## 2 Problem Formulation
### The Relationship between Convergence Rate and \(\alpha\)
For convenience, denote \(\Lambda(\alpha;\widehat{\theta}_{N})=N(\widehat{\theta}_{N}-\bar{\theta})( \widehat{\theta}_{N}-\bar{\theta})^{T}\) and define the average value of the matrix trace for multiple trajectories as \(\Omega_{N,b}=\operatorname{tr}(\tilde{\Lambda}(\alpha_{b};\widehat{\theta}_{N,A_{b} })),\ b=1,2,\ldots,B\), where \(B\) is the number of different values of \(\alpha\). Through the above method, a set of convergence rate data \(\{(\alpha_{1},\Omega_{N,1}),(\alpha_{2},\Omega_{N,2}),\ldots,\)\((\alpha_{B},\Omega_{N,B})\}\) can be obtained under the same \(m\) and unknown-but-consistent packet loss conditions for different values of \(\alpha\). The data are used for subsequent model fitting and solving optimization problems. The data generation process is illustrated in Figure 2:  
(3) Model order determination  
Use polynomial fitting to model the relationship between the index and \(\alpha\). Since the specific order of the relationship expression is unknown, the Akaike information criterion (AIC) is used to determine the order of polynomial fitting. Suppose the fitting expression is  
\[\Omega_{N}(\alpha)=c_{h}\alpha^{h}+c_{h-1}\alpha^{h-1}+\cdots+c_{1}\alpha+c_{0}, \tag{40}\]  
where \(h\) is the order of the polynomial. Denote \(\
u_{h}=[c_{0},c_{1},\ldots,c_{h}]\) as the polynomial coefficients,  
\[X_{h}= \begin{bmatrix}1&\alpha_{1}&\cdots&\alpha_{1}^{h}\\ \vdots&\vdots&\ddots&\vdots\\ 1&\alpha_{B}&\cdots&\alpha_{B}^{h}\end{bmatrix} ,\quad Y= \begin{bmatrix}\Omega _{N,1}\\ \vdots\\ \Omega_{N,B}\end{bmatrix} .\]  
Then, the least squares solution for the polynomial coefficients can be obtained as \(\widehat{\
u}_{h}=(X_{h}^{T}X_{h})^{-1}X_{h}^{T}Y\). Denote the sum of the squared residuals:  
\[e_{h}=\sum_{b=1}^{B}\Big{(}\Omega_{N,b}-(\widehat{c}_{h}\alpha_{b}^{h}+\widehat {c}_{h-1}\alpha_{b}^{h-1}+\cdots+\widehat{c}_{1}\alpha_{b}+\widehat{c}_{0}) \Big{)}^{2}.\]  
In the calculation of AIC, the likelihood function is inversely proportional to the sum of the squared residuals. Therefore, replacing the likelihood function with the sum of the squared residuals, the calculation formula becomes  
\[B\ln\Big{(}\frac{e_{h}}{B}\Big{)}+2(h+1).\]  
Figure 2: Data generation process. For each order, use the least squares method to find the best polynomial coefficients, and calculate the AIC values for different orders. The final chosen fitting order is  
\[h^{*}=\arg\min_{h\geq 1}\{B\ln\Bigl{(}\frac{e_{h}}{B}\Bigr{)}+2(h+1)\}. \tag{41}\]  
The corresponding polynomial coefficients are  
\[\widehat{v}_{h^{*}}=(X_{h^{*}}^{T}X_{h^{*}})^{-1}X_{h^{*}}^{T}Y. \tag{42}\]  
Using the above method, the relationship between the trace of the approximate covariance matrix and \(\alpha\) is obtained as  
\[\Omega_{N}(\alpha)=\widehat{v}_{h^{*}}[1,\alpha,\ldots,\hat{a}^{h^{*}}]^{T}.\]  
Denote the minimum value of \(\Omega_{N}(\alpha)\) in the interval \((0,1)\) as \(D\). From the identification algorithm (25)-(27), it is known that the parameter estimate fails to converge to the true value when \(\alpha\) is 0 or 1. So, when \(\Omega_{N}(\alpha)\) in the \((0,1)\) interval of the minimum value does not exist--that is, when the minimum value is obtained approaching the endpoints--there is no solution. Then, the solution that makes the convergence rate optimal is  
\[\alpha_{N}^{*}=\left\{ \begin{array}{ll}\text{No solution},&\text{if $D$ does not exist;}\\ D,&\text{if $D$ exists.}\end{array} \right.\]