# Communication-Efficient Federated Learning with Adaptive Consensus ADMM
## 4 Analysis
 \tag{27}\]  
\[ \begin{split}&\big{\|}y^{t}-y^{*}\big{\|}^{2}+\gamma(\rho^{t})^{2} \big{\|}\theta^{t}-\theta^{*}\big{\|}^{2}\geq\Big{\|}y^{t+1}-y^{*}\Big{\|}^{2} +\gamma(\rho^{t+1})^{2}\big{\|}\theta^{t}-\theta^{*}\big{\|}^{2}+\gamma(2- \gamma)(\rho^{t+1})^{2}\\ &\times\Big{\|}w_{i}^{t+1}-\theta^{t+1}\Big{\|}^{2}+\gamma(\rho^ {t+1})^{2}\Big{\|}\theta^{t}-\theta^{t+1}\Big{\|}^{2}+2\gamma\rho^{t+1}(w_{i} ^{t+1}-\theta^{t+1})^{\top}(\theta^{t+1}-\theta^{t})\end{split}  \tag{28}\]  
**Lemma 2**. : _For \(t\geq 2\),we obtain_  
\[\rho^{t+1}(w_{i}^{t+1}-\theta^{t+1})^{\top}(\theta^{t+1}-\theta^{t})\geq(1- \gamma)\rho^{t}(w_{i}^{t}-\theta^{t})^{\top}(\theta^{t+1}-\theta^{t}) \tag{29}\]  
_We define \(\gamma\in(0,(1+\sqrt{5})/2)\), and therefore, \(1+\gamma-\gamma^{2}>0\). Assume \(T=2-(1/3)(1+\gamma-\gamma^{2})\), and we can obtain:_  
\[2-T=(1/3)(1+\gamma-\gamma^{2}) \tag{30}\]  
\[T-\gamma=(1/3)(\gamma^{2}-4\gamma+5) \tag{31}\]  
**Theorem 1**. : _Let \(\left\{w_{i}^{*},y^{*},\theta^{*}\right\}\) denote a solution to the variable inequality problem. Then, there exists a constant \(t_{0}\geq 2\), if satisfied \(\forall t\geq t_{0}\):_  
\[\left\|y^{t+1}-y^{*}\right\|^{2}+\gamma(\rho^{t+1})^{2}\Big{\|} \theta^{t+1}-\theta^{*}\Big{\|}^{2}+\gamma(T-\gamma)\times(\rho^{t+1})^{2} \Big{\|}w_{i}^{t+1}-\theta^{t+1}\Big{\|}^{2}\leq \tag{32}\] \[(1+\tau_{i})^{2}(\left\|y^{k}-y^{*}\right\|^{2}+\gamma(\rho^{t})^{ 2}\Big{\|}\theta^{k}-\theta^{*}\Big{\|}^{2}+\gamma(T-\gamma)(\rho^{t})^{2} \big{\|}w_{i}^{t}-\theta^{t}\big{\|}^{2})-\gamma(2-T)\] \[\times(\rho^{t+1})^{2}(\left\|w_{i}^{t+1}-\theta^{t+1}\right\|^{ 2}+\left\|\theta^{t+1}-\theta^{t}\right\|\Big{\|}^{2})\]  
**Proof of Theorem 1. ** Combining (28), (29) and the Cauchy-Schwartz inequality, we obtain:  
\[\left\|y^{t+1}-y^{*}\right\|^{2}+\gamma(\rho_{t+1})^{2}\Big{\|} \theta^{t+1}-\theta^{*}\Big{\|}^{2}+\gamma(T-\gamma)\] (33) \[\times(\rho^{t+1})^{2}\Big{\|}w_{i}^{t+1}-\theta^{t+1}\Big{\|}^  
As a result, there exists a constant \(C>0\) such that  
\[ \begin{split}&\Big{\|}y^{+1}-y^{*}\Big{\|}^{2}+\gamma(\rho^{t+1})^{ 2}\Big{\|}\theta^{t+1}-\theta^{*}\Big{\|}^{2}+\gamma(T-\gamma)\\ &\times(\rho^{t+1})^{2}\Big{\|}w_{i}^{t+1}-\theta^{t+1}\Big{\|}^ {2}\leq C,\quad\forall t\geq 0\end{split}  \tag{37}\]  
Combine (32) and (37), we obtain  
\[ \begin{split}&\sum_{t=t_{0}}^{\infty}\gamma(2-T)(\rho^{t+1})^{2}( \Big{\|}w_{i}^{t+1}-\theta^{t+1}\Big{\|}^{2}+\Big{\|}\theta^{t+1}-\theta^{t} \Big{\|}\Big{\|}^{2})\\ &\leq\big{\|}y^{t_{0}}-y^{*}\big{\|}^{2}+\gamma(\rho^{t_{0}})^{2} \Big{\|}\theta^{t+1}-\theta^{*}\big{\|}^{2}+\gamma(T-\gamma)\\ &\times(\rho^{t_{0}})^{2}\Big{\|}w_{i}^{t+1}-\theta^{t+1}\Big{\|} ^{2}\sum_{t=t_{0}}^{\infty}(2\tau_{i}+\tau_{i}^{2})(\big{\|}y^{t}-y^{*}\big{\|} ^{2}\\ &+\gamma(\rho^{t})^{2}\big{\|}\theta^{t}-\theta^{*}\big{\|}^{2}+ \gamma(T-\gamma)(\rho^{t})^{2}\big{\|}w_{i}^{t}-\theta^{t}\big{\|}^{2})\\ &\leq(1+C_{s})C\end{split}  \tag{38}\]  
and then  
\[\lim_{t\to\infty}\gamma(2-T)(\rho^{t+1})^{2}(\Big{\|}w_{i}^{t+1}-\theta^{t+1} \Big{\|}^{2}+\Big{\|}\theta^{t+1}-\theta^{t}\Big{\|}^{2})=0 \tag{39}\]  
This means that the sequence \(\{\rho^{t}\}\) is both an upper bound and a lower bound away from zero, which means that we have  
\[\inf\{\rho^{t}\}_{1}^{\infty}:=B_{L}>0,\quad\sup\{\rho^{t}\}_{1}^{\infty}:=B_{ U}<\infty.\]  
and from (30), Theorem 3 is proved. If the sequence \(\{\rho^{t}\}\) is generated by (8), then it satisfies the following two conditions:  
**Condition C1. **\(\inf\{\rho^{t}\}_{1}^{\infty}=B_{L}>0\) and \(\sum_{t=1}^{\infty}\eta_{t}^{2}<+\infty\), where  
\[ \begin{split}&\theta_{t}=\begin{cases}\sqrt{1-\left(\rho^{t}/ \rho^{t+1}\right)^{2}},&\text{if $t\in T_{D}$,}\\ 0,&\text{otherwise}\;,\end{cases}\\ & K_{D}:=\Big{\{}t\mid\rho^{t+1}<\rho^{t}\Big{\}}.\end{split}  \tag{40}\]  
**Condition C2. **\(\sum_{t=1}^{\infty}\theta_{t}^{2}<+\infty\), where  
\[