Decentralized EM Algorithm for Gaussian Mixtures under Data Heterogeneity and Partial Labeling

We systematically study several network-based Expectation-Maximization (EM) algorithms for the Gaussian mixture model within decentralized federated learning (DFL). Our theoretical investigation shows that directly extending the classic EM algorithm to DFL leads to a biased estimator when data are heterogeneously distributed across sites. To address this, we introduce a momentum network EM (MNEM) algorithm, which integrates information from both current and historical estimators from previous DFL iterations. We further develop a semi-supervised MNEM (semi-MNEM) algorithm, which utilizes information provided by partially labeled data. Rigorous theoretical analysis demonstrates that the MNEM estimator can achieve the same asymptotic efficiency as the whole-sample estimator under appropriate regularity conditions, even with heterogeneous data. Moreover, the semi-MNEM estimator significantly improves the convergence speed of the MNEM algorithm, even if different mixture components are poorly separated. Extensive simulations are conducted, and a widely used chest X-ray dataset is analyzed to demonstrate the finite-sample performance of the proposed methods.