\section{Conclusion}
% \paragraph{Conclusion.}
We propose and analyze two differentially private algorithms under multi-party setting for linear regression, and theoretically both of them are asymptotically optimal with increasing dataset size.
Empirically, \methodtwoshort{} has the best performance on both synthetic datasets and real-world datasets, while extremely large training set size $n$ is necessary for \methodoneshort{}.
% Our lesson is that when enforcing privacy in the multi-party setting, it is perhaps more predictable to focus on how to carefully design additive noise added to the data rather than how to modify the model training stage. Because the latter might lead to training instability and is harder to control. \kevin{add some meaning in conclusion as well.}
% 
We hope our work can bring more attention to the need for multi-party data release algorithms and we believe that ML practitioners would benefit from such effort in the era of privacy.

\paragraph{Future work.}
We focus on linear regression only, and one future direction is to extend our algorithm to classification, e.g. logistic regression, while achieving the same asymptotic optimality.
In addition, we assume different parties own the same set of data subjects. Another future direction is to relax this assumption: the set of subjects owned by different parties might be slightly different. %\jiankai{In this paper, we only focus on the setting of linear regression. One future direction is to extend our algorithm to classification problems while achieving the same asymptotic optimality.
% In addition, we would like to relax our assumption that different parties own the same set of data subjects and apply our algorithms to scenarios that the set of subjects owned by different parties are different.}


