## 3 Proposed Framework
### Reducing Cross-Domain Discrepancy Based on Sinkhorn Distance
To address this, an entropy regularization term is added to the equation:  
\[d(a,b)=\sum_{i,j}S_{i,j}L_{i,j}+\beta H(S) \tag{5}\]  
where, \(\beta\) is the entropy regularization parameter, and \(H(S)\) is the entropy function of \(S\):  
\[H(S)=-\sum_{i,j}S_{i,j}\log(S_{i,j}) \tag{6}\]  
Approximating Sinkhorn distance typically involves maximizing a loss function \(L_{ud}\) concerning the parameter \(\theta_{w}\). This loss function is designed as a dual problem that minimizes Sinkhorn distance. Specifically, for the representation distributions \(P\) and \(Q\) of the source and target domains, along with their corresponding data \(X\) and \(Y\), maximizing the Sinkhorn distance loss function with respect to the parameter \(\theta_{w}\) can be expressed as:  
\[L_{ud}(\theta_{w})=\max_{u,v}(\sum_{i}u_{i}+\sum_{j}u_{j}-\epsilon\sum_{i,j}u_{ i}e^{-c_{i,j}}v_{j}) \tag{7}\]  
where \(u\) and \(v\) are scaling coefficients. \(\epsilon\) is the regularization parameter, and \(c_{i,j}\) represents the cost or distance between samples \(X_{i}\) and \(Y_{j}\). During the process of maximizing \(L_{ud}\), the scaling coefficients \(u\) and \(v\) are iteratively updated to approximate the actual Sinkhorn distance. Implementing a gradient penalty on the parameters is an effective approach to enforcing Lipschitz constraints [58]. This is particularly useful in DL-based models, controlling the model's Lipschitz constant through gradient clipping or regularization, enhancing stability, and mitigating issues like gradient explosion or vanishing during training [59]. In the context of Sinkhorn distance, enforcing Lipschitz constraints is typically achieved by adding a gradient penalty term to the loss function. The gradient penalty term can be in the form of parameter norms or other forms to ensure that the model's gradients do not become excessively large during training. To ensure the model's Lipschitz continuity, we choose the gradient penalty term \(L_{grad}(\theta_{w})\) as the L2 norm of the model parameters:  
\[L_{grad}(\theta_{w})=\frac{1}{2}|\theta_{w}|_{2}^{2} \tag{8}\]  
Due to the Sinkhorn distance being nearly continuously differentiable, the initial step involves training SDRL. After reaching optimal parameters for SDRL, these parameters are fixed, and the estimation of Sinkhorn distance is minimized. This process enables the mBERT network to learn domain-shared feature representations, accomplished by solving an optimization problem involving both maximization and minimization:  
\[L_{adv}=\min_{\theta_{mBERT}}\max_{\theta_{w}}\{L_{ud}-\lambda L_{grad}\} \tag{9}\]  
where \(\theta_{mBERT}\) represents the parameters of the feature extraction layer. \(\lambda\) is the penalty coefficient, set to 0 during the minimization process, ensuring the gradient penalty does not guide the learning process. The optimization goal is to learn domain-shared feature representations by minimizing the Sinkhorn distance loss function \(L_{\textit{ind}}\). During training, the minimization of \(L_{\textit{ind}}\) is achieved by adjusting the parameters \(\theta_{mBERT}\) of the feature extractor network through solving a maximization-minimization problem. The estimation parameters \(\theta_{w}\) for Sinkhorn distance are determined by maximizing \(L_{\textit{ind}}\). The overall optimization problem involves two components:  
1. Minimizing Sinkhorn distance: Adjusting the parameters \(\theta_{mBERT}\) of the feature extractor network to reduce the loss of Sinkhorn distance. 2. Maximizing gradient penalty: Tuning the estimated parameters \(\theta_{w}\) for Sinkhorn distance to increase the gradient penalty term \(\lambda L_{\textit{grad}}\). The ultimate optimization goal is to strike a balance between these two components, achieving a model with both domain-shared feature representation and compliance with Lipschitz constraints. This type of optimization problem is commonly referred to as adversarial training, where one network (\(mBERT\)) aims to minimize the loss, while the other network (\(SDRL\)) seeks to maximize it.