Unsupervised learning has been widely used in many real-world applications. One of the simplest and most important unsupervised learning models is the Gaussian mixture model (GMM). In this work, we study the multi-task learning problem on GMMs, which aims to leverage potentially similar GMM parameter structures among tasks to obtain improved learning performance compared to single-task learning. We propose a multi-task GMM learning procedure based on the EM algorithm that not only can effectively utilize unknown similarity between related tasks but is also robust against a fraction of outlier tasks from arbitrary distributions. The proposed procedure is shown to achieve minimax optimal rate of convergence for both parameter estimation error and the excess mis-clustering error, in a wide range of regimes. Moreover, we generalize our approach to tackle the problem of transfer learning for GMMs, where similar theoretical results are derived. Finally, we demonstrate the effectiveness of our methods through simulations and real data examples. To the best of our knowledge, this is the first work studying multi-task and transfer learning on GMMs with theoretical guarantees.

翻译：无监督学习已广泛应用于许多实际场景。其中最简单且最重要的无监督学习模型之一便是高斯混合模型（GMM）。本文研究了GMM上的多任务学习问题，其目标是通过利用任务间可能相似的GMM参数结构，相较于单任务学习获得更优的学习性能。我们提出了一种基于EM算法的多任务GMM学习流程，该方法不仅能有效利用相关任务间的未知相似性，还能对任意分布中的异常值任务比例具有鲁棒性。理论分析表明，在广泛的任务场景下，所提方法在参数估计误差和超额误聚类误差上均能达到极小化最优收敛速率。此外，我们将该方法推广至GMM的迁移学习问题，并推导出相似的理论结果。最后，通过仿真实验和真实数据案例验证了方法的有效性。据我们所知，这是首项为GMM上的多任务与迁移学习提供理论保证的研究工作。