# Distributed Interference-Aware Power Optimization for Multi-Task Over-the-Air Federated Learning
## 2 System Model
### Communication Model  
Each cell adopts the federated stochastic gradient descent (FedSGD) for model training, where devices compute and transmit gradients to the AP via AirComp-enabled uplink aggregation. The training is conducted over \(N\) global communication rounds, indexed by \(n\in\mathcal{N}\triangleq\{1,\dots,N\}\). At the beginning of round \(n=1\), the aggregation center broadcasts a common initialization \(\mathbf{e}^{(1)}\) to all APs, which then distribute it to their associated devices, setting their local model parameters \(\mathbf{e}_{k}^{(1)}=\mathbf{e}^{(1)},\forall k\in\mathcal{K}_{m}\). We assume that the downlink is error-free. During each round \(n\), device \(k\in\mathcal{K}_{m}\) computes its local gradient estimate \(\mathbf{a}_{k}^{(n)}\) over a randomly sampled mini-batch \(\widetilde{\mathcal{D}}_{k}^{(n)}\subset\mathcal{D}_{k}\), expressed as  
\[\mathbf{a}_{k}^{(n)}=\frac{1}{|\widetilde{\mathcal{D}}_{k}^{(n)}|}\sum_{\mathbf{x}_{i },\mathbf{\xi}_{i}\in\widetilde{\mathcal{D}}_{k}^{(n)}}\
abla H_{k}(\mathbf{e}_{k}^{( n)},\mathbf{x}_{i},\mathbf{\tilde{\mathbf{x}}}_{i}). \tag{4}\]  
To align power levels and mitigate signal distortion, each device applies gradient normalization, defined as  
\[\mathbf{a}_{k}^{(n)} =\frac{1}{C}\sum_{c=1}^{C}a_{k,c}^{(n)}, \tag{5}\] \[(v_{k}^{(n)})^{2} =\frac{1}{C}\sum_{c=1}^{C}\Big{(}a_{k,c}^{(n)}-\bar{a}_{k}^{(n)} \Big{)}^{2},\] (6) \[s_{k,c}^{(n)} =\frac{a_{k,c}^{(n)}-\bar{a}_{k}^{(n)}}{v_{k}^{(n)}}, \tag{7}\]  
where \(\bar{a}_{k}^{(n)}\in\mathbb{R}\) and \(v_{k}^{(n)}\in\mathbb{R}_{+}\) denote the mean and standard deviation of the \(C\) dimensional local gradient \(\mathbf{a}_{k}^{(n)}\), respectively. \(s_{k,c}^{(n)}\) is the normalized signal to be transmitted, satisfying \(\mathbb{E}[s_{k,c}^{(n)}]=0\) and \(\mathbb{E}[(s_{k,c}^{(n)})^{2}]=1\). Gradients from different devices are assumed to be statistically independent. Each device transmits its signal over a quasi-static channel. Let \(h_{m,k}^{(n)}\) denote the channel from device \(k\in\mathcal{K}_{m}\) to its AP \(m\in\mathcal{M}\), and \(h_{m,i}^{(n)}\) denote the channel from interfering device \(i\in\mathcal{K}_{j}\), \(j\
eq m\) to AP \(m\). The transmission coefficient is set as \(b_{k}^{(n)}=\sqrt{p_{k}^{(n)}}\frac{(h_{m,k}^{(n)})^{H}}{|h_{m,k}^{(n)}|}\) with power \(p_{k}^{(n)}\leq P_{k}^{max}\), where \(P_{k}^{max}\) is the instantaneous power budget on each device. The signal received at AP \(m\in\mathcal{M}\) for the \(c\)-th model parameter is expressed as  
\[r_{m,c}^{(n)}=\sum_{k\in\mathcal{K}_{m}}\sqrt{p_{k}^{(n)}}|h_{m,k}^{(n)}|s_{k,c }^{(n)}+Z_{m,c}^{(n)}+\sum_{j\in\mathcal{M}\setminus\{m\}}\sum_{i\in\mathcal{K }_{j}}\frac{\sqrt{p_{i}^{(n)}}h_{m,i}^{(n)}(h_{j,i}^{(n)})^{H}}{|h_{j,i}^{(n) }|}s_{i,c}^{(n)}, \tag{8}\]  
where \(Z_{m,c}^{(n)}\sim\mathcal{CN}(0,\sigma_{m}^{2})\) denotes the additive white Gaussian noise (AWGN) with power \(\sigma_{m}^{2}\) at the receiver of AP \(m\in\mathcal{M}\). The denoising factor \(\theta_{m}^{(n)}\) is used to equalize signal power and suppress noise/interference.