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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.