## 3 Distributed Stochastic Approximation Algorithm
A5.2 There exists a continuously differentiable function \(v(\cdot):\mathbb{R}^{l}\to\mathbb{R}\) such that  
\[O_{i,k+1}=f_{i}(x_{i,k})+\varepsilon_{i,k+1}, \tag{45}\]0  
for any \(\Delta>\delta>0\), where \(v_{x}(\cdot)\) denotes the gradient of \(v(\cdot)\) and \(d(x,J)=\min_{y}\{\|\cdot x-y\|:y\in J\}\),  
\(\text{(b)}\ v(J)\triangleq\{v(x):x\in J\}\) is nowhere dense,  
\(\text{(c)}\ \|x^{*}\|<c_{0}\) and \(v(x^{*})<\inf_{\|x\|=c_{0}}v(x)\) for some positive constant \(c_{0}\), where \(x^{*}\) is used in (47) and (48). A5.3 The local functions \(f_{i}(\cdot)\ \ \forall i\in\mathcal{V}\) are continuous. A5.4 \(\text{(a)}\ W(k)\ \ \forall k\geq 0\) are doubly stochastic matrices (A matrix is said to be doubly stochastic if all entries being nonnegative and the sum of entries in each row and each column being \(1\). );  
\(\text{(b)}\) There exists a constant \(0<\eta<1\) such that  
\[O_{i,k+1}=f_{i}(x_{i,k})+\varepsilon_{i,k+1}, \tag{45}\]1  
\(\text{(c)}\) The digraph \(\mathcal{G}_{\infty}=\{\mathcal{V},\mathcal{E}_{\infty}\}\) is strongly connected, where \(\mathcal{E}_{\infty}\) is the set of edges \((j,i)\) such that \(j\) is a neighbor of \(i\) which communicates with \(i\) infinitely often, that is,  
\[O_{i,k+1}=f_{i}(x_{i,k})+\varepsilon_{i,k+1}, \tag{45}\]2  
\(\text{(d)}\) There exists a positive integer \(B\) such that for every \((j,i)\in\mathcal{E}_{\infty}\), agent \(j\) sends information to the neighbor \(i\) at least once every \(B\) consecutive time slots, that is,  
\[O_{i,k+1}=f_{i}(x_{i,k})+\varepsilon_{i,k+1}, \tag{45}\]3  
for all \((j,i)\in\mathcal{E}_{\infty}\) and any \(k\geq 0\). **Remark 7**. : _Assumptions A5.1-A5.3 are similar to those conditions used in the centralized SAAWET. Since the DSAAWET is in a distributed setting, assumption A5.4 is a commonly used condition describing the topology of the network [34]._  
Next, we introduce the condition on the observation noises for each agent. A5.5 For the sample path \(\omega\) under consideration and for each agent \(i\),  
(a) \(\gamma_{k}\epsilon_{j,k+1}\underset{k\rightarrow\infty}{\longrightarrow}0\), and  
(b) \(\lim_{T\to 0}\limsup_{k\rightarrow\infty}\frac{1}{T}\parallel\sum_{m =m_{k}}^{m(n_{k},k_{k})}\gamma_{m}\epsilon_{i,m+1}I_{\parallel\{x_{i,m} \parallel\leq K\}}\parallel=0\ \forall t_{k}\in[0,T]\) along indices \(\{n_{k}\}\) whenever \(\{x_{i,n_{k}}\}\) converges. We now give the main results of DSAAWET. Define the vectors \(X_{k}\overset{\Delta}{=}\operatorname{col}\{x_{1,k},\cdots,x_{N,k}\}\), \(\epsilon_{k}\overset{\Delta}{=}\operatorname{col}\{\epsilon_{1,k},\cdots, \epsilon_{N,k}\}\), \(F(X_{k})\overset{\Delta}{=}\operatorname{col}\{f_{1}(x_{1,k}),\cdots,f_{N}(x _{N,k})\}\). Denote by \(X_{\perp,k}\overset{\Delta}{=}D_{\perp}X_{k}\) the disagreement vector of \(X_{k}\) with \(D_{\perp}\triangleq(\mathbf{I}_{N}-\frac{\mathbf{1}\mathbf{1}^{T}}{N}) \otimes\mathbf{I}_{l}\), and by \(x_{k}=\frac{1}{N}\sum_{i=1}^{N}x_{i,k}\) the average of the estimates derived at all agents at time \(k\). **Theorem 5**. : _([44]) Let \(\{x_{i,k}\}\) be produced by (46)-(49) with an arbitrary initial value \(x_{i,0}\). Assume A5.1-A5.4 hold. Then on the sample path \(\omega\) where A5.5 holds, the following assertions take place: (i) \(\{x_{i,k}\}\) is bounded and there exists a positive integer \(k_{0}\) possibly depending on \(\omega\) such that_  
\[O_{i,k+1}=f_{i}(x_{i,k})+\varepsilon_{i,k+1}, \tag{45}\]4  
_or in the compact form:_  
\[O_{i,k+1}=f_{i}(x_{i,k})+\varepsilon_{i,k+1}, \tag{45}\]5  
(ii) \(X_{\perp,k}\underset{k\rightarrow\infty}{\longrightarrow}\mathbf{0}\) _and_ \(d(x_{k},J)\underset{k\rightarrow\infty}{\longrightarrow}0\)_;_ (iii) _There exists a connected subset_ \(J^{*}\subset J\) _such that_  
\[O_{i,k+1}=f_{i}(x_{i,k})+\varepsilon_{i,k+1}, \tag{45}\]6  
Proof. : The proof can be obtained by first proving the finite number of truncations, then the consensus of estimates, and finally, the convergence of estimates. The detailed proof can be found in [44].