## 3 Method
### Approach Statement  
We assume that there are two data owners \(\mathbf{PartyA\&B}\) training model \(A\) and model \(B\). Firstly, the model needs feature extraction to generate two implicit representations \(u_{i}^{A}={}^{\prime}{}_{A}\big{(}x_{i}^{A}\big{)}\) and \(u_{i}^{B}={}^{\prime}{}_{B}\big{(}x_{i}^{B}\big{)}\), where \(u^{A}\in\mathbb{R}^{N_{A}\times d}\) and \(u^{B}\in\mathbb{R}^{N_{B}\times d}\), and \(d\) represent the dimensions of the hidden layer. \(\varphi(x;w,b)=\frac{1}{1+\exp(-w^{B}x)}\) is the feature transformation function, where matrix-vector pair \((w,b)\in\mathbb{R}^{N}\times\mathbb{R}\). The neural network \(\varphi_{A}\) and \(\varphi_{B}\) transform their features, and project the features of \(\mathbf{PartyA}\) and \(\mathbf{PartyB}\) into a common feature subspace, where they can exchange knowledge. Then, the source domain data of feature space is classified by the prediction layer, and the correct labels are separated as far as possible. The prediction function \(\psi(u_{i}^{B})=\psi(u_{1}^{A},y_{1}^{A}...u_{N_{A}}^{A},y_{N_{A}}^{A},u_{j}^ {B})\),  
where \(\psi(\mathbf{a})=\left[\frac{\exp(a_{i})}{\sum_{j=1}^{A}\exp(a_{j})}\right]_{i=1}^ {|\mathbf{a}|}\). Thus, we obtain the first optimization goal:  
\[\underset{\Theta^{x},\Theta^{y}}{\text{argmin}}\mathcal{L}_{1}=\frac{1}{n}\sum _{i=1}^{T}\ell_{1}\left(y_{i}^{A},\psi\left(u_{i}^{B}\right)\right) \tag{4}\]  
where \(\Theta^{x},\Theta^{y}\) are training parameters of layers \(\varphi_{A},\psi\). \(\ell_{1}\) is classification loss, using the negative log-probability of the correct labels. The network also needs a regularization term, that is, a metric criterion so that the source domain and the target domain in the feature space are as close as possible. We use H-divergence to measure [42; 43; 44]:  
\[\hat{d}_{\mathcal{H}}(S,T)= 2\Bigg{(}1-\underset{\eta\in\mathcal{H}}{\text{min}}\Bigg{[} \frac{1}{n}\sum_{i=1}^{n}I[\eta(\mathbf{x}_{i})=0] \tag{5}\] \[+\frac{1}{n^{\prime}}\sum_{i=n+1}^{N}I[\eta(\mathbf{x}_{i})=1] \Bigg{]}\Bigg{)}\]  
where \(\mathcal{H}\) is a hypothetical class, binary classifiers \(\eta:X\rightarrow\{0,1\}\). According to formula (5), the empirical H-divergence between the source domain and the target domain can be calculated by the following formula:  
\[\hat{d}_{\mathcal{H}}\Big{(}D^{A},D^{B}\Big{)}= 2\Bigg{(}1-\underset{\eta\in\mathcal{H}}{\text{min}}\Bigg{[} \frac{1}{n}\sum_{i=1}^{n}I\Big{[}\eta\left(\omega\left(u_{i}^{B}\right)\right)= 0\Big{]} \tag{6}\] \[+\frac{1}{n^{\prime}}\sum_{i=n+1}^{N}I\Big{[}\eta\left(\omega \left(u_{i}^{B}\right)\right)=1\Big{]}\Bigg{]}\Bigg{)}\]  
It is usually difficult to calculate \(\hat{d}_{\mathcal{H}}\big{(}D^{A},D^{B}\big{)}\). According to the inspiration of [PERSON]'s research [42; 43], we can define the approximate solution of the classification problem of the source domain and target domain, that is, domain classifier. It classifies data in feature space and tries to distinguish which domain the data comes from. Domainclassifier \(\omega(u_{j}^{B})=\frac{1}{1+\exp(-u_{j}^{B})}\), and we can make the feature distribution of the data in the source domain and the target domain as close as possible by approximately calculating the empirical H-divergence between the two samples. We can obtain the second optimization target:  
\[\underset{\Theta^{y},\Theta^{z}}{\operatorname{argmax}}\mathcal{L}_{2}= -\frac{1}{n}\sum_{i=1}^{n}\ell_{2}\Big{(}z_{i}^{A},\omega\left(u_{i}^{B }\right)=0\Big{)} \tag{7}\] \[-\frac{1}{n^{\prime}}\sum_{i=n+1}^{N}\ell_{2}\Big{(}z_{i}^{A}, \omega\left(u_{i}^{B}\right)=1\Big{)}\]  
where \(\Theta^{y},\Theta^{z}\) are training parameters of layers \(\psi,\omega\). \(z_{i}\) is the domain classification label. \(\ell_{2}\) is the loss of domain classification layer, and the loss of feature alignment is calculated by cross entropy.