# An Effective Adaptive Combination Strategy for Distributed Learning Network
## 5 Simulation Results  
We evaluate herein the MSD of the proposed algorithm. Without loss of generality, the unknown weight vector \(\mathbf{w}_{o}\) is set to be \(\mathbf{1}_{M}/M\) with \(M=5\). The initial weight vector estimations are \(\mathbf{w}_{k}(0)=\mathbf{0}_{M}\) for each node \(k\). The constant \(\varepsilon\) used in (22) is set to be \(0.5\times 10^{-4}\) and the forgetting factor \(\lambda=0\) or \(\lambda=0.95\). We consider \(\mathbf{D}_{k}(t)=\text{diag}\{\mathbf{b}_{k}(t)\}^{\gamma}\) in terms of the proposed AC, where \(\gamma\) can be \(0,-1\) or \(-2\). We use the empirical MSDs as the performance metrics. Both the transient and steady-state empirical network MSDs hereinafter are obtained by averaging \(\mathcal{L}=500\) independent trials over all nodes of the network,  
\[\mathcal{MSD}(t) \triangleq\frac{1}{\mathcal{LN}}\sum_{k=1}^{N}\sum_{\ell=1}^{ \mathcal{L}}\|\tilde{\mathbf{w}}_{k}^{(\ell)}(t)\|_{2}^{2}, \tag{27}\] \[\mathcal{MSD} \triangleq\frac{1}{\mathcal{LN}}\sum_{k=1}^{N}\sum_{\ell=1}^{ \mathcal{L}}\|\tilde{\mathbf{w}}_{k}^{(\ell)}(\infty)\|_{2}^{2}, \tag{28}\]  
where \(\tilde{\mathbf{w}}_{k}^{(\ell)}(t)\triangleq\mathbf{w}_{\text{o}}-\mathbf{w}_{k}^{(\ell)} (t)\) with \(\mathbf{w}_{k}^{(\ell)}(t)\) denoting the transient weight estimate of the \(\ell\)th trial and \(\|\tilde{\mathbf{w}}_{k}^{(\ell)}(\infty)\|_{2}^{2}\) is obtained by averaging \(\|\tilde{\mathbf{w}}_{k}^{(\ell)}(t)\|_{2}^{2}\) of 100 iterations after convergence. **Example 1**. : _We consider the same topology of the network with \(N=15\) nodes as ([12], Figure 5) ([17], Figure 6a), illustrated in Figure (a)a. The measurement noise is real Gaussian white noise with its variance \(\sigma_{v,k}^{2}\) at each node \(k\) presented in Figure (b)b. According to the mean convergence condition, we set the step size to \(\mu_{k}=0.01\) for each node \(k\). We herein consider \(\gamma=0\). Each regressor \(\mathbf{u}_{k}(t)\) at each node \(k\) is real Gaussian white noise sequence with their covariance matriceswith \(\sigma^{2}_{u,k}=1\) for all \(k\). The noise power of node \(5,12,13\) at the 1500 th iteration suddenly goes up to 5._  
As demonstrated in Figure 2, the proposed AC strategy outperforms the classic AC [12] and the uniform combination [1] in terms of superior steady-state performance and similar convergence rates, and outperforms LS-based AC [17] and the optimal AC [13] in terms of accelerated convergence rate. We also observe that the forgetting factor can further enhance the performance of the proposed AC. After the sudden change of the noise power of some nodes, the proposed AC exhibits rather high reconvergence rate and good steady-state performance. **Example 2**. : _The initial measurement noise variance \(\sigma^{2}_{v,k}\) at each node \(k\) is presented in Figure 1b. In this simulation, we compare the proposed AC scheme with the uniform combination and the classic AC in terms of the steady-state MSD at different noise variances \(\tau\sigma^{2}_{v,k}\) at each node \(k\). The other simulation conditions are same as Example 1. The steady-state MSDs with respect to the noise variances are illustrated in Table 1._  
The fairness of this experiment is endorsed by the relatively approximate convergence behavior of the transient MSD curves plotted in Figure 3. As shown in Table 1, with the noise variances increasing, the ATC algorithm incorporated with the proposed AC exhibits  
Figure 1: Network topology and noise profile: (**a**) network topology; (**b**) noise profile. Figure 2: The MSD learning curves. superior performance compared to the ATC algorithm with the uniform combination or the classic AC strategy. Additionally, we can also find that the performance gain brought by the forgetting factor is limited when the noise variances increase to some certain limit.