  The Expectation-Maximization (EM) algorithm is a popular choice for learning
latent variable models. Variants of the EM have been initially introduced,
using incremental updates to scale to large datasets, and using Monte Carlo
(MC) approximations to bypass the intractable conditional expectation of the
latent data for most nonconvex models. In this paper, we propose a general
class of methods called Two-Timescale EM Methods based on a two-stage approach
of stochastic updates to tackle an essential nonconvex optimization task for
latent variable models. We motivate the choice of a double dynamic by invoking
the variance reduction virtue of each stage of the method on both sources of
noise: the index sampling for the incremental update and the MC approximation.
We establish finite-time and global convergence bounds for nonconvex objective
functions. Numerical applications on various models such as deformable template
for image analysis or nonlinear models for pharmacokinetics are also presented
to illustrate our findings.
