Algorithm 1 PLDP-FL

Require: N is the number of local clients, $\gamma$ is the local client selection factor, $0<\gamma \le 1$, E is the number of local epochs, B is the local mini-batch size, $\eta$ is the learning rate, $\mathcal{L}$ is the loss function. Ensure: The trained model W.  1:ServerUpdate:  2:Initialize the weight parameter $w_0$ and send it to all clients;  3:for each round $t=1,2,\dots ,T$ do  4:   The server randomly selects $\gamma ·N$ local clients ${\mathcal{P}}_t$;  5:   for each client $P_{i\in \left[N\right]}$ in parallel do  6:     $w_{i,t}\leftarrow$ ClientUpdate $(P_i,{\overline{w}}_{t-1})$ and sends the result to the server;  7:   end for  8:   ${\overline{w}}_t\leftarrow \frac{1}{|{\mathcal{P}}_t|}{\sum }_i{w}_{i,t};$  9:   return ${\overline{w}}_t$.  10:end for  11:ClientUpdate$(P_i,{\overline{w}}_{t-1})$:  12:$t\leftarrow t+1$;  13:Receive the latest updates from the server: $w_{i,t}\leftarrow {\overline{w}}_{t-1}$;  14:for each local epoch $j=1,2,\dots ,E$ do  15:   for each batch $b\in B$ do  16:      $w_{i,t}\leftarrow w_{i,t}-\eta ·\nabla \mathcal{L}(w_{i,t};b)$;  17:   end for  18:end for  19:${\tilde{w}}_{i,t}\leftarrow$Personalized($w_{i,t}$).