## 6 E-DDTMD Algorithm Based on ADMM
### E-Ddtmd Algorithm Based on ADMM  
According to [32], we have the augmented Lagrangian function of problem (32) as  
\[ \begin{array}{l}L_{p}(\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\}_{i\in\mathcal{ I}},\{\mathbf{b},\mathbf{d}\},\{\mathbf{\delta^{i}},\mathbf{\sigma^{i}}\}_{i\in\mathcal{I}})\\ =\sum\limits_{i\in\mathcal{I}}v_{i}(a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}})\\ +\sum\limits_{i\in\mathcal{I}}\sum\limits_{k\in\mathcal{I},j\in\mathcal{J}} \delta^{i}_{k,j}(\hat{b}^{i}_{k,j}-b_{k,j})+\sum\limits_{i\in\mathcal{I}}\sum \limits_{k\in\mathcal{I},j\in\mathcal{J}}\sigma^{i}_{k,j}(\hat{d}^{i}_{k,j}-d _{k,j})\\ +\frac{\rho}{2}\sum_{i\in\mathcal{I}}\sum_{k\in\mathcal{I},j\in\mathcal{J}} \left\|\hat{b}^{i}_{k,j}-b_{k,j}\right\|^{2}+\frac{\rho}{2}\sum_{i\in\mathcal{ I}}\sum_{k\in\mathcal{I},j\in\mathcal{J}}\left\|\hat{d}^{i}_{k,j}-d_{k,j} \right\|^{2},\end{array}  \tag{33}\]  
where the \(\mathbf{\sigma^{i}}=\left\{\sigma^{i}_{k,j}\right\}\) and \(\mathbf{\delta^{i}}=\left\{\sigma^{i}_{k,j}\right\}\) and is problem (33) of the Lagrange multiplier, \(\rho\) is punish coefficient [30]. E-DDTMD based on ADMM used in through an iterative update \(\left\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\right\}_{i\in\mathcal{I}}, \left\{\mathbf{b},\mathbf{d}\right\}\) and \(\left\{\mathbf{\delta^{i}},\mathbf{\sigma^{i}}\right\}_{i\in\mathcal{I}}\) to solve (33), set \(\left\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\right\}_{i\in\mathcal{I}}^{[t]}, \left\{\mathbf{b},\mathbf{d}\right\}^{[t]}\) and \(\left\{\mathbf{\delta^{i}},\mathbf{\sigma^{i}}\right\}_{i\in\mathcal{I}}^{[t]}\) corresponding to the first \(t\) iteration \(\left\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\right\}_{i\in\mathcal{I}}, \left\{\mathbf{b},\mathbf{d}\right\}\) and \(\left\{\mathbf{\delta^{i}},\mathbf{\sigma^{i}}\right\}_{i\in\mathcal{I}}\). Below, according to the following steps, we update the \((t+1)\) iteration when \(\left\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\right\}_{i\in\mathcal{I}}, \left\{\mathbf{b},\mathbf{d}\right\}\), and \(\left\{\mathbf{\delta^{i}},\mathbf{\sigma^{i}}\right\}_{i\in\mathcal{I}}\) value. (1) SN local variables update \(\left\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\right\}_{i\in\mathcal{I}}\) update:  
In this step, we need to update the local variables. In a given \(\left\{\mathbf{b},\mathbf{d}\right\}^{[t]}\) and \(\left\{\mathbf{\delta^{i}},\mathbf{\sigma^{i}}\right\}_{i\in\mathcal{I}}^{[t]}\) the local variable \(\left\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\right\}_{i\in\mathcal{I}}^{[t +1]}\) is updated; the equivalent to the following questions needs to be solved:  
\[\left\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\right\}_{i\in\mathcal{I}}^{[t+ 1]}=\underset{\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\}}{\min}L_{\rho}( \left\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\right\}_{i\in\mathcal{I}}, \left\{\mathbf{b},\mathbf{d}\right\}^{[t]},\left\{\mathbf{\delta^{i}},\mathbf{\sigma^{i}} \right\}_{i\in\mathcal{I}}^{[t]}). \tag{34}\]  
For problem (34), we can decompose it into I subproblems, each of which is solved locally and independently by each SN in the optimization layer.