  We develop a communication-efficient distributed learning algorithm that is
robust against Byzantine worker machines. We propose and analyze a distributed
gradient-descent algorithm that performs a simple thresholding based on
gradient norms to mitigate Byzantine failures. We show the (statistical)
error-rate of our algorithm matches that of Yin et al.~\cite{dong}, which uses
more complicated schemes (coordinate-wise median, trimmed mean). Furthermore,
for communication efficiency, we consider a generic class of
$\delta$-approximate compressors from Karimireddi et al.~\cite{errorfeed} that
encompasses sign-based compressors and top-$k$ sparsification. Our algorithm
uses compressed gradients and gradient norms for aggregation and Byzantine
removal respectively. We establish the statistical error rate for non-convex
smooth loss functions. We show that, in certain range of the compression factor
$\delta$, the (order-wise) rate of convergence is not affected by the
compression operation. Moreover, we analyze the compressed gradient descent
algorithm with error feedback (proposed in \cite{errorfeed}) in a distributed
setting and in the presence of Byzantine worker machines. We show that
exploiting error feedback improves the statistical error rate. Finally, we
experimentally validate our results and show good performance in convergence
for convex (least-square regression) and non-convex (neural network training)
problems.
