## 4 Discussion  
In this paper, we have constructed and tested the DP-BCD algorithm. We have demonstrated that it comes with a utility bound, that bounds the loss of \(R^{2}\) as a function of the privacy parameters and the \(R^{2}\) of the BCD solution. This is a new concept in DP learning. However, in its current form its practical relevance is very small, since the bound is too wide. The lower bound in this form deteriorates quickly with an increasing number of parties and iterations.  
The simulations in Sections 3.2 and 3.3 demonstrate that the weights obtained with DP-BCD are similar to BCD, also for correlated data. Nonetheless, the predictive power is lower, especially for problems with low \(R^{2}\). Nonetheless, the median \(R^{2}\) values observed are similar to the BCD, albeit with larger deviations and some outliers. We find that the predictive power is considerably lower for small values of \(\varepsilon\), high values of \(\gamma\) or small sample sizes, provided that \(\gamma\) is chosen big enough not to abort.  
Both \(\gamma\) and \(\varepsilon\) have a strong impact on the predictive power. Therefore, the \(\gamma\) value should be set as low as possible, as there is no benefit to having a high \(\gamma\). With respect to \(\varepsilon\), the algorithm retained predictive power even for single-digit privacy budgets. Though not incorporated in the simulation, the number of parties is also expected to impact \(R^{2}\), as it makes the BCD procedure more challenging and has a significant impact on the utility bound.  
Unbiased estimation of \(\beta\) parameters is a more challenging task than retaining predictive power. With the current procedure, this is not feasible with the amount of noise required. Particularly for highly correlated variables, the number of iterations may exceed the point where the increased precision as a result of iterations is overshadowed by the accumulated noise. For large sample sizes and large values of \(\varepsilon\), it is possible to obtain \(\beta\) parameters similar to the BCD procedure. This was also visualized in the forest fire analysis, where \(\varepsilon=20\) led to parameter estimates closer, though not identical, to the centralized and BCD setting.  
A fixed number of iterations has been used in the experiments. In this way, a clearer presentation of the performance of DP-BCD can be given. However, a convergence criterion, as described in Section 2.7, makes it possible to explicitly decide each round whether the improved utility is worth the consumed privacy budget. In this way, algorithms with better performance in terms of privacy budget and utility can be constructed.  
The problem of federated linear regression on vertically partitioned data is also studied in [15]. Based on the used techniques, it is our estimate that their solution can provide a higher utility on average, since it only requires a single round of noise addition. On the other hand, the use of secure multiplication techniques will probably lead to a longer  
Figure 8: \(R^{2}\) over \(\varepsilon\) for the Garment employee productivity data set (5 iterations, 1000 repetitions, and \(\gamma=1.2\)).  
learning phase, we expect our solution to be faster. Since we had no access to their code, we can only compare the solutions qualitatively.  
The approach chosen in this article may work as well for logistic regression. The BCD algorithm can also be used for logistic regression [4] on vertically partitioned data. It has been demonstrated that objective perturbation works well for logistic regression [23]. However, it is not clear whether it is possible to provide a utility guarantee of a similar nature for logistic regression.