  We propose a communication- and computation-efficient distributed
optimization algorithm using second-order information for solving ERM problems
with a nonsmooth regularization term. Current second-order and quasi-Newton
methods for this problem either do not work well in the distributed setting or
work only for specific regularizers. Our algorithm uses successive quadratic
approximations, and we describe how to maintain an approximation of the Hessian
and solve subproblems efficiently in a distributed manner. The proposed method
enjoys global linear convergence for a broad range of non-strongly convex
problems that includes the most commonly used ERMs, thus requiring lower
communication complexity. It also converges on non-convex problems, so has the
potential to be used on applications such as deep learning. Initial
computational results on convex problems demonstrate that our method
significantly improves on communication cost and running time over the current
state-of-the-art methods.
