On the Global Convergence of (Fast) Incremental Expectation Maximization Methods

The EM algorithm is one of the most popular algorithm for inference in latent data models. The original formulation of the EM algorithm does not scale to large data set, because the whole data set is required at each iteration of the algorithm. To alleviate this problem, Neal and Hinton [1998] have proposed an incremental version of the EM (iEM) in which at each iteration the conditional expectation of the latent data (E-step) is updated only for a mini-batch of observations. Another approach has been proposed by Cappe and Moulines [2009] in which the E-step is replaced by a stochastic approximation step, closely related to stochastic gradient. In this paper, we analyze incremental and stochastic version of the EM algorithm as well as the variance reduced-version of [Chen et al., 2018] in a common unifying framework. We also introduce a new version incremental version, inspired by the SAGA algorithm by Defazio et al. [2014]. We establish non-asymptotic convergence bounds for global convergence. Numerical applications are presented in this article to illustrate our findings.