  There is growing interest in large-scale machine learning and optimization
over decentralized networks, e.g. in the context of multi-agent learning and
federated learning. Due to the imminent need to alleviate the communication
burden, the investigation of communication-efficient distributed optimization
algorithms - particularly for empirical risk minimization - has flourished in
recent years. A large fraction of these algorithms have been developed for the
master/slave setting, relying on a central parameter server that can
communicate with all agents. This paper focuses on distributed optimization
over networks, or decentralized optimization, where each agent is only allowed
to aggregate information from its neighbors. By properly adjusting the global
gradient estimate via local averaging in conjunction with proper correction, we
develop a communication-efficient approximate Newton-type method Network-DANE,
which generalizes DANE to the decentralized scenarios. Our key ideas can be
applied in a systematic manner to obtain decentralized versions of other
master/slave distributed algorithms. A notable development is
Network-SVRG/SARAH, which employs variance reduction to further accelerate
local computation. We establish linear convergence of Network-DANE and
Network-SVRG for strongly convex losses, and Network-SARAH for quadratic
losses, which shed light on the impacts of data homogeneity, network
connectivity, and local averaging upon the rate of convergence. We further
extend Network-DANE to composite optimization by allowing a nonsmooth penalty
term. Numerical evidence is provided to demonstrate the appealing performance
of our algorithms over competitive baselines, in terms of both communication
and computation efficiency. Our work suggests that performing a certain amount
of local communications and computations per iteration can substantially
improve the overall efficiency.
