  For massive data, the family of subsampling algorithms is popular to downsize
the data volume and reduce computational burden. Existing studies focus on
approximating the ordinary least squares estimate in linear regression, where
statistical leverage scores are often used to define subsampling probabilities.
In this paper, we propose fast subsampling algorithms to efficiently
approximate the maximum likelihood estimate in logistic regression. We first
establish consistency and asymptotic normality of the estimator from a general
subsampling algorithm, and then derive optimal subsampling probabilities that
minimize the asymptotic mean squared error of the resultant estimator. An
alternative minimization criterion is also proposed to further reduce the
computational cost. The optimal subsampling probabilities depend on the full
data estimate, so we develop a two-step algorithm to approximate the optimal
subsampling procedure. This algorithm is computationally efficient and has a
significant reduction in computing time compared to the full data approach.
Consistency and asymptotic normality of the estimator from a two-step algorithm
are also established. Synthetic and real data sets are used to evaluate the
practical performance of the proposed method.
