## 3 Methodology
### Problem Formulation  
In implicit feedback recommendation (a common approach in recommendation systems where user actions like clicks or views, rather than explicit ratings, are used as feedback), we solely consider whether the user has interacted with an item, where 1 denotes previous interaction and 0 denotes no interaction [26]. A set of interaction records of the user \(u\) with items is \(R_{u}=\{(u,i,r_{ui});i\in I\}\), where \(I\) is the set of items in the FRS and \(r_{ui}\in\{0,1\}\) represents the interaction case. FRSs aim to predict future interactions of user \(u\) with an item \(i\) that has not been interacted with. In FRSs, each user is treated as a client who utilizes his local data to model himself and the items for prediction. Typically, models are developed individually for clients and items. At the beginning of FL, the server selects FL clients and distributes initialized model parameters to participants for parallel local updates. On client \(u\)'s device, the client's user model parameter \(\theta_{u}\) and the model parameter of item \(i\), \(\delta_{i}\), are input into the client \(u^{\prime}\) score function \(score_{u}()\) to obtain the predicted value of client \(u\) to item \(i\), as Equation (1) shows. \[\hat{r}_{ui}=score_{u}(\theta_{u},\delta_{i}) \tag{1}\]  
In FedMF, \(\theta_{u}\) and \(\delta_{i}\) are user and item embeddings, respectively, and \(score_{u}()\) means calculating the inner product of them. The loss function of client \(u\) is defined as the binary cross-entropy loss for implicit feedback recommendation, shown as Equation (2). \[L_{u}=-\sum_{i\in I_{u}}\log\hat{r}_{ui}-\sum_{i^{\prime}\in I_{u}^{\prime}} \log(1-\hat{r}_{ui^{\prime}}) \tag{2}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Definition** \\ \hline \(I\) & The whole set of items in the FRS \\ \(I_{u}\) & The set of items that ever interacted with client \(u\) \\ \(I_{u}^{{}^{\prime}}\) & The negative instances set of client \(u\) \\ \(U\) & The whole set of clients in the FRS \\ \(U_{t}\) & Federated clients in communication round \(t\) \\ \(U_{t}^{\prime}\) & Communication-constrained clients in communication round \(t\) \\ \(R_{u}\) & Interaction records of user \(u\) \\ \(r_{ui}\in\{0,1\}\) & Interaction case of client \(u\) with item \(i\) \\ \(r_{ui}\) & The predicted value of client \(u\) to positive item \(i\) \\ \(\hat{r}_{ui^{\prime}}\) & The predicted value of client \(u\) to negative item \(i^{\prime}\) \\ \(\hat{\theta}_{u}\) & Client model parameters of client \(u\) \\ \(\delta_{i}\) & Global model parameters of item \(i\) \\ \(\delta_{ui}\) & Local model parameters of item \(i\) trained by client \(u\) \\ \(score_{u}()\) & The score function of client \(u\) \\ \(\eta\) & Learning rate \\ \(t\) & The index of communication round \\ \(\mathcal{B}\) & Mini-batches of size \(B\) \\ \(E\) & Epoch parameter of local update \\ \(e\) & Index of local epoch \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notations. where \(I_{u}\) is the set of items that ever interacted with client \(u\) and \(I_{u}^{{}^{\prime}}\) is the negative instances set of client \(u\). On the client side, client and item model parameters are alternately updated by Equation (3). \[\theta_{u}\leftarrow\theta_{u}-\eta\
abla_{\theta_{u}}L_{u}\qquad\delta_{ui} \leftarrow\delta_{ui}-\eta\
abla_{\delta_{ui}}L_{u} \tag{3}\]  
where \(\eta\) is the learning rate. \(\delta_{ui}\) is item \(i\)'s local model parameters trained by client \(u\). If the parameters of the score function also require updating, perform Equation (3) together with Equation (4). \[score_{u}\gets score_{u}-\eta\
abla_{score_{u}}L_{u} \tag{4}\]  
In the FedMF model, the function \(score_{u}()\) is defined as a fixed inner-product calculation operation. In contrast, in FedNCF, the \(score_{u}()\) consists of a Hadamard product operation and two shallow neural networks. The neural networks require the execution of Equation (4) for parameter updates.