## 1 Introduction  
There are many circumstances where organizations need to use each other's data to perform tasks, such as data analysis or prediction [1; 2; 3]. For example, different parties may own different data sets that can be combined for improved predictive performance or inference. When these data contain personal information, the required data exchange can be problematic. In these situations, federated learning can be used to facilitate such collaborations. It is a privacy-preserving technique that keeps the data local during the analysis and ensures that no other party gains access to it. Within the research field dedicated to federated learning, there is increasing attention towards solutions for vertically partitioned data. One speaks of vertically partitioned data when different parties owning different attributes on the same subjects. An analysis that researchers often seek to perform on vertically partitioned data is regression analysis. Recently, an approach has been presented for this scenario: Block coordinate descent (BCD) [4]. BCD is a promising and fast way to perform federated learning for generalized linear models. One of its strengths is that it avoids computationally expensive cryptographic operations to secure the computations. Although no raw data is shared during BCD, the information that is exchanged can still leak information, as no privacy guarantees are in place. There are several examples of this in the context of federated learning [5; 6]. To limit the possibility of information leakage, we supplement BCD with differential privacy (DP). Differential privacy was introduced by [7] and is widely referred to as the state-of-the art approach to privacy preservation. Essentially, it involves adding uncertainty to the data analysis such that similar data sets will likely lead to similar results. Since its introduction, differential privacy has observed some application, but not yet widespread adoption. One of the reasons for this is that the DP parameters quantifying the privacy guarantees often cannot be made as small as hoped while preserving utility. The result is a noisy learning algorithm with reduced performance that theoretically could reveal information about data in the data set with considerable certainty. The motivation for this project is twofold. The first is to extend BCD with privacy guarantees to make it applicable in a wider range of use cases. The second motivation is to improve the practicality of DP in realistic use cases. To do so, we make some optimistic choices in our set-up. This means that less noise has to be added and a better performance is obtained. This clearly reduces the amount of protection DP offers. However, we believe that in this way we provide more meaningful privacy guarantees that correspond better to the data analyst's practice. Related WorkMultiple approaches have been presented to add differential privacy to a linear regression problem in the centralized setting (i.e., with one party) [8, 9, 10, 11]. For a federated setting, there has been more focus on horizontally partitioned data [12, 13]. For vertically partitioned data, only a few solutions are known [14, 15]. A problem similar to ours is treated in [15], where it is approached using techniques from multi-party computation. Although the algorithm performs well, it does not provide a utility guarantee or expectation. Utility expectations of differentially private learning have been presented for other applications. Examples can be found in differentially private empirical risk minimization [8, 16, 17, 18]. Since such utility bounds are typically asymptotic, large numbers of iterations are required for such bounds to become reliable. For learning techniques with a small number of iterations, such as ours, these are not practical. The approaches above apply differential privacy over the entire universe of data sets. The concept of locally sensitive differential privacy has been studied before [19, 20, 21], albeit under various names.