  We develop a distributed second order optimization algorithm that is
communication-efficient as well as robust against Byzantine failures of the
worker machines. We propose COMRADE (COMunication-efficient and Robust
Approximate Distributed nEwton), an iterative second order algorithm, where the
worker machines communicate only once per iteration with the center machine.
This is in sharp contrast with the state-of-the-art distributed second order
algorithms like GIANT [34] and DINGO[7], where the worker machines send
(functions of) local gradient and Hessian sequentially; thus ending up
communicating twice with the center machine per iteration. Moreover, we show
that the worker machines can further compress the local information before
sending it to the center. In addition, we employ a simple norm based
thresholding rule to filter-out the Byzantine worker machines. We establish the
linear-quadratic rate of convergence of COMRADE and establish that the
communication savings and Byzantine resilience result in only a small
statistical error rate for arbitrary convex loss functions. To the best of our
knowledge, this is the first work that addresses the issue of Byzantine
resilience in second order distributed optimization. Furthermore, we validate
our theoretical results with extensive experiments on synthetic and benchmark
LIBSVM [5] data-sets and demonstrate convergence guarantees.
