  Distributed optimization often consists of two updating phases: local
optimization and inter-node communication. Conventional approaches require
working nodes to communicate with the server every one or few iterations to
guarantee convergence. In this paper, we establish a completely different
conclusion that each node can perform an arbitrary number of local optimization
steps before communication. Moreover, we show that the more local updating can
reduce the overall communication, even for an infinity number of steps where
each node is free to update its local model to near-optimality before
exchanging information. The extra assumption we make is that the optimal sets
of local loss functions have a non-empty intersection, which is inspired by the
over-paramterization phenomenon in large-scale optimization and deep learning.
Our theoretical findings are confirmed by both distributed convex optimization
and deep learning experiments.
