# Federated Pseudo-Sample Clustering Algorithm:
## 4 Lpfl-Gd
### Convergence Analysis
So, the probability that classifier \(h_{2}^{\prime}\) observes a disagreement rate smaller than \(h\) on \(D_{2}^{\prime}\) is less than  
\[C_{s_{2}+g}^{F}d\big{(}h_{2}^{\prime},D_{2}^{\prime}\big{)}^{F}\big{[}1-d \big{(}h_{2}^{\prime},D_{2}^{\prime}\big{)}\big{]}^{s_{2}+g-F} \tag{15}\]  
Further,  
\[d\big{(}h_{i},h_{j}\big{)}=d\big{(}h_{j},h_{i}\big{)},d(h_{i},h_{i})=0 \tag{7}\]0  
\[d\big{(}h_{i},h_{j}\big{)}=d\big{(}h_{j},h_{i}\big{)},d(h_{i},h_{i})=0 \tag{7}\]1  
Since that  
\[d\big{(}h_{i},h_{j}\big{)}=d\big{(}h_{j},h_{i}\big{)},d(h_{i},h_{i})=0 \tag{7}\]2  
If \(d(h_{2}^{\prime},D_{2}^{\prime})\geq b_{1}\), then  
\[ \begin{array}{l}d(h_{2}^{\prime},D_{2}^{\prime})\ =\frac{s_{2}d\big{(}h_{2}^{\prime},h\big{)}+gd \big{(}G^{\prime},h_{2}^{\prime}\big{)}}{s_{2}+g}\\ \ \geq\frac{s_{2}b_{1}+gd\big{(}G^{\prime},h_{2}^{\prime}\big{)}}{s_{2}+g}\\ \ \geq\frac{s_{2}b_{1}+gc+gb_{0}(1-c)}{s_{2}+g}\\ \ >\frac{F}{s_{2}+g}\end{array}  \tag{19}\]  
When \(\frac{F}{s_{2}+g}<d(h_{2}^{\prime},D_{2}^{\prime})<1\) and \(d(h_{2}^{\prime},D_{2}^{\prime})\geq b_{1}\), the derivative of Formula (16) leads to the conclusion that Formula (16) monotonically increases with decreasing \(d(h_{2}^{\prime},D_{2}^{\prime})\). Therefore,  
\[ \begin{array}{l}P[d(h_{2}^{\prime},h)\geq d(D_{2}^{\prime},h_{2}^{\prime})] <\\ C_{s_{2}+g}^{F}d(h_{2}^{\prime},D_{2}^{\prime})^{F}[1-d(h_{2}^{\prime},D_{2}^{ \prime})]^{s_{2}+g-F}<\\ C_{s_{2}+g}^{F}\big{(}\frac{s_{2}b_{1}+g}{s_{2}+g}\big{)}^{F}\Big{(}1-\frac{s_ {2}b_{1}+g}{s_{2}+g}\Big{)}^{s_{2}+g-F}\end{array}  \tag{20}\]  
According to the Poisson theorem, Formula (20) can be approximated as:  
\[ \begin{array}{l}P[d(h_{2}^{\prime},h)\geq d(D_{2}^{\prime},h_{2}^{\prime})] <\\ C_{g-g}^{F}d(h_{2}^{\prime},D_{2}^{\prime})^{F}[1-d(h_{2}^{\prime},D_{2}^{ \prime})]^{s_{2}+g-F}\approx\\ \frac{[s_{2}b_{1}+gc+gb_{0}(1-c)]^{F}}{F!}e^{-[s_{2}b_{1}+gc+gb_{0}(1-c)]} \end{array}  \tag{21}\]  
Assuming \(e^{F}\sqrt{F! }-F\geq s_{2}b_{0}\), then  
\[\frac{[s_{2}b_{1}+gc+gb_{0}(1-c)]^{F}}{F!}e^{-[s_{2}b_{1}+gc+gb_{0}(1-c)]}\leq e ^{-s_{2}b_{0}} \tag{22}\]  
When \(s_{2}\geq\frac{\ln\big{(}\frac{|H|}{c}\big{)}}{b_{0}}\), \(e^{-s_{2}b_{0}}\leq\frac{c}{|H|}\)In conclusion, when the generalization error of \(|H|-1\) classifiers is not less than \(b_{1}\), and their disagreement rate with \(D_{2}^{\prime}\) is lower than \(h\), then  
\[ \begin{array}{ll}P[d(h_{2}^{\prime},h)\geq b_{1}]&\leq(|H|-1)e^{-s_{2}b_{0}}\\ &\leq(|H|-1)\frac{\xi}{|H|}\\ &<\xi\end{array}  \tag{22}\]  
Theorem 1 is proved. \(\square\)  
It is not difficult to deduce based on Formulas (17) and (12) that when \(d(G^{\prime},h_{2}^{\prime})\) is sufficiently large, \(b_{1}<b_{0}\), meaning that in the same confidence level, the upper bound of the generalization error of classifier \(h_{2}^{\prime}\) is smaller than that of classifier \(h_{2}\). Therefore, it can be proven that the experiment has further converged. Although \(G^{\prime}\) is obtained by label correction through \(h_{2}\) on \(G\), the samples in \(G^{\prime}\) are still generated by a GAN network composed of classifier \(h_{1}\) and a generator. Then, \(h_{2}^{\prime}\) is obtained by further training \(D_{2}\) and \(G^{\prime}\) combined on the basis of \(h_{2}\). Therefore, \(d(G^{\prime},h_{2}^{\prime})\) can map the differences between \(h_{1}\) and \(h_{2}\), and further map the differences between the training datasets \(D_{1},\ D_{2}\) for \(h_{1},\ h_{2}\). For personalized federated learning, it is common for there to be significant differences between the datasets of different clients. So, the condition for performance improvement is usually met.