  To fast approximate maximum likelihood estimators with massive data, this
paper studies the Optimal Subsampling Method under the A-optimality Criterion
(OSMAC) for generalized linear models. The consistency and asymptotic normality
of the estimator from a general subsampling algorithm are established, and
optimal subsampling probabilities under the A- and L-optimality criteria are
derived. Furthermore, using Frobenius norm matrix concentration inequalities,
finite sample properties of the subsample estimator based on optimal
subsampling probabilities are also derived. Since the optimal subsampling
probabilities depend on the full data estimate, an adaptive two-step algorithm
is developed. Asymptotic normality and optimality of the estimator from this
adaptive algorithm are established. The proposed methods are illustrated and
evaluated through numerical experiments on simulated and real datasets.
