## 4 Construction of the Cascaded Model and Training
### Cascaded Model and Training
\begin{array}{ll}SBR_{1}(t)=F_{Trans}[x_{11}(t),x_{12}(t),\cdots,\\ x_{1N_{P}}(t),y_{1I}(t),y_{1R}(t)]\\ SBR_{1}=[SBR_{1}(t-t_{0}),\cdots,\\ SBR_{1}(t),\cdots,SBR_{1}(t+t_{1})]\end{array} \right.\, \tag{11}\]  
where \(F_{Trans}\) denotes the Transformer model being utilized, \(SBR_{1}(t)\) represents the SBR corresponding to the input feature vector of the channel at a specific time point, and \(SBR_{1}\) represents the SBR sequence of the channel. \(t_{0}\) and \(t_{1}\) are the left and right boundaries of the time range, respectively. By feeding the SBR into the subsequent DNN model with distinct parameters, the prediction of equalizer parameters can be achieved via Equation (12):  
\[\mathbf{Y}^{c}=F_{DNN}(\mathbf{SBR}_{1},\mathbf{SBR}_{2},\cdots,\mathbf{SBR}_{N_{C}}) \tag{12}\]  
By integrating the equalizer parameters with the input featuring spectral characteristics, an augmented matrix \(SBR_{N_{C}}^{eq}(t)\) is constructed for the prediction of post-equalizationSBR. This matrix, via Equation (13), is then processed into a time-dependent sequence following the methodology presented in Equation (11). \[\left\{ \begin{array}{l}SBR_{N_{C}}^{eq}(t)=F_{Trans}[x_{N_{C1}}(t),\cdots,x_{N_{ C}N_{P}}(t),\\ \mathbf{y}_{N_{C1}}(t),y_{N_{C}R}(t),\mathbf{Y}_{N_{C}}^{*}(t)]\\ \mathbf{S}\mathbf{B}\mathbf{R}_{N_{C}}^{eq}=[SBR_{N_{C}}^{eq}(t-t_{0}),\cdots, \\ SBR_{N_{C}}^{eq}(t),\cdots,SBR_{N_{C}}^{eq}(t+t_{1})]\\ \mathbf{Y}_{N_{C}}^{e}(t)=[y_{FFE-N_{C}}^{Taps}(t),y_{CTLE-N_{C}}^{DecGain}(t ),y_{DFE-N_{C}}^{Taps}(t)]\end{array} \right. \tag{13}\]  
Finally, the obtained post-equalization SBR is concatenated with \(\mathbf{X}^{f}\) to form the final input features for predicting the COM value. Before formally training the cascaded model, we standardized the input data and employed a pre-training strategy. Initially, the DNN models \(F_{DNN}^{freq}(\mathbf{X})\) and \(F_{DNN}^{eq}(\mathbf{X})\) designed for predicting spectral features and equalizer parameters, respectively, were trained independently, and their corresponding parameters were saved. These pre-trained DNN models were then integrated with the necessary Transformer model to construct the complete cascaded model for predicting the equalizer parameters and COM, as depicted in Figure 9. For training the cascaded model, Smooth L1 was used as the cost function and the Adam algorithm was applied to update the model parameters. With the help of the optuna toolkit, model structuring and hyperparameter optimization (HPO) were performed via Bayesian optimization using the GP [37; 38]. The chosen acquisition function was the Expected Improvement (EI) function, with an initial set of 10 observation points, allowing for effective HPO within a computationally feasible scope. Compared to the global grid search, random grid search, and halving search methods, Bayesian optimization is more efficient for HPO and requires less optimization time. Additionally, the K-Fold cross-validation method is employed in this paper to enhance the reliability of the model evaluation. After completing the model's training, its predictive capabilities can be assessed with several different metrics. One of these metrics, shown in Equation (14), is the RMSE. The squaring operation in RMSE increases its sensitivity to errors, making it particularly responsive to outliers. This increased sensitivity can also make it overly responsive to outliers, but the square-root operation cannot intuitively reflect the actual magnitude of error. \[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(y_{n}-\hat{y}_{n}\right)^{2}} \tag{14}\]  
MAE is calculated according to the average value of absolute differences between the predicted and actual values. Compared to RMSE, MAE is less sensitive to outliers and provides a more intuitive reflection of the actual magnitude of errors. However, this can lead to an underestimation of the impact of the errors.