## 2 Centralized Stochastic Approximation Algorithm
### Ordinary Differential Equation Method  
We first introduce two technical lemmas.  
**Lemma 1**.: _(Arzela-Ascoli, [47]) Let \(\{f_{\lambda}(t),\ t\in[0,+\infty),\ \lambda\in\Lambda\}\) be a function class satisfying the condition of uniform boundedness and equi-continuity. Then there exists a continuous function \(f(t)\) and a subsequence \(\{f_{\lambda_{k}}(t)\}_{k\geq 1}\) such that \(\{f_{\lambda_{k}}(t)\}_{k\geq 1}\) converges to \(f(t)\) uniformly on any fixed bounded interval in \([0,+\infty)\)._  
**Remark 3**.: _For the function class \(\{f_{\lambda}(t),\ t\in[0,+\infty),\ \lambda\in\Lambda\}\), the equi-continuity means that for any \(\varepsilon>0\), there exists \(\delta>0\) such that for any \(|t-s|<\delta\),_  
\[if_{\lambda}(t)-f_{\lambda}(s)|<\varepsilon,\ \ \forall\ \lambda\in\Lambda. \tag{21}\]  
The following result is well-known for the Lyapunov stability of ODE.  
**Lemma 2**.: _Consider the ODE_  
\[\dot{x}(t)=f(x(t)),\ t\geq 0 \tag{22}\]  
_with an equilibrium point \(x^{0}\), that is, \(f(x^{0})=0\). If there exists a continuously differentiable \(v(x)\) such that_  
_(i)_ \(v(x^{0})=0\)_;_  
_(ii)_ \(v(x)>0,\ \forall\ x\
eq x^{0}\)_;_ \(v(x)\underset{\|x\|\to\infty}{\longrightarrow}\infty\)_;_  
_(iii)_ \(\triangledown v(x)^{T}f(x)<0,\ \ \forall\ x\
eq x^{0}\)_;_  
_then \(x^{0}\) is the globally stable solution of (22)._  
Before introducing the conditions and the results for the ODE method, we first introduce a crucial notation to be used in this paper. For any \(T>0\), define  
\[m(n,T)\triangleq\max\Biggl{\{}m:\sum_{i=n}^{m}a_{i}\leq T\Biggr{\}}. \tag{23}\]  
From the definition in (23) we can see that \(m(n,T)\) is the maximal number of steps starting from \(n\) with the sum of stepsizes not exceeding \(T\). Noting that \(a_{k}\to 0\) as \(k\to\infty\), we have that \(m(n,T)\to\infty\) as \(n\to\infty\).  
We make the following assumptions.  
A3.1\(a_{k}>0,\ a_{k}\longrightarrow_{k\to\infty}0,\ \sum_{k=0}^{\infty}a_{k}=\infty\).  
A3.2 There exists a twice-differentiable function \(v(\cdot):\mathbb{R}^{l}\to\mathbb{R}\) such that  
**Theorem 2**.: _([6,7,21]) Consider the sample path \(\omega\) where A3.3 takes place. If A3.1, A3.2, and A3.4 hold on the sample path \(\omega\) and \(\{x_{k}(\omega)\}_{k\geq 0}\) is bounded, then_  
\[x_{k}(\omega)\underset{k\rightarrow\infty}{\longrightarrow}x^{0}. \tag{25}\]  
Proof.: We sketch the proof.  
Define \(t_{k}\triangleq\sum_{i=0}^{k}a_{i}\) with \(t_{0}=0\). By interpolating the estimates generated from (2), we have a sequence of continuous functions  
\[x_{t}^{0}\triangleq\frac{t_{k}-t}{a_{k}}x_{k-1}+\frac{t-t_{k-1}} {a_{k}}x_{k},\ \ t\in[t_{k-1},t_{k}], \tag{26}\] \[x_{k}(t)=x_{1+k}^{0},\ \ k\geq 1, \tag{27}\]  
which can be proved to be bounded by the boundedness of \(\{x_{k}\}_{k\geq 0}\) and uniform continuity by A3.3.  
Then by Lemma 1, we can prove that there exists a subsequence \(\{x_{n_{k}}(t)\}_{k\geq 0}\) converging a continuous function \(x(t)\) satisfying the following ODE  
\[\dot{x}(t)=f(x(t)). \tag{28}\]  
By A3.2 and Lemma 2 we can further prove that the equilibrium point \(x^{0}\) is globally stable and the estimates \(\{x_{k}\}_{k\geq 0}\) converge to \(x^{0}\) on the sample path \(\omega\).  
The essential idea of the above proof is to transform the convergence of a recursive algorithm to the stability of an associated ODE, and thus the analysis method is called the ODE method. From assumption A3.3 we can see that the ODE method has a wider application potential than the PM method. However, the boundedness assumption on the estimate sequence is still restrictive. Aiming at removing this condition and further relaxing the technical assumptions on the noise and the root-seeking function, in the next section we will introduce the SAA with expanding truncations (SAAWET) and its general convergence theorem.