  The problem of saddle-point avoidance for non-convex optimization is quite
challenging in large scale distributed learning frameworks, such as Federated
Learning, especially in the presence of Byzantine workers. The celebrated
cubic-regularized Newton method of \cite{nest} is one of the most elegant ways
to avoid saddle-points in the standard centralized (non-distributed) setup. In
this paper, we extend the cubic-regularized Newton method to a distributed
framework and simultaneously address several practical challenges like
communication bottleneck and Byzantine attacks. Note that the issue of
saddle-point avoidance becomes more crucial in the presence of Byzantine
machines since rogue machines may create \emph{fake local minima} near the
saddle-points of the loss function, also known as the saddle-point attack.
Being a second order algorithm, our iteration complexity is much lower than the
first order counterparts. Furthermore we use compression (or sparsification)
techniques like $\delta$-approximate compression for communication efficiency.
We obtain theoretical guarantees for our proposed scheme under several settings
including approximate (sub-sampled) gradients and Hessians. Moreover, we
validate our theoretical findings with experiments using standard datasets and
several types of Byzantine attacks, and obtain an improvement of $25\%$ with
respect to first order methods in iteration complexity.
