  The Expectation-Maximization (EM) algorithm is an iterative method to
maximize the log-likelihood function for parameter estimation. Previous works
on the convergence analysis of the EM algorithm have established results on the
asymptotic (population level) convergence rate of the algorithm. In this paper,
we give a data-adaptive analysis of the sample level local convergence rate of
the EM algorithm. In particular, we show that the local convergence rate of the
EM algorithm is a random variable $\overline{K}_{n}$ derived from the data
generating distribution, which adaptively yields the convergence rate of the EM
algorithm on each finite sample data set from the same population distribution.
We then give a non-asymptotic concentration bound of $\overline{K}_{n}$ on the
population level optimal convergence rate $\overline{\kappa}$ of the EM
algorithm, which implies that $\overline{K}_{n}\to\overline{\kappa}$ in
probability as the sample size $n\to\infty$. Our theory identifies the effect
of sample size on the convergence behavior of sample EM sequence, and explains
a surprising phenomenon in applications of the EM algorithm, i.e. the finite
sample version of the algorithm sometimes converges faster even than the
population version. We apply our theory to the EM algorithm on three canonical
models and obtain specific forms of the adaptive convergence theorem for each
model.
