High-Dimensional Variance-Reduced Stochastic Gradient Expectation-Maximization Algorithm


Rongda Zhu, Lingxiao Wang, Chengxiang Zhai, Quanquan Gu ;
Proceedings of the 34th International Conference on Machine Learning, PMLR 70:4180-4188, 2017.


We propose a generic stochastic expectation-maximization (EM) algorithm for the estimation of high-dimensional latent variable models. At the core of our algorithm is a novel semi-stochastic variance-reduced gradient designed for the $Q$-function in the EM algorithm. Under a mild condition on the initialization, our algorithm is guaranteed to attain a linear convergence rate to the unknown parameter of the latent variable model, and achieve an optimal statistical rate up to a logarithmic factor for parameter estimation. Compared with existing high-dimensional EM algorithms, our algorithm enjoys a better computational complexity and is therefore more efficient. We apply our generic algorithm to two illustrative latent variable models: Gaussian mixture model and mixture of linear regression, and demonstrate the advantages of our algorithm by both theoretical analysis and numerical experiments. We believe that the proposed semi-stochastic gradient is of independent interest for general nonconvex optimization problems with bivariate structures.

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