning
## 3 Convergence Analysis and Problem Formulation
### Problem Formulation  
In this subsection, building on the convergence analysis from the previous subsection, we derive the optimality gap expression and further formulate a power control optimization problem aimed at minimizing the optimality gap in each cell. From the derived upper bound, it is evident that the initial gap and gradient variance terms remain constant once the system configuration is fixed. Therefore, our attention centers on the aggregation error, which varies with the power control variables and dominates the effective optimality gap to be minimized, formulated as  
\[\sum_{n\in\mathcal{N}}\sum_{c=1}^{\mathbb{C}}J_{m}^{(n)}L_{m}(\eta_{m}^{(n)}) ^{2}\mathbb{E}\bigg{[}\Re\{\mathbf{e}_{m,c}^{(n)}\}^{2}\bigg{]}. \tag{19}\]  
To enable tractable optimization in parallel multi-task scenarios, we decouple the problem into \(N\) subproblems, each corresponding to a single communication round. Furthermore, for clarity, we focus on the aggregation error in a single dimension and omit the superscript \(n\) without loss of generality. Based on Equation (9), the MSE for the \(c\)-th model parameter in cell \(m\) is given by  
\[\mathbb{E}\Big{[}\Re\{\mathbf{e}_{m,c}\}^{2}\Big{]}=\frac{1}{K^{2}} \Bigg{(}\sum_{k\in\mathcal{K}_{m}}\bigg{(}\frac{\sqrt{p_{k}}|h_{m,k}|}{\sqrt{ \theta_{m}}}-v_{k}\bigg{)}^{2}+\sum_{j\in\mathcal{M}\setminus\{m\}}\sum_{i\in \mathcal{K}_{j}}\frac{p_{i}\Re\{h_{m,i}(h_{j,j})^{H}\}^{2}}{\theta_{m}|h_{j,i} |^{2}}+\frac{\sigma_{m}^{2}}{2\theta_{m}}\Bigg{)}. \tag{20}\]Substituting this expression into (19), we obtain the following formulation for the effective optimality gap as a function of the power control variables, written as  
\[\Phi(\{p_{k}\},\{\theta_{m}\})\triangleq E_{m}\left(\sum_{k\in\mathcal{K}_{m}}\left(\frac{\sqrt{p_{k}}|h_{m,k}|}{ \sqrt{\theta_{m}}}-v_{k}\right)^{2}+\sum_{j\in\mathcal{M}\backslash\{m\}}\sum_ {i\in\mathcal{K}_{j}}\frac{p_{i}\Re\{h_{m,j}h_{j,i}^{H}\}^{2}}{\theta_{m}|h_{ j,i}|^{2}}+\frac{\sigma_{m}^{2}}{2\theta_{m}}\right), \tag{21}\]  
where \(E_{m}=\frac{I_{m}L_{m}\eta_{m}^{2}}{K^{2}}\). Based on this, the optimality gap minimization problem is formulated as  
\[(\mathrm{P0}): \min_{\{p_{k},\theta_{m}\}}\{\Phi(\{p_{k}\},\{\theta_{m}\})\}\] (22) s.t. \[0\leq\theta_{m}, \tag{23}\] \[0\leq p_{k}\leq p_{k}^{max},\forall k\in\mathcal{K}_{m},m\in \mathcal{M}. \tag{24}\]  
Minimizing the objective function \(\Phi(\{p_{k}\},\{\theta_{m}\})\) effectively reduces the optimality gap for each cell. However, in multi-cell wireless systems, independently optimizing each cell may inadvertently amplify inter-cell interference, thus degrading the learning performance of neighboring cells.  
To address this challenge and ensure fairness across distributed tasks, it is essential to introduce a theoretical framework that captures the global trade-offs among all cells' learning performances. To this end, the next subsection introduces the concept of the Pareto boundary to characterize the optimal trade-off set of optimality gaps among cells under shared resource constraints, thereby laying a theoretical foundation for subsequent coordinated power control strategies across cells.