## 2 Proposed Method
### Model
#### 2.1.3 Tuning Parameters Using Multi-Task Learning  
Multi-task learning is a technique for optimizing multi-objective models employed in many deep learning tasks [24]. Multi-task learning tries to enhance learning efficiency and prediction accuracy by learning numerous targets from a joint representation instead of constructing a separate model for each task [25]. It can be thought of as a way to increase inductive knowledge transfer by allowing complementary tasks to share domain information. The auto-encoder task and the score-guided network task are two subtasks of the approach provided in this paper. The loss functions of the auto-encoder and the score-guided network are \(L_{RE}(X,\hat{X})\) and \(L_{SE}(X_{i},\hat{X}_{i},s)\), respectively, according to Equations (3) and (7).  
In multi-task learning, the optimal training effect can only be achieved by weighting the loss function of the subtasks appropriately. The loss function of the algorithm in this paper adopts the typical loss function expression as follows:  
\[L_{total}=\lambda_{RE}L_{RE}(X,\hat{X})+\lambda_{SE}\frac{1}{n}\sum_{i=1}^{n }L_{SE}(X_{i},\hat{X}_{i},s) \tag{9}\]  
where \(\lambda_{RE}\) is the weights of the loss function of the auto-encoder; \(\lambda_{SE}\) is the weights of the loss function of the score-guided network. The model's performance depends heavily on the choice of these two parameters. We introduce uncertainty to determine these two parameters [26] automatically. The core idea of this method is to give lower loss function weights to tasks with higher uncertainty to reduce the difficulty of training.  
This strategy is based on a Bayesian model that includes two types of uncertainty: Epistemic uncertainty and Aleatoric uncertainty. Epistemic uncertainty is a type of model uncertainty that describes what our model does not know owing to a lack of training data.  
Our uncertainty about information that our data cannot explain is captured by Aleatoric uncertainty. There are two types of Aleatoric uncertainty: data-dependent uncertainty and task-dependent uncertainty. Data-dependent uncertainty is expected as a model output and is dependent on the input data. Task-dependent uncertainty is not dependent on the input data. It is not a model output; instead, it is a quantity that stays constant for all input data and varies between different tasks. Task-dependent uncertainty measures the relative confidence between tasks in a multi-task environment, representing the uncertainty inherent in the regression or classification task. Hence, we use task-dependent uncertainty as the basis for weighted loss.  
After introducing noise (uncertainty) parameters \(\sigma_{1}\) and \(\sigma_{2}\) for auto-encoder and score-guided networks. The final loss function of the proposed model is:  
\[L(X,\hat{X},s)=\frac{1}{2\sigma_{1}^{2}}L_{RE}(X,\hat{X})+\frac{1}{2\sigma_{2 }^{2}}\frac{1}{n}\sum_{i=1}^{n}L_{SE}(X_{i},\hat{X}_{i},s)+log\sigma_{1}\sigma _{2} \tag{10}\]  
We can see that as the noise \(\sigma\) continues to increase, the task's weight decreases. \(log\sigma_{1}\sigma_{2}\) is the standard term of this loss function, which aims to prevent a certain \(\sigma\) from being too large and causing serious imbalance in model training. Parameters \(\sigma_{1}\) and \(\sigma_{2}\) are optimized by the back error propagation algorithm.  
\[\sigma_{1}=\sigma_{1}-\eta\frac{\partial L(X,\hat{X},s)}{\partial\sigma_{1}} \tag{11}\]  
\[\sigma_{2}=\sigma_{2}-\eta\frac{\partial L(X,\hat{X},s)}{\partial\sigma_{2}} \tag{12}\]