# An Effective Multi-Label Feature Selection Model Towards Eliminating Noisy Features
## 4 Multi-Task Optimization for ELC
Then, minimizing \(\mathcal{L}(\mathbf{U},\mathbf{W},\mathbf{p},\mathbf{V})\) is equivalent to the following two subproblems; i.e.,  
\[\bullet\quad\min_{\mathbf{U}}\mathcal{L}_{1}(\mathbf{U})=\min_{ \mathbf{U}}\frac{\beta}{2}\left\|\mathbf{W}-\mathbf{U}\right\|^{2}+\lambda \left\|\mathbf{U}\right\|_{2,1}+tr(\mathbf{V}^{T}\mathbf{U});\] \[\bullet\quad\min_{\mathbf{W},\mathbf{p}}\mathcal{L}_{2}(\mathbf{W },\mathbf{p})=\min_{\mathbf{W},\mathbf{p}}\frac{1}{\beta}\left\|\mathbf{\hat{A}} diag(\mathbf{p})\mathbf{W}-\mathbf{\Gamma}^{*}\right\|_{F}^{2}+\left\|\mathbf{W}- \mathbf{U}\right\|^{2}-\frac{2}{\beta}tr(\mathbf{V}^{T}\mathbf{W}).\]  
Note that the Lagrangian multiplier \(\mathbf{V}\) is fixed through the above analysis, mainly for simplifying the solution process. We further tackle this problem in the popular PADM framework as illustrated in Algorithm 1. In this framework, \(\mathbf{V}\) can be updated as  
\[\mathbf{V}^{[t+1]}=\mathbf{V}^{[t]}-\beta\left(\mathbf{W}^{[t+1]}-\mathbf{U}^{[t +1]}\right). \tag{15}\]  
```
0:\(\mathbf{F}=\left\{\mathbf{f}_{1},\ldots,\mathbf{f}_{d}\right\},\mathbf{Y}, \mathbf{S},k,\beta,\tau,\lambda\)
0:\(\mathbf{p}^{[t]}\)
1:begin
2:\(t=0,\mathbf{W}^{[0]}=\mathbf{0}_{d\times l},\mathbf{U}^{[0]}=\mathbf{0}_{d \times l},\mathbf{V}^{[0]}=\frac{1}{d}\mathbf{1}_{d\times l}\);
3:find top-\(k\) features \(\hat{\mathbf{z}}_{1}^{[0]},\ldots,\hat{\mathbf{z}}_{k}^{[0]}\) that minimize Equation (1), and set \(p_{i}^{[0]}=\left\{ \begin{array}{l}1,\ \mathbf{f}_{i}\in\left\{\hat{\mathbf{z}}_{1}^{[0]}, \ldots,\hat{\mathbf{z}}_{k}^{[0]}\right\}\\ 0,\text{ otherwise}\end{array} \right.\)
4:while \"not converged\" do
5: optimize \(\mathbf{U}^{[t+1]}\) according to Equation (9);
6: optimize \(\mathbf{W}^{[t+1]}\) according to Equation (13);
7: find top-\(k\) features \(\hat{\mathbf{z}}_{1}^{[t+1]},\ldots,\hat{\mathbf{z}}_{k}^{[t+1]}\) which minimize Equation (14), and set \(p_{i}^{[t+1]}=\left\{ \begin{array}{l}1,\ \mathbf{f}_{i}\in\left\{\hat{\mathbf{z}}_{1}^{[t+1]}, \ldots,\hat{\mathbf{z}}_{k}^{[t+1]}\right\}\\ 0,\text{ otherwise}\end{array} \right.\)
8: update \(\mathbf{V}^{[t+1]}\) according to Equation (15);
9:\(t=t+1\);
10:endwhile;
11:return\(\mathbf{p}^{[t]}\);
12:end;
```  
**Algorithm 1** ELC. ELC in Algorithm 1 is implemented in the regression framework PADM, which is a fast alternating approach for the well-known alternating direction method (ADM) framework. PADM is effective and efficient in solving the minimization problem of the augmented Lagrangian function, and is able to converge to a certain solution \(\left\{\mathbf{W}^{*},\mathbf{U}^{*}\right\}\) from any starting point \(\left\{\mathbf{W}^{[0]},\mathbf{U}^{[0]}\right\}\) for any \(\beta>0\)[40]. In terms of the complexity of ELC, it only takes \(O(k\log d)\) time to find \(k\) remarkable features from the \(d\) candidates. Thus, the time consumption for line 3 is \(O(ndl^{2}+k\log d)\). The cost of the while loop in Algorithm 1 mainly lies in lines 6 and 7, which is \(O(d^{2}l^{2}+ndl^{2}+k\log d)\). As this iteration process is repeated for \(t\) times, its total cost is \(O(t(d^{2}l^{2}+ndl^{2}+k\log d))\). Suppose \(t\gg 1\). Then, the total complexity of ELC is approximately equal to \(O(t(d^{2}l^{2}+ndl^{2}+k\log d))\), where \(d,n,l,k,t\) are the numbers of features, instances, labels, selected features, and iterations for convergence, respectively.