The application of secure multiparty computation (MPC) in machine learning, especially privacy-preserving neural network training, has attracted tremendous attention from the research community in recent years. MPC enables several data owners to jointly train a neural network while preserving the data privacy of each participant. However, most of the previous works focus on semi-honest threat model that cannot withstand fraudulent messages sent by malicious participants. In this paper, we propose an approach for constructing efficient $n$-party protocols for secure neural network training that can provide security for all honest participants even when a majority of the parties are malicious. Compared to the other designs that provide semi-honest security in a dishonest majority setting, our actively secure neural network training incurs affordable efficiency overheads of around 2X and 2.7X in LAN and WAN settings, respectively. Besides, we propose a scheme to allow additive shares defined over an integer ring $\mathbb{Z}_N$ to be securely converted to additive shares over a finite field $\mathbb{Z}_Q$, which may be of independent interest. Such conversion scheme is essential in securely and correctly converting shared Beaver triples defined over an integer ring generated in the preprocessing phase to triples defined over a field to be used in the calculation in the online phase.