# Distributed Interference-Aware Power Optimization for Multi-Task Over-the-Air Federated Learning
## 4 Proposed Method
### Distributed Scheme
Future work may consider extending the framework to incorporate robust optimization techniques for time-varying or uncertain CSI, thereby improving resilience under mobility and fading.) Building upon this foundation, an iterative collaborative algorithm is proposed, wherein APs dynamically adjust mutual \(\Pi\) values through peer-to-peer signaling interactions. In each iteration, the system selects a pair of APs for local collaborative updates. These updates are designed to ensure that the optimality gap of the local cell does not increase, nor does the performance of other cells degrade, thereby guaranteeing non-decreasing overall system performance. To implement this mechanism, it is assumed that APs possess basic backhaul link support for essential information sharing and synchronization of interference parameters. However, determining whether the system achieves Pareto optimality under arbitrary \(\Pi\) configurations remains challenging. To address this, a lemma is proposed to characterize the necessary constraints between \(\Pi\) and the cell optimality gap under Pareto optimality conditions. This lemma provides a theoretical foundation for subsequent algorithm design. **Lemma 1** (Necessary Condition for Pareto Optimality). : _Under any given \(\Pi\) configuration \(\mathbf{\Gamma}\), if the optimality gap \(\overline{\Delta_{m}}(\mathbf{\Gamma}_{m})\) has reached a Pareto optimal state, then for any pair of APs \(m\) and \(j\), the determinant of the following \(2\times 2\) matrix \(\mathbf{T}_{j,m}\) must be zero, given as [34]_  
\[\mathbf{T}_{j,m}=\left[ \begin{array}{cc}\frac{\partial\overline{\Delta_{m}}( \mathbf{\Gamma}_{m})}{\partial\Gamma_{j,m}}&\frac{\partial\overline{\Delta_{m}} (\mathbf{\Gamma}_{m})}{\partial\Gamma_{m,j}}\\ \frac{\partial\overline{\Delta_{j}}(\mathbf{\Gamma}_{j})}{\partial\Gamma_{j, m}}&\frac{\partial\overline{\Delta_{j}}(\mathbf{\Gamma}_{j})}{\partial \Gamma_{m,j}}\end{array} \right]. \tag{67}\]  
Building upon the previously established analytical framework, we now delve into the derivation of the matrix elements within the \(\Pi\) optimization context. By solving both the primal and dual formulations of the problem, we obtain explicit expressions for each component of the matrix \(\mathbf{T}_{j,m}\), which encapsulates the sensitivity of the optimality gaps to variations in \(\Pi\) parameters. Specifically, the partial derivatives of the optimality gap \(\overline{\Delta_{m}}(\mathbf{\Gamma}_{m})\) with respect to the \(\Pi\) parameters \(\Gamma_{j,m}\) and \(\Gamma_{m,j}\) are given by  
\[\frac{\partial\overline{\Delta_{m}}(\mathbf{\Gamma}_{m})}{\partial\Gamma_{j, m}}=-\lambda_{j,m}^{*}\gamma_{m^{\prime}}^{*}\frac{\partial\overline{\Delta_{j}}( \mathbf{\Gamma}_{j})}{\partial\Gamma_{m,j}}=-\lambda_{m,j}^{*}\gamma_{j}^{*}, \tag{68}\]where \(\left\{\lambda_{j,m}^{*},\lambda_{m,j}^{*}\right\}\) respectively denote the optimal dual multipliers associated with the IT constraints in the primal problems of cell \(m\) and cell \(j\), and \(\left\{\gamma_{m}^{*},\gamma_{j}^{*}\right\}\) are the corresponding optimal inverse denoising factors. Additionally, the derivatives with respect to the reciprocal IT parameters are written as  
\[\frac{\partial\overline{\Delta_{m}}(\mathbf{\Gamma}_{m})}{\partial\Gamma_{m,j}}=E_ {m}\gamma_{m}^{*},\frac{\partial\overline{\Delta_{j}}(\mathbf{\Gamma}_{j})}{ \partial\Gamma_{j,m}}=E_{j}\gamma_{j}^{*}. \tag{69}\]  
To iteratively refine the IT vector \(\mathbf{\Gamma}\), we focus on updating only the mutual IT parameters between a selected pair of APs \(m\) and \(j\), while keeping the rest unchanged, which is defined as  
\[\left[\Gamma_{j,m}^{\prime},\Gamma_{m,j}^{\prime}\right]^{T}=\left[\Gamma_{j, m},\Gamma_{m,j}\right]^{T}+\delta_{j,m}\cdot\mathbf{t}_{j,m}, \tag{70}\]  
where \(\mathbf{\Gamma}^{\prime}\) represents the updated IT vector; \(\delta_{j,m}\) denotes a suitably small step size; and \(\mathbf{t}_{j,m}\) is a vector ensuring that the product \(\mathbf{T}_{j,m}\mathbf{t}_{j,m}<0\), thereby guaranteeing a reduction in the optimality gaps.