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.