Algorithm 1: L1-DKMSE

Input: Initialize the training set $S_j:=({\mathit{x}}_j,{\mathit{y}}_j)$ for node $j\in J$, iterations $k=0$, ${\overline{f_{B_j}}}^k({\mathit{x}}_j)={\mathit{y}}_j$, and $p_j^k({\mathit{x}}_j)=\mathbf{0}$, choose the Radial Basis Function (RBF) as the kernel function, and initialize the kernel parameter $\sigma$ and the regularization parameter $\lambda$.

Output: the sparse model $f_j^{*}(\cdot ),\forall j\in J$.

Repeat:

Step 1: each node $j\in J$ obtains its sparse model $f_j^k(\cdot )$ by iterations of Equations (27)–(29) using its training examples. Then, it broadcasts its sparse model to its one-hop neighboring nodes in $B_j$.

Step 2: each node $j\in J$ receives $f_i^k(\cdot ),i\in B_j$ and adds the key examples in $f_i^k(\cdot ),i\in B_j$ to its local training set.

Step 3: each node $j\in J$ predicts its local training examples by using $f_i^k(\cdot ), i\in B_j$ and then computes ${\overline{f_{B_j}}}^k({\mathit{x}}_j)$ and $p_j^k({\mathit{x}}_j)$ using Equations (23) and (24), respectively.

Step 4: If the models $f_j^k(\cdot )$ on each node are all stable, stop; otherwise, increment $k$ ($k=k+1$) and return to Step 1.