# Federated Pseudo-Sample Clustering Algorithm:
## 4 Lpfl-Gd
### Convergence Analysis  
In this context, we define \(x\) as the input space and \(y\) as the output space. The input \(x\) follows the distribution \(D\). \(H:x\to y\) represents the hypothesis space, \(|H|\leq\infty\), and the manually labeled results are represented by \(h\in H\), which means the generalization error of \(h\) as 0. **Definition 1**. : _Assume that there are two classifiers \(h_{i},h_{j}\in H\). In this paper, the generalization disagreement between them is defined as:_  
\[d\big{(}h_{i},h_{j}\big{)}=d\big{(}h_{i},h_{j}\big{|}x\big{)}=P\big{[}h_{i}(x) \
eq h_{j}(x)\big{|}x\in X\big{]} \tag{6}\]  
_Consequently, for \(h_{i},h_{j}\):_  
\[d\big{(}h_{i},h_{j}\big{)}=d\big{(}h_{j},h_{i}\big{)},d(h_{i},h_{i})=0 \tag{7}\]  
_The generalization error of the classifier \(h_{i}\) can be expressed as \(d(h_{i},h)\). Assuming, \(\exists\varepsilon>0,\ \xi>0\), if:_  
\[P[d(h_{i},h)\geq\varepsilon]<\xi \tag{8}\]  
_This means that the probability of the disagreement between classifiers \(h_{i}\) and \(h\) being less than \(\varepsilon\) is less than \(\xi\). When the value of \(\xi\) is determined, optimizing the classifier's performance is equivalent to minimizing \(\varepsilon\)._  
**Theorem 1**. : _Given two initialized datasets \(D_{1},\ D_{2}\), their sizes are, respectively, \(s_{1},\ s_{2}\). The classifiers obtained through training on \(D_{1},\ D_{2}\), denoted as \(h_{1},\ h_{2}\), have generalization errors \(a_{0}<0.5,\ b_{0}<0.5\), respectively, with the confidence parameter \(\xi>0\). Existing methods [36; 37] can achieve a classification generalization error within 0.5. The generative adversarial network (GAN) composed of classifier \(h_{1}\) and generator is utilized to generate a dataset \(G\) with size \(g\). After clustering and label correcting \(G\), the dataset \(G^{\prime}\) is obtained, with a generalization error of \(c\). The combination of \(G^{\prime}\) and \(D_{2}\) results in \(D^{\prime}_{2}\). Training \(h_{2}\) on \(D^{\prime}_{2}\) results in \(h^{\prime}_{2}\), with a generalization error of \(b_{1}\)._  
_If:_  
\[s_{1}\geq\frac{\ln\Big{(}\frac{|H|}{\xi}\Big{)}}{a_{0}} \tag{9}\]  
\[s_{2}\geq\frac{\ln\Big{(}\frac{|H|}{\xi}\Big{)}}{b_{0}} \tag{10}\]  
\[e^{\mathrm{F}}\sqrt{F}-F\geq s_{2}b_{0} \tag{11}\]  
_where \(F=gc+gb_{0}(1-c)\), then_  
\[P\big{[}d\big{(}h^{\prime}_{2},h\big{)}\geq b_{1}\big{]}<\xi \tag{12}\]  
_where \(b_{1}=\max\Big{\{}b_{0}+g\frac{c+b_{0}(1-c)-d\big{(}G^{\prime},h^{\prime}_{2} \big{)}}{s_{2}},0\Big{\}}\)._Proof. : We use \(d(h_{i},h)\) to denote the disagreement rate of \(h_{i}\). Similarly, we can use \(d\big{(}D_{i}^{\prime},h\big{)}\) to denote the disagreement rate of \(D_{i}^{\prime}\), then:  
\[d\big{(}D_{2}^{\prime},h\big{)}=\frac{gd(G^{\prime},h)}{s_{2}+g} \tag{13}\]  
\[d\big{(}D_{2}^{\prime},h_{2}^{\prime}\big{)}=\frac{s_{2}d(h_{2}^{\prime},h)+gd (G^{\prime},h_{2}^{\prime})}{s_{2}+g} \tag{14}\]  
The purpose of further training is to obtain a classifier \(h_{2}^{\prime}\) with a low disagreement rate from the dataset. In order to obtain a classifier with better performance, the dataset \(D_{2}^{\prime}\) should ensure that the possibility of a classifier \(h_{2}^{\prime}\), which has a generalization error greater than \(b_{1}\), having a lower disagreement rate than \(h\) on \(D_{2}^{\prime}\) is smaller than \(\xi\). Due to the upper bound of the generalization error of classifier \(h_{2}\) being \(b_{0}\), and the generalization error of clustering being \(c\), we can obtain \(d(D_{2}^{\prime},h)_{\text{max}}=\frac{gd(G^{\prime},h)}{s_{2}+g}=\frac{F}{s_ {2}+g}\).