Optimal subsampling algorithms for big data regressions

In order to quickly approximate maximum likelihood estimators from massive data, this study examines 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, the finite-sample properties of the subsample estimator based on optimal subsampling probabilities are also derived. Because the optimal subsampling probabilities depend on the full data estimate, an adaptive two-step algorithm is developed. The asymptotic normality and optimality of the estimator from this adaptive algorithm are established. The proposed methods are illustrated and evaluated using numerical experiments on simulated and real data sets.