  This paper develops an efficient distributed inference algorithm, which is
robust against a moderate fraction of Byzantine nodes, namely arbitrary and
possibly adversarial machines in a distributed learning system. In robust
statistics, the median-of-means (MOM) has been a popular approach to hedge
against Byzantine failures due to its ease of implementation and computational
efficiency. However, the MOM estimator has the shortcoming in terms of
statistical efficiency. The first main contribution of the paper is to propose
a variance reduced median-of-means (VRMOM) estimator, which improves the
statistical efficiency over the vanilla MOM estimator and is computationally as
efficient as the MOM. Based on the proposed VRMOM estimator, we develop a
general distributed inference algorithm that is robust against Byzantine
failures. Theoretically, our distributed algorithm achieves a fast convergence
rate with only a constant number of rounds of communications. We also provide
the asymptotic normality result for the purpose of statistical inference. To
the best of our knowledge, this is the first normality result in the setting of
Byzantine-robust distributed learning. The simulation results are also
presented to illustrate the effectiveness of our method.
