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Cooperative learning, that enables two or more data owners to jointly train a model, has been widely adopted to solve the problem of insufficient training data in machine learning. Nowadays, there is an urgent need for institutions and organizations to train a model cooperatively while keeping each other's data privately. To address the issue of privacy-preserving in collaborative learning, secure outsourced computation and federated learning are two typical methods. Nevertheless, there are many drawbacks for these two methods when they are leveraged in cooperative learning. For secure outsourced computation, semi-honest servers need to be introduced. Once the outsourced servers collude or perform other active attacks, the privacy of data will be disclosed. For federated learning, it is difficult to apply to the scenarios where vertically partitioned data are distributed over multiple parties. In this work, we propose a multi-party mixed protocol framework, ABG$^n$, which effectively implements arbitrary conversion between Arithmetic sharing (A), Boolean sharing (B) and Garbled-Circuits sharing (G) for $n$-party scenarios. Based on ABG$^n$, we design a privacy-preserving multi-party cooperative learning system, which allows different data owners to cooperate in machine learning in terms of data security and privacy-preserving. Additionally, we design specific privacy-preserving computation protocols for some typical machine learning methods such as logistic regression and neural networks. Compared with previous work, the proposed method has a wider scope of application and does not need to rely on additional servers. Finally, we evaluate the performance of ABG$^n$ on the local setting and on the public cloud setting. The experiments indicate that ABG$^n$ has excellent performance, especially in the network environment with low latency.