## 2 Centralized Stochastic Approximation Algorithm
### SAAWET and Trajectory-Subsequence Method  
We first give an example. **Example 3**. : _([5]) Consider the root-seeking of the function \(f(x)=-(x-10)^{3}\) and assume that the observation noise \(\varepsilon_{k}\equiv 0\)._  
_If we choose the initial value_ \(x_{0}=0\) _and the stepsize_ \(a_{k}=\frac{1}{k+1}\)_, it can be directly verified that the estimates generated from (_2_)_  
\[x_{0}=0,\ \ x_{1}=1000,\ \ x_{2}=-485,148,500,\ \ x_{3}\approx 3.8\times 10^{ 25},\cdots\]  
_If we choose the initial value_ \(x_{0}=0\) _and the stepsize_ \(a_{k}=\frac{1}{k+10^{3}}\)_, it can be directly verified that the estimates generated from (_2_) is convergent, that is,_ \(x_{k}\underset{k\rightarrow\infty}{\longrightarrow}10\)_._  
For Example 3, neither the growth rate condition on \(f(\cdot)\) used by PM nor the boundedness condition on the estimate sequence used by the ODE method take place. On the other hand, if we can choose the stepsize in an adaptive manner, estimates generated from SAA may still converge. The core idea of SAAWET consists in adaptively choosing the stepsizes. Let us describe it. Denote by \(J\) the root set of an unknown function \(f(\cdot):\mathbb{R}^{l}\rightarrow\mathbb{R}^{l}\), that is, \(f(x^{0})=0\)\(\forall\)\(x^{0}\in J\). Choose \(\{M_{k}\}_{k\geq 0}\) a positive sequence increasingly diverging to infinity, \(M_{k}\underset{k\rightarrow\infty}{\longrightarrow}\infty\). With an arbitrary initial value \(x_{0}\in\mathbb{R}^{l}\), the estimate sequence \(\{x_{k}\}_{k\geq 1}\) is generated by the following SAAWET:  
\[x_{k+1}= (x_{k}+a_{k}y_{k+1})I_{[\|x_{k}+a_{k}y_{k+1}\|\leq M_{\alpha_{k}}]}+x ^{*}I_{[\|x_{k}+a_{k}y_{k+1}\|>M_{\alpha_{k}}]}, \tag{29}\] \[\sigma_{k}= \sigma_{k-1}+I_{[\|x_{k-1}+a_{k-1}y_{k}\|>M_{\alpha_{k-1}}]},\quad \sigma_{0}=0, \tag{30}\]  
where  
\[y_{k+1}=f(x_{k})+\varepsilon_{k+1}, \tag{31}\]  
\(y_{k+1}\) is the observation of \(f(\cdot)\) at \(x_{k}\), \(\varepsilon_{k+1}\)is the observation noise, \(a_{k}\) is the stepsize and \(x_{k}\) is the estimate for the root set of \(f(\cdot)\) at time \(k\). From (29) and (30), we can see that if the magnitude of \(x_{k}+a_{k}y_{k+1}\) is located in the truncation bound, then the algorithm evolves as the classical SAA (2), while if the magnitude of \(x_{k}+a_{k}y_{k+1}\) exceeds the truncation bound, then \(x_{k+1}\) is pulled back to \(x^{*}\) with the truncation bound being enlarged and the stepsize being reduced for the next recursion. We first list conditions for theoretical analysis. A4.1 \(a_{k}>0\), \(a_{k}\mathop{\longrightarrow}\limits_{k\to\infty}0\), \(\sum\limits_{k=1}^{\infty}a_{k}=\infty\); A4.2 There exists a continuously differentiable function \(v(\cdot):\mathbb{R}^{l}\to\mathbb{R}\) such that  
* \(\sup\limits_{\delta\leq d(x,l)\leq\Delta}\triangledown v(x)^{T}f(x)<0\), \(\forall\ 0<\delta<\Delta\),
* The set \(v(J)\triangleq\{v(x),x\in J\}\) is nowhere dense,
* For \(x^{*}\in\mathbb{R}^{l}\) in (29), there is a \(c_{0}>0\) such that\(\|x^{*}\|<c_{0}\) and \(v(x^{*})<\inf\limits_{\|x\|=c_{0}}v(x)\),  
A4.3 On the sample path \(\omega\), for any convergent subsequence \(\{x_{n_{k}}\}_{k\geq 1}\) generated from (29) and (30), it holds that  
\[\lim\limits_{T\to 0}\limsup\limits_{k\to\infty}\frac{1}{T}\left\|\sum \limits_{i=n_{k}}^{m(n_{k},T_{k})}a_{i}\varepsilon_{k+1}\right\|=0,\quad \forall\ T_{k}\in(0,T], \tag{32}\]  
with \(m(n_{k},T_{k})\triangleq\max\left\{m:\sum\limits_{i=n_{k}}^{m}a_{i}\leq T_{k }\right\}\); A4.4 The function \(f(\cdot)\) is measurable and locally bounded. **Remark 5**. : _A set being nowhere dense means that the set of the interior points of its closure is empty. It is clear that a set with a single point is nowhere dense. Note that for the noise condition A3.3, it only requires to verify (32) on a fixed sample path and along the index of any convergent subsequence, which compared with (24) is much easier to be verified in practice. In this regard, the analysis method for SAAWET is called the trajectory-subsequence (TS) method._  
**Theorem 3**.