## 4 Experiments
### Experimental Results
#### 4.2.1 RQ1: Impacts on Convergence Speed and Recommendation Performance  
We first focus on the relevant results of I2 CG. Figure 3 and Table 4 show the impact of I2 CG on FRSs with or without DPM. From the results, we observe that, whether FRSs adopt DPM or not, in most cases, the I2 CG approach can effectively accelerate the convergence of HR and NDCG. For example, on Foursquare, when PFedMF employs I2 CG, it only requires 84 communication rounds to reach convergence, which represents a 64.71% reduction in the number of communication rounds required compared to when I2 CG is not employed, which necessitates 238 communication rounds. In a few instances, such as FedRECON on Lastfm-2K, the introduction of I2 CG does not reduce the communication rounds needed for convergence (\(104\to 110\)). Introducing the I2 CG strategy enables the utilization of local computation resources from communication-constrained clients, facilitating the comprehensive training of these clients, thereby accelerating overall model convergence rate. The impact of I2 CG on the model performance demonstrates variability across diverse FRSs, datasets and evaluation metrics. Specifically, on Lastfm-2K, the introduction of I2 CG in PFedMF enhances HR from 0.8196 to 0.8540, representing a 5.2% increase. However,  
Figure 3: Learning curves of PFedMF/FedMF with or without I2 CG. (**a–d**) PFedMF-I2 CG: (**a**) MovieLens-100K. (**b**) MovieLens-1M. (**c**) Lastfm-2K. (**d**) Foursquare. (**e**–h**) FedMF-I2 CG: (**e**) MovieLens-100K. (**f**) MovieLens-1M. (**g**) Lastfm-2K. (**h**) Foursquare. on MovieLens-100K, the introduction of I2 CG in PFedMF results in a minor decrease in HR, from 0.7243 to 0.7211, a decline of 0.44%. Fortunately, the introduction of I2 CG in experiments has never led to a significant decline in the recommendation performance. The reduction can be alleviated through the early-stop strategy which means removing I2 CG from the system after proper rounds. Additionally, early-stop aids in minimizing the extra computation burden imposed by the I2 CG approach. Further elaboration on this matter will be provided in Section 4.2.3. On one hand, I2 CG can be seen as semi-federated learning whose clients do not upload the updated item models to the server for aggregation, resulting in a slight \"drift\" of the global item models compared to the global item models in federated learning incorporating both federated and I2 CG clients. On the other hand, abstaining from uploading item models for aggregation could lead to a \"delay\" in global item models' updating process compared to client models. During the next round of training, the \"drift\" and \"delay\" may have regularizing or negative effects on model performance. The direction of the impact may vary depending on the FRSs, datasets and evaluation metrics. Now, we turn our attention to I2 CL. Figure 4a-d and Table 5 show the impact of I2 CL on FRSs with DPM. As Figure 4a-d and Table 5 show, the I2 CL strategy markedly boosts the convergence speed of FRSs with DPM. For instance, on Lastfm-2K, the incorporation of I2 CL significantly reduces the training rounds required for PFedRECON to reach convergence, decreasing from 198 to 47 rounds (\(-\)76.26%). In exceptional cases, such as on Foursquare, the training rounds needed of PFedMF remain unchanged.