## 5 Trade-Off between Communication Rate and Convergence Performance
### Trade-Off between Convergence Performance and Communication Rate  
From Theorem 3 and Section 4.2, it is clear that in scenarios where \(p\) is unknown both the communication rate and convergence rate are related to \(\alpha\). To represent their relationship with \(\alpha\), the communication rate is denoted as \(\overline{\gamma}=\overline{\gamma}(\alpha)\). \(\Omega(\alpha)\) serves as an indicator to describe the performance of the algorithm (25)-(27), which represents the trace of the covariance matrix \(N\mathbb{E}(\widehat{\theta}_{N}-\theta)(\widehat{\theta}_{N}-\theta)^{T}\), as \(N\) tends to infinity. What is the optimal selection of \(\alpha\) that minimizes \(\overline{\gamma}(\alpha)\) while meeting the performance requirements of the algorithm? This can be regarded as a constrained optimization problem, as follows:  
\[\min_{\alpha} \overline{\gamma}(\alpha),\] (46) s.t. \[\Omega(\alpha)\leq\epsilon,\;\alpha\in(0,1), \tag{47}\]  
where \(\epsilon>0\) is a constant to bound the algorithm performance. Define the solution set of \(\Omega(\alpha)\leq\epsilon\) in \((0,1)\) as \(A_{\epsilon}\
eq\emptyset\) and denote \(\alpha^{*}\) as the solution to (46) and (47). **Remark 5**. : _The index \(\Omega(\alpha)\) that portrays the convergence rate can be obtained approximately by using the method in Section 4.2 or by employing other nonlinear modeling methods, such as exponential and logarithmic curve fitting._  
**Theorem 4**. : _Under the conditions of Theorem 1, the solution to the optimization problem (46) and (47) is provided by_  
\[\alpha^{*}=\min A_{\epsilon}, \tag{48}\]  
_where \(\min A_{\epsilon}\) denotes the minimum value in the set \(A_{\epsilon}\). At this point, the communication rate of the improved CSTCS (23) and (24) is_  
\[\overline{\gamma}(\alpha^{*})=\alpha^{*}+\frac{1-\alpha^{*}}{mn}\sum_{i=1}^{n }F_{i}. \tag{49}\]  
Proof. : From Theorem 3, it is known that  
\[\frac{\mathrm{d}(\overline{\gamma}(\alpha))}{\mathrm{d}\alpha}=1-\frac{1}{mn }\sum_{i=1}^{n}F_{i}.\]  
Combining \(m\geq 1\) and \(F_{i}<1,\;i=1,2,\ldots,n\), we have \(\frac{\mathrm{d}(and the data packet loss situation in the interval [950, 1000]. At the moments when data are utilized for estimating \(p\), the value of \(\gamma_{k}^{c}\) is equal to 1. The subset \(Z_{N}\) is selected as described in Section 4.2, and the data proportion used for estimating the packet loss probability is set to \(\alpha=0.5\). The parameter \(\theta\) is estimated under the unknown and known packet loss probability, respectively. Figures 4 and 5 show the convergence of the parameter under the above values of \(m,p,\alpha\), with both \(\widehat{\theta}_{N}\) converging gradually to the true value \([5,20]^{T}\). This verifies the conclusions of Theorems 1 and 2. Figure 6 shows the relationship between convergence performance and \(p\) in the context of known packet loss probabilities, particularly when the packet loss probabilities are set at \(0.1,0.3,0.5\). The performance measurement standard is based on the average value of \(||\widehat{\theta}_{N}-\theta||\) generated from 200 trajectories. The higher the probability of packet loss, the worse the convergence performance. Figure 4: Convergence of \(\widehat{\theta}_{N}\) when \(p\) is known. Figure 3: Data packet loss and CSTCS in the time interval [950, 1000]. Based on Theorem 3, the calculated communication rates for \(\alpha=0.1,0.3,0.5\) are 0.2207, 0.3939, and 0.5671, respectively. From Figure 9, it is clear that as \(N\) increases, the average communication rate progressively approaches the theoretical value. Furthermore, the larger \(\alpha\) results in the higher communication rate, which is consistent with the result of Theorem 3. Adjusting \(m\) to 10, the relationship between the convergence rate and \(\alpha\) is obtained according to the subset selection, data generation, and model fitting methods described in Section 4.2. Figure 10 shows that the order of the model is determined to be 5 according to the AIC criterion.