## 2 Literature Review
### GBDT Algorithm  
After secondary dimensionality reduction, the choice of applying the gradient-boosted decision tree (GBDT) algorithm for prediction is due to several advantages. As a powerful machine learning algorithm [32], GBDT effectively improves the accuracy of prediction by iteratively training multiple decision tree models. Its feature importance evaluation capability identifies key influencing factors, and its robustness and generalization capabilities ensure the stability and generalization performance of the model in real data. In addition, GBDT is applicable to complex non-linear relationships, and its wide range of applications in different fields enhances its usefulness. Therefore, in the context of dealing with large-scale satisfaction data, prediction by GBDT can support analysis with high accuracy, an in-depth understanding of key influencing factors and guidance for decision making.  
GBDT is a kind of tree-based boosting model [33] based on the idea of integrated learning. It employs a sequential technique, where each iteration selects a weak learner. In this process, a new decision tree is constructed in the direction of the gradient of the residuals of the reduced residuals. The joint decision making is then executed by a group of associated decision trees [34]. The principle is shown in Figure 2.  
The GBDT model parameters mainly include the number of iterations (number of trees), learning rate and loss function [35]. Then, \(x_{i}\) is defined as the sample value, \(K\) is the total number of trees, and \(f_{k}\) is the kth tree, where \(y_{i}\) is the prediction result of the \(x_{i}\) and can be expressed as  
Figure 2: Schematic diagram of GBDT.  
\[\hat{y_{i}}=\sum_{k=1}^{K}f_{k}(x_{i}) \tag{1}\]  
The prediction model follows the forward-distributed addition method, which generates a new regression tree at each iteration, and the new tree will continuously fit the residuals of the previous tree, constantly repairing the previous experimental results, so as to construct a learning model with a higher accuracy rate. The strategy is as follows.  
\[\hat{y_{i}}(t)=\sum_{k-1}^{t}f_{t}(x_{i})=\hat{y_{i}}(t-1)+f_{t}(x_{i}) \tag{2}\]  
where \(t\) is the combined t-tree, \(\hat{y_{i}}(t)\) is the prediction of the combined \(t\)-tree model for the sample of \(x_{i}\), \(\hat{y_{i}}(t-1)\) is the prediction of the combined \((t-1)\)-tree model for the sample of \(x_{i}\), and \(f_{t}(x_{i})\) is the estimate of the \(t\)th tree model for the current round of losses.  
In each iteration of GBDT, the negative gradient of the loss function under the current model is used to fit the loss estimate (i.e., residual estimate) of the current round [36]. In this way, the loss function can be reduced as fast as possible in each round of training and converge to the local or global optimal solution as soon as possible [37]. And \(r_{ti}\) is a negative gradient of the loss function for the \(i\)th sample in round t that can be determined by  
\[r_{ti}=-\left|\frac{l(y_{i},\hat{y_{i}})}{\hat{y_{i}}}\right|_{f(x)=f_{i-1}(x_ {i})} \tag{3}\]  
The \(C_{ij}\) is the best-fit value that minimizes the loss function at each leaf node, \(r_{ti}\) is summed, and \(f_{t}(\text{X}_{i})\) is the \(t\)th tree model's estimate of the current round's loss that can be expressed as  
\[f_{t}(x_{i})=\sum_{j=1}^{j}c_{ij}I\big{(}x_{i}\in R_{ij}\big{)} \tag{4}\]