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. (2018), 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. (2019) 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 Karimireddi et al. 2019) 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.
