2019/08/28 by Yihong Wu, Harrison H. Zhou, Wu, Yihong +1 · 2 citations
Computer Science · #Algorithms and Data Compression #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Machine Learning (stat.ML) #Machine Learning and Algorithms #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1908.10935
openalex publication_date 2019/08/28 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We analyze the classical EM algorithm for parameter estimation in the symmetric two-component Gaussian mixtures in d dimensions. We show that, even in the absence of any separation between components, provided that the sample size satisfies n=Ω(d log3 d), the randomly initialized EM algorithm converges to an estimate in at most O(√(n)) iterations with high probability, which is at most O(((d log3 n)/(n))1/4) in Euclidean distance from the true parameter and within logarithmic factors of the minimax rate of ((d)/(n))1/4. Both the nonparametric statistical rate and the sublinear convergence rate are direct consequences of the zero Fisher information in the worst case. Refined pointwise guarantees beyond worst-case analysis and convergence to the MLE are also shown under mild conditions. This improves the previous result of Balakrishnan et al \citeBWY17 which requires strong conditions on both the separation of the components and the quality of the initialization, and that of Daskalakis et al \citeDTZ17 which requires sample splitting and restarting the EM iteration.