## 3 Distributed Stochastic Approximation Algorithm  
In this section, we will introduce the distributed version of SAAWET (DSAAWET). The key difference from the existing distributed SAA, see for example, [40], lies in the expanding truncation mechanism that adaptively defines enclosing bounds. The theoretical results of DSAAWET guarantee that estimates generated by DSAAWET for all agents converge almost surely to a consensus set, which is contained in the root set of the sum function. We first introduce the problem formulation. Consider the case where all agents in a network cooperatively search the root of the sum function given by  
\[f(\cdot)=\tfrac{1}{N}\sum_{i=1}^{N}f_{i}(\cdot), \tag{44}\]  
where \(f_{i}(\cdot):\mathbb{R}^{l}\to\mathbb{R}^{l}\) is the local objective function which can only be observed by agent \(i\). Denote the root set of \(f(\cdot)\) by \(I\triangleq\{x\in\mathbb{R}^{l}:f(x)=0\}\). For any \(i\in\mathcal{V}\), denote by \(x_{i,k}\in\mathbb{R}^{l}\) the estimate for the root of \(f(\cdot)\) given by agent \(i\) at time \(k\). Since at time \(k+1\), agent \(i\) only has its local noisy observation  
\[O_{i,k+1}=f_{i}(x_{i,k})+\varepsilon_{i,k+1}, \tag{45}\]  
where \(\varepsilon_{i,k+1}\) is the observation noise, to estimate the root of \(f(\cdot)\), all agents need to exchange information with adjacent agents via the topology of the network. The topology of the network at time \(k\) is described by a digraph \(\mathcal{G}(k)=\{\mathcal{V},\mathcal{E}(k)\}\), where \(\mathcal{V}=\{1,\cdots,N\}\) is the index set of all agents, and \(\mathcal{E}(k)\subset\mathcal{V}\times\mathcal{V}\) is the edge set with \((j,i)\in\mathcal{E}(k)\) representing the information flow from agent \(j\) to agent \(i\) at time \(k\). Denote the adjacency matrix of the network at time \(k\) by \(W(k)=[\omega_{ij}(k)]_{i,j=1}^{N}\), where \(\omega_{ij}(k)>0\) if, and only if \((j,i)\in\mathcal{E}(k)\), and \(\omega_{ij}(k)=0\), otherwise. Denote by \(N_{i}(k)=\{j\in\mathcal{V}:(j,i)\in\mathcal{E}(k)\}\) the set of neighbors of agent \(i\) at time \(k\). We apply the idea of expanding truncation to the distributed estimation. Denote by \(x_{i,k}\) the estimate for agent \(i\) at time \(k\) and by \(\sigma_{i,k}\) the number of truncations for agent \(i\) up to time \(k\). DSAAWET is given by (46)-(49) with any initial values \(x_{i,0}\), where \(O_{i,k+1}\) is defined by (45), \(\{\gamma_{k}\}\) is the stepsize, \(x^{*}\in\mathbb{R}^{l}\) is a vector known to all agents, \(\{M_{k}\}_{k\geq 0}\) is a positive sequence increasingly diverging to infinity with \(M_{0}\geq\|x^{*}\|\). \[\sigma_{i,0}=0,\ \ \ \partial_{i,k}\triangleq\max_{j\in N_{i}(k)}\sigma_{j,k}, \tag{46}\] \[x^{\prime}_{i,k+1}=\Big{(}\sum_{j\in N_{i}(k)}\omega_{ij}(k)(x_{j,k} I_{[\sigma_{j,k}=\varrho_{i,k}]}+x^{*}I_{[\sigma_{j,k}<\varrho_{i,k}]})+\gamma_{k}O_{i,k+ 1}\Big{)}I_{[\sigma_{i,k}=\varrho_{i,k}]}+x^{*}I_{[\sigma_{i,k}<\varrho_{i,k}]},\] (47) \[x_{i,k+1}=x^{*}I_{[\|x^{\prime}_{i,k+1}\|>M_{\varrho_{i,k}}]}+x^{ \prime}_{i,k+1}I_{[\|x^{\prime}_{i,k+1}\|\leq M_{\varrho_{i,k}}]},\] (48) \[\sigma_{i,k+1}=\partial_{i,k}+I_{\big{[}|x^{\prime}_{i,k+1}\|>M_{ \varrho_{i,k}}\big{]}}, \tag{49}\]  
Denote by  
\[\sigma_{k}\triangleq\max_{i\in\mathcal{V}}\sigma_{i,k} \tag{50}\]  
the largest truncation number among all agents at time \(k\). If \(N=1\), then by denoting \(\vartheta_{i,k}=\sigma_{i,k}\triangleq\sigma_{k}\), \(x_{i,k}\triangleq x_{k}\), \(x^{\prime}_{i,k}\triangleq x^{\prime}_{k}\), \(O_{i,k}\triangleq O_{k}\), (46)-(49) becomes  
\[x^{\prime}_{k+1}=x_{k}+\gamma_{k}O_{k+1},\] \[x_{k+1}=x^{*}I_{[\|x^{\prime}_{k+1}\|>M_{\varrho_{k}}]}+x^{\prime }_{k+1}I_{[\|x^{\prime}_{k+1}\|\leq M_{\varrho_{k}}]},\] \[\sigma_{k+1}=\sigma_{k}+I_{\big{[}\|x^{\prime}_{k+1}\|>M_{ \varrho_{k}}\big{]}},\]  
which is the SAAWET in the centralized setting. We first introduce assumptions to be used. A5.1 \(\gamma_{k}>0\), \(\gamma_{k}\underset{k\to\infty}{\longrightarrow}0\), and \(\sum_{k=1}^{\infty}\gamma_{k}=\infty\).