## Appendix B  
Proof of Lemma 5.: For this proof the following variable will be introduced \(q_{i}(k)=\sum_{j=1}^{N}w_{ij}\lambda_{j}(k)\). Then it can be written that  
\[||\lambda_{i}(k+1)-u||^{2}=||q_{i}(k)-u||^{2}-2\gamma_{i}(k)\frac{\
abla\Phi_ {i}\big{(}\lambda_{i}(k)\big{)}^{T}\big{(}q_{i}(k)-u\big{)}}{y_{ii}(k)}+\gamma _{i}^{2}(k)\frac{||\
abla\Phi_{i}(\lambda_{i}(k))||^{2}}{y_{ii}^{2}(k)}. \tag{1}\]  
It is worth highlighting here that the gradient calculation weighted by the \(y_{ii}(k)\) term appears here. By subsequently considering the row stochasticity of the matrix \(W\) and the convexity of the term \(||\cdot||^{2}\) and \(||q_{i}(k)-u||^{2}\), the term can be bounded as follows:  
\[||q_{i}(k)-u||^{2}\geq\sum_{j=1}^{N}w_{ij}||\lambda_{j}(k)-u||^{2}. \tag{2}\]  
Subsequently, when considering Equation (1) where the term \(\frac{2}{y_{ii}(k)}\) will be neglected, the following form will be obtained:\[-\gamma_{i}(k)\
abla\Phi_{i}\big{(}\lambda_{i}(k)\big{)}^{T}\cdot \big{(}q_{i}(k)-u\big{)}\leq la(k)||q_{i}(k)-u||-\frac{1}{k+1}\big{(}\
abla\Phi( \lambda(k))\big{)}^{T}\big{(}q_{i}(k)-u\big{)}\] \[\leq la(k)\sum_{j=1}^{N}w_{ij}||\lambda_{j}(k)-u||-\frac{1}{k+1} \big{(}\
abla\Phi(\overline{\lambda}(k))-\Phi_{i}(u)\big{)}+\frac{l}{k+1}\sum_ {j=1}^{N}w_{ij}||\lambda_{j}(k)-\overline{\lambda}(k)|| \tag{10}\] \[+\frac{2l}{k+1}||\lambda_{i}(k)-\overline{\lambda}(k)||.\]  
The first inequality in Equation (10) follows \(|\gamma_{i}(k)-\frac{1}{k+1}|\leq a(k)\) and at the same time \(|\
abla\Phi|\leq l\) and the second inequality is valid because \(\Phi_{i}(\lambda)\) is convex and \(l-\)Lipschitz continuous. (Lipschitz continuity is a strong form of uniform function continuity. Lipschitz's continuous function is limited in how fast it can change. A real number exists such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant.) In the next step, Equations (10) and (11) will be substituted into Equation (11), resulting in the following prescription:  
\[||\lambda_{i}(k+1)-u||^{2}\leq\sum_{j=1}^{N}w_{ij}||\lambda_{j}(k)-u||^{2}- \frac{2\big{(}\Phi(\overline{\lambda}(k))-\Phi_{i}(u)\big{)}}{(k+1)y_{ii}(k)} +l^{2}l^{2}\gamma_{i}^{2}(k) \tag{11}\]  
\[+\frac{2 lh}{k+1}\sum_{j=1}^{N}w_{ij}||\lambda_{j}(k)-\overline{\lambda}(k)||+ \frac{4 lh}{k+1}||\lambda_{i}(k)-\overline{\lambda}(k)||+2 lha(k)\sum_{j=1}^{N }w_{ij}||\lambda_{j}(k)-u||.\]  
Now, multiplying \(\pi_{i}\), both sides of Equation (11), the following will be obtained:  
\[ \begin{split}\sum_{i=1}^{N}\pi_{i}||&\lambda_{i}(k +1)-u||^{2}\leq\sum_{j=1}^{N}\pi_{i}||\lambda_{i}(k)-u||^{2}-\frac{2}{k+1} \sum_{i=1}^{N}\frac{\pi_{i}}{y_{ii}(k)}\big{(}\Phi_{i}(\overline{\lambda}(k)) -\Phi_{i}(u)\big{)}\\ &+\frac{6 lh}{k+1}\sum_{i=1}^{N}\pi_{i}||\lambda_{i}(k)-\overline {\lambda}(k)||+2 lha(k)\sum_{i=1}^{N}\pi_{i}||\lambda_{i}(k)-u||+h^{2}l^{2} \sum_{i=1}^{N}\pi_{i}\gamma_{i}^{2}(k).\end{split}  \tag{12}\]  
In Equation (12), the fact \(\pi^{T}\mathsf{W}=\pi^{T}\) will be used, thanks to which it will be simplified. Next, the second term will be considered on the right-hand side in Equation (12), for which  
\[ \begin{split}-\frac{2}{k+1}\sum_{i=1}^{N}\frac{\pi_{i}}{y_{ii}(k )}\big{(}\Phi_{i}(\overline{\lambda}(k))-\Phi_{i}(u)\big{)}\\ \leq-\frac{2}{k+1}\big{(}\Phi(\overline{\lambda}(k))-\Phi(u) \big{)}+\frac{2}{k+1}\sum_{i=1}^{N}\Big{|}\frac{y_{ii}(k)-\pi_{i}}{y_{ii}(k)} \Big{|}\cdot|\Phi_{i}(\overline{\lambda}(k))-\Phi_{i}(u)|\\ \leq-\frac{2}{k+1}\big{(}\Phi(\overline{\lambda}(k))-\Phi(u) \big{)}+\frac{2n\hbar\beta^{k}}{k+1}||\overline{\lambda}(k)-u||.\end{split}  \tag{13}\]  
If Equation (13) is substituted into (12), this substitution will lead to the shape given in Equation (31). The combination of Equations (A8)-(A10) implies that \(\sum_{k=0}^{\infty}c(k)<\infty\). Thanks to this, all the conditions stated in **Lemma 3** are fulfilled, and on this basis, two main results can be determined.