# Distributed Interference-Aware Power Optimization for Multi-Task Over-the-Air Federated Learning
## 4 Proposed Method
### Distributed Scheme
Assuming \(\mathbf{T}_{j,m}= \begin{bmatrix}a&b\\ c&d\end{bmatrix} \), a feasible solution for \(\mathbf{t}_{j,m}\) is computed as  
\[\mathbf{t}_{j,m}=\text{sign}(bc-ad)\cdot\left[\ell_{j,m}d-b,a-\ell_{j,m}e\right]^{T}, \tag{71}\]  
where \(\ell_{j,m}\) is a ratio parameter that modulates the relative reduction rate of the optimality gap between APs \(m\) and \(j\); \(\text{sign}(\cdot)\) denotes the sign function. By adjusting \(\ell_{j,m}\), we can control the balance of improvement between the two APs. Specifically, setting \(\ell_{j,m}\geq 1\) prioritizes AP \(m\), while \(\ell_{j,m}\leq 1\) favors AP \(j\). Through the careful selection of \(\delta_{j,m}\) and \(\ell_{j,m}\), we can navigate the Pareto boundary to identify configurations where no AP performance can be improved without degrading the others. This iterative approach culminates in a distributed power control algorithm that leverages IT optimization to enhance overall system efficiency. The complete algorithm is encapsulated in Algorithm 2, which systematically updates the IT parameters to converge towards Pareto-optimal configurations. (The proposed distributed scheme currently assumes uniform data and computational resources across devices. In practice, edge devices may exhibit heterogeneity in dataset sizes, local data distributions, and processing capabilities. Our framework can accommodate such variations by incorporating per-device weighting or fairness constraints in the optimization objectives. Extending the algorithm to explicitly model non-IID data and device-level heterogeneity is a promising direction for future research). The distributed algorithm outlined above incrementally minimizes the global optimality gap across all APs through iterative, pairwise updates of IT parameters. (Although our distributed power optimization framework shares a conceptual similarity with some interference-constrained optimization methods (e.g., [35]), there are key distinctions. In [35], the objective is to enhance location privacy and spectrum efficiency in cognitive radio networks through spectrum sharing policies, typically under single-task settings. In contrast, our work addresses a fundamentally different problem of minimizing the learning optimality gap in multi-task Air-FL systems. Furthermore, our framework integrates gradient aggregation distortion, Pareto boundary trade-offs, and an iterative IT update mechanism that collectively enable decentralized learning coordination, which is different from [35].) In each iteration, a selected pair of APs adjusts their mutual IT levels to reduce their respective optimality gaps. Crucially, these adjustments are designed to ensure that the optimality gaps of other APs in the network remain unaffected, thereby preserving overall system stability. ```
1:Input: IT vector \(\mathbf{\Gamma}\), convergence threshold \(\iota\). 2:for any pair of APs \(m\) and \(j\), \(j\in\mathcal{M}\backslash\{m\}\)do
3:if\(\left|\mathbf{T}_{j,m}\right|>\iota\)then
4: APs \(m\) and \(j\) synchronously update their current IT values via a reliable backhaul link. 5: APs \(m\) and \(j\) solve problem \((\mathrm{P7})\) to obtain optimal solution \(\left\{p_{k}^{\ast}\right\}_{k\in\mathcal{K}_{m}}\), \(\left\{p_{i}^{\ast}\right\}_{i\in\mathcal{K}_{j}}\), \(\left\{\theta_{m}^{\ast},\theta_{j}^{\ast}\right\}\), \(\left\{\lambda_{j,m}^{\ast},\lambda_{m,j}^{\ast}\right\}\). 6: APs \(m\) and \(j\) update the elements in \(\mathbf{T}_{j,m}\) based on (68) and (69)
7: APs \(m\) and \(j\) exchange elements in \(\mathbf{T}_{j,m}\) via the backhaul link, reconstruct \(\mathbf{T}_{j,m}\) via (67), and compute \(\mathbf{t}_{j,m}\) via (71). 8: APs \(m\) and \(j\) update their IT values \(\Gamma_{j,m}^{\prime}\) according to (70). 9:endif
10:endfor
11:Output: Optimal device transmit power \(p_{k}^{\ast}\) and optimal denoising factor \(\theta_{m}^{\ast}\).