# MFCDR: A Meta-Learning-Based Model for Federated Personalized Cross-Domain Recommendation
## 3 Model
### Overall Procedure
For assessment metrics, we considered RMSE, MSE, MAE, and Top-K. MSE is the abbreviation of mean square error, which is the average value of the square of the difference between the predicted value and the actual value. It is a measure of the difference between the predicted value and the actual value. The smaller the MSE, the smaller the difference between the predicted result and the real result, indicating that the model's prediction effect is better. RMSE is less sensitive to outliers than MSE because it operates on the square root. MAE maximizes user satisfaction but does not minimize prediction error. Top-K measures how many items in a given recommendation the recommendation algorithm matches with the user's true preferences, often with a long tail of users problem. The MSE is sensitive to outliers and computationally efficient, and it can be effectively applied to our model. Therefore, we use MSEs as performance indicators; they are used as assessment metrics in many related works, and the formula for the performance indicator MSE is:

\[MSE=\frac{1}{N}\sum_{(i,j,y_{i,j})\in R_{test}}\left(y_{i,j}-\hat{y}_{i,j}\right) ^{2} \tag{11}\]

where \(R_{test}\) is the test set used for performance comparison or the cold-start user validation set used for parameter selection, \(N\) is the number of ratings in the test set, \(y_{i,j}\) is the true rating of item \(t_{j}\) by user \(u_{i}\), and \(\hat{y}_{i,j}\) is the predicted rating of item \(t_{j}\) by user \(u_{i}\). According to the definition of MSE, a smaller MSE number suggests higher model performance. ### Model Comparison (RQ1 and RQ2)

According to the MSE, we examine the performance of several models with varying amounts of overlapping users. Figure 4 is a graphical representation of the experimental results, and Table 2 provides a summary of the outcomes for all three cross-domain recommendation tasks. It can be seen from the data that MFPCRR performs better than other baseline models in three cross-domain recommendation scenarios. The mean MSE of MFPCR in the first scenario is 1.086, and the median is 1.081. The mean MSE of MFPCRR in the second scenario is 1.126, and the median is 1.128. The mean MSE of MFPCRR in the third scenario is 0.8896, and the median is 0.895. The results yielded the following observations. Figure 4: Impact of the proportion of overlapping users (\(\eta\) represents the proportion of overlapping users). Firstly, MFPCDR outperformed all methods for all cross-domain recommendation tasks, and MFPCDR outperformed all methods with different proportions of overlapping users. In Scenario 1, when the proportion of overlapping users reaches approximately 20%, the fold starts to level off gradually. This phenomenon is also observed in Scenario 2 and Scenario 3 when the proportion of overlapping users reaches approximately 10%. We find that it is at the beginning of the model, as the proportion of overlapping users increases, that the model is more likely to capture the personalized transferable preferences of users, which helps our model solve the user cold-start problem. Additionally, as the proportion of overlapping to users continues to increase, the discounting starts to level off, which we find is due to several reasons. Firstly, it is due to the overlap between users who may be interested in the same products; secondly, when the proportion of overlapping users increases, more users in the MFPCDR model will generate similar behaviors; and, finally, when the proportion of overlapping users is high, the MFPCDR model is susceptible to the influence of popular recommendations. Compared to ANR, MFPCDR improved performance in MSE by 3.53% to 4.35% when the proportion of overlapping users was 60%. As the proportion of overlapping users becomes larger, the performance improvement also becomes larger. Both MFPCDR and ANR outperform CMF, demonstrating the effectiveness of the deep learning approach. CMF performs the worst of all evaluations because it learns the user representation by decomposing the union matrix only.