## 3 Materials and Method
### Experimental Setup
#### 3.4.2 Optimizer Configuration and Hyperparameter Settings  
The Adam optimizer, known for its superior performance in deep learning, was selected for this study due to its ability to achieve fast convergence and stability. Adam combines momentum and adaptive learning rates, dynamically adjusting the step size based on the first- and second-moment estimates of the gradients during each parameter update. Specifically, the update rules for Adam are as follows:  
\[m_{t}=\beta_{1}m_{t-1}+(1-\beta_{1})g_{t},\quad v_{t}=\beta_{2}v_{t-1}+(1-\beta _{2})g_{t}^{2}, \tag{29}\]  
\[\hat{m}_{t}=\frac{m_{t}}{1-\beta_{1}^{t}},\quad\hat{v}_{t}=\frac{v_{t}}{1- \beta_{2}^{t}}, \tag{30}\]  
\[\theta_{t+1}=\theta_{t}-\eta\frac{\hat{m}_{t}}{\sqrt{\theta_{t}}+\epsilon}, \tag{31}\]where \(m_{t}\) and \(v_{t}\) represent the first and second-moment estimates of the gradients, \(\beta_{1}\) and \(\beta_{2}\) are the decay rates, and \(\eta\) is the learning rate, which was set to 0.001 for this study. The adaptive nature of Adam makes it particularly effective at handling sparse gradients or noisy data. The hyperparameters \(\beta_{1}\) and \(\beta_{2}\) were set to 0.9 and 0.999, respectively, following standard settings commonly used for the Adam optimizer. To ensure the model's robustness across different data distributions, the dataset was divided and preprocessed appropriately. The dataset was split into training, validation, and test sets with an 80:10:10 ratio, with 80% of the data used for model training, 10% for validation to adjust the hyperparameters, and the remaining 10% for evaluating the model's final performance. The dataset used for training consisted of 10,000 samples, the validation set contained 1250 samples, and the test set also included 1250 samples. During training, 5-fold cross-validation was employed, where the training data were divided into five subsets. In each round, one subset was used as the validation set, while the remaining subsets were used for training. This process was repeated five times, and the final model performance was evaluated based on the average performance across all five folds. The average results from cross-validation help mitigate errors arising from a single split, thus improving the model's generalization ability. To ensure consistent evaluation across different tasks, cross-validation was conducted for both the medical image dataset and the financial dataset separately, allowing for task-specific adjustments. The specific hyperparameter settings involved grid search optimization for the obfuscation strength parameter. The obfuscation strength \(\alpha\), which determines the level of perturbation applied in the data obfuscation strategy, was searched within the range of \([0.1,1.0]\) with a step size of 0.1. For each value of \(\alpha\), the model was trained and validated, and the optimal value was selected based on performance metrics on the validation set. In each experiment, the value of \(\alpha\) was applied during model training, and the optimal value was selected based on performance metrics (e.g., accuracy and precision) on the validation set. The grid search was conducted over a total of 10 different values of \(\alpha\). Grid search systematically explores the parameter space, ensuring the optimal configuration for the specific task. In addition, the hyperparameters \(\lambda_{1}\) and \(\lambda_{2}\), which control the trade-off between the information entropy loss and the probability density estimation loss, were also tuned through grid search. The values of \(\lambda_{1}\) and \(\lambda_{2}\) were searched within the range of \([0.1,1.0]\) with a step size of 0.1. The optimal values of \(\lambda_{1}\) and \(\lambda_{2}\) were determined based on the performance of the model on the validation set, ensuring a balanced loss function that effectively incorporates both privacy protection and task performance.