  In modern data science, it is common that large-scale data are stored and
processed parallelly across a great number of locations. For reasons including
confidentiality concerns, only limited data information from each parallel
center is eligible to be transferred. To solve these problems more efficiently,
a group of communication-efficient methods are being actively developed. We
propose two communication-efficient Newton-type algorithms, combining the
M-estimator and the gradient collected from each data center. They are created
by constructing two Fisher information estimators globally with those
communication-efficient statistics. Enjoying a higher rate of convergence, this
framework improves upon existing Newton-like methods. Moreover, we present two
bias-adjusted one-step distributed estimators. When the square of the
center-wise sample size is of a greater magnitude than the total number of
centers, they are as efficient as the global $M$-estimator asymptotically. The
advantages of our methods are illustrated by extensive theoretical and
empirical evidences.
