# Local Differential Privacy-Based Federated Learning under Personalized Settings
## 3 System Model
### Client Side  
There are \(N\) clients participating in a federated learning task. The local training data of clients are either IID or non-IID. In each round, each client will update its local model parameter \(\widetilde{G}_{i}^{(r)}\), based on the federated model sent from the server. The client uses stochastic gradient descent (SGD) [24] to optimize the model's parameter, and obtain the local model parameter. Then, the client uses the perturbation mechanism \(M\), to add some random noise or make some perturbation on the local model parameter. Clients can configure their individualized privacy budget as they want. Parameter \(\varepsilon_{i}\), is the privacy budget of client \(i\) in the mechanism. We define that \(\varepsilon_{i}\) satisfies \(\varepsilon_{low}\leq\varepsilon_{i}\leq\varepsilon_{high}\), where \(\varepsilon_{low}\) is usually set to 1, and \(\varepsilon_{high}\) is usually set to 10 [25; 26]. The privacy budget \(\varepsilon_{i}\), of each client \(i\), is known to the server. Next, we will describe the local training process in the \(r\)-th round, in detail. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Description** & **Symbol** & **Description** \\ \hline \(\mathbb{R}\) & Real number field & \(M\) & Perturbation mechanism \\ \hline \(f\) & Aggregation rule & \(N\) & Total number of clients \\ \hline \(m\) & Number of selected clients & \(i\) & The \(i\)-th client \\  & by using probability parameter & & \\ \hline \(R\) & Maximum number of training rounds & \(r\) & The \(r\)-th round (\(r=1,2,\ldots,R\)) \\ \hline \(d\) & Dimension of model parameter & \(\eta\) & Learning rate of SGD \\ \hline \(\lambda l\) & Maximum number of logistic regression rounds & \(q\) & Random sampling probability \\ \hline \(\omega\) & A uniformly distributed random number & \(D_{i}\) & Training dataset of the \(i\)-th client \\  & between 0 and 1, generated by the server & & \\ \hline \(\varepsilon_{i}\) & Privacy budget of the \(i\)-th client & \(c_{i}\) & Clipping threshold of the \(i\)-th client \\ \hline \(\delta_{i}\) & Inverse of training dataset size (default: \(10^{-5}\)) & \(\sigma_{i}\) & Standard deviation of the \(i\)-th client \\ \hline \(\alpha_{i}\) & Weighting coefficient of the \(i\)-th client & \(\rho_{i}\) & Inverse of the standard deviation \\  & in the weighted average method & & of the Gaussian distribution of the \(i\)-th client \\ \hline \(P_{i}\) & Probability parameter of the \(i\)-th client & \(W^{(0)}\) & Initial federated model parameter \\ \hline \(W^{(r)}\) & Federated model parameter & \(g_{i}^{(r)}\) & Gradient of the \(i\)-th client in the \(r\)-th round \\  & generated in the \(r\)-th round & & \\ \hline \(\Delta_{i}^{(r)}\) & Local model update of the \(i\)-th client & \(G_{i}^{(r)}\) & Local model parameter of the \(i\)-th client \\  & in the \(r\)-th round & & generated in the \(r\)-th round \\ \hline \(\widehat{G}_{i}^{(r)}\) & Clipped local model parameter of the \(i\)-th client & \(\widetilde{G}_{i}^{(r)}\) & Perturbed local model parameter of the \(i\)-th client \\  & generated in the \(r\)-th round & & generated in the \(r\)-th round \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations. **Train the local model \(G_{i}^{(r)}\). ** Given the federated model parameter \(W^{(r-1)}\) from the previous round (i.e., the \(r-1\)-th round), each client applies the SGD algorithm on the local training dataset \(D_{i}\), and obtains the local model parameter \(G_{i}^{(r)}\):  
\[G_{i}^{(r)}=W^{(r-1)}-\eta g_{i}^{(r)}, \tag{3}\]  
where \(\eta\) denotes the learning rate of the SGD algorithm, and \(g_{i}^{(r)}\) denotes the gradient of the \(i\)-th client in the \(r\)-th round. It should be noted that, when the clients' training data are IID, each client \(i\), will sample from its own training dataset \(D_{i}\), with the same sampling probability \(q\), before training the local model, and then each client applies the SGD algorithm on the sampled dataset \(D_{i}^{\prime}\). **Model clipping.