  In order to perform machine learning among multiple parties while protecting
the privacy of raw data, privacy-preserving machine learning based on secure
multi-party computation (MPL for short) has been a hot spot in recent. The
configuration of MPL usually follows the peer-to-peer architecture, where each
party has the same chance to reveal the output result. However, typical
business scenarios often follow a hierarchical architecture where a powerful,
usually privileged party, leads the tasks of machine learning. Only the
privileged party can reveal the final model even if other assistant parties
collude with each other. It is even required to avoid the abort of machine
learning to ensure the scheduled deadlines and/or save used computing resources
when part of assistant parties drop out.
  Motivated by the above scenarios, we propose pMPL, a robust MPL framework
with a privileged part}. pMPL supports three-party training in the semi-honest
setting. By setting alternate shares for the privileged party, pMPL is robust
to tolerate one of the rest two parties dropping out during the training. With
the above settings, we design a series of efficient protocols based on vector
space secret sharing for pMPL to bridge the gap between vector space secret
sharing and machine learning. Finally, the experimental results show that the
performance of pMPL is promising when we compare it with the state-of-the-art
MPL frameworks. Especially, in the LAN setting, pMPL is around $16\times$ and
$5\times$ faster than TF-encrypted (with ABY3 as the back-end framework) for
the linear regression, and logistic regression, respectively. Besides, the
accuracy of trained models of linear regression, logistic regression, and BP
neural networks can reach around 97%, 99%, and 96% on MNIST dataset
respectively.
