# Distributed Interference-Aware Power Optimization for Multi-Task Over-the-Air Federated Learning
## 3 Convergence Analysis and Problem Formulation
### Optimality Gap vs. Aggregation Error  
Building upon the preceding assumptions, this subsection investigates the convergence behavior of parallel multi-task Air-FL, with a particular focus on how aggregation errors influence the optimality gap.  
Let \(\mathbf{e}_{m}^{(n+1)}\) denote the updated global model parameter for cell \(m\in\mathcal{M}\) after global communication round \(n\). The corresponding optimality gap is defined as \(\mathbb{E}\Big{[}H_{m}\Big{(}\mathbf{e}_{m}^{(n+1)}\Big{)}\Big{]}-H_{m}^{\star}\). Let \(\mathbb{E}\Big{[}\Re\big{(}\mathbf{e}_{m,\mathbf{c}}^{(n)}\big{)}^{2}\Big{]}\) represent the MSE of the global gradient estimate. With an appropriately designed diminishing learning rate, we establish the following result to characterize the convergence bound for each task.  
**Theorem 1**.: _Consider a parallel multi-task Air-FL setup where each cell \(m\in\mathcal{M}\) adopts a diminishing learning rate of the form \(0\leq\eta_{m}^{(n)}=\frac{u_{m}}{n+v_{m}}\leq\frac{1}{L_{m}}\leq\frac{1}{\mu_{m}},\forall n\in\mathcal{N}\) with constants \(v_{m}>0,u_{m}>\frac{1}{\mu_{m}}\). Suppose that each device employs a fixed mini-batch size \(u_{b}=N\). Under these settings, the expected optimality gap for each cell \(m\in\mathcal{M}\) after \(N\) communication rounds is bounded by_  
\[\mathbb{E}\Big{[}H_{m}\Big{(}\mathbf{e}_{m}^{(N+1)}\Big{)}\Big{]}-H_{ m}^{\star}\leq\prod_{n\in\mathcal{N}}D_{m}^{(n)}\Big{(}\mathbb{E}\Big{[}H_{m} \mathbf{e}_{m}^{1}\Big{)}\Big{]}-H_{m}^{\star}\Big{)}\] \[+\sum_{n\in\mathcal{N}}J_{m}^{(n)}(\eta_{m}^{(n)}B_{m}+L_{m}(\eta _{m}^{(n)})^{2}\sum_{c=1}^{\mathbb{C}}\mathbb{E}\big{[}\Re\{\mathbf{e}_{m,c}^{(n)} \}^{2}\big{]}), \tag{18}\]  
_where \(D_{m}^{(n)}\triangleq 1-\mu_{m}\eta_{m}^{(n)}\) satisfies \(0<D_{m}^{(n)}<1,\forall n\in\mathcal{N}\), \(J_{m}^{(n)}\triangleq\frac{\sum\limits_{m=1}^{N}D_{m}^{(n)}}{2D_{m}^{(n)}}\), and the gradient variance is defined as \(B_{m}\triangleq\frac{1}{K}\sum\limits_{k\in\mathcal{K}_{m}}\frac{\phi_{k}^{2}} {N}\)._  
**Remark 1**.: _The proof of Theorem 1 is similar to reference [19]. Please refer to reference [19] for more details. The convergence upper bound of the optimality gap in Theorem 1 consists of three distinct components. The first component is the initial optimality gap \(D_{m}^{(n)}\mathbb{E}\left[H_{m}\mathbf{e}_{m}^{1}\right]-H_{m}^{\star}\), determined by the discrepancy between the initial point of the global loss function and its optimal value in cell \(m\in\mathcal{M}\). The second component is the gradient variance term \(J_{m}^{(n)}\eta_{m}^{(n)}B_{m}\), capturing the influence of stochastic gradient noise arising from statistical heterogeneity within the local datasets of cell \(m\). The third component is the aggregation error term \(J_{m}^{(n)}L_{m}(\eta_{m}^{(n)})^{2}\sum\limits_{c=1}^{\mathbb{C}}\mathbb{E} \Big{[}\Re\{\mathbf{e}_{m,c}^{(n)}\}^{2}\Big{]}\), which quantifies the cumulative effect of model aggregation errors induced by channel noise, channel fading, and inter-cell interference. In real-world deployments, the number of communication rounds is finite, and the learning rate cannot be reduced indefinitely. As a result, each cell's optimization process converges to a neighborhood of the global minimum, characterized by a small but non-zero steady-state optimality gap._