  While supervised federated learning approaches have enjoyed significant
success, the domain of unsupervised federated learning remains relatively
underexplored. Several federated EM algorithms have gained popularity in
practice, however, their theoretical foundations are often lacking. In this
paper, we first introduce a federated gradient EM algorithm (FedGrEM) designed
for the unsupervised learning of mixture models, which supplements the existing
federated EM algorithms by considering task heterogeneity and potential
adversarial attacks. We present a comprehensive finite-sample theory that holds
for general mixture models, then apply this general theory on specific
statistical models to characterize the explicit estimation error of model
parameters and mixture proportions. Our theory elucidates when and how FedGrEM
outperforms local single-task learning with insights extending to existing
federated EM algorithms. This bridges the gap between their practical success
and theoretical understanding. Our numerical results validate our theory, and
demonstrate FedGrEM's superiority over existing unsupervised federated learning
benchmarks.
