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This paper develops an efficient distributed inference algorithm, which isrobust against a moderate fraction of Byzantine nodes, namely arbitrary andpossibly adversarial machines in a distributed learning system. In robuststatistics, the median-of-means (MOM) has been a popular approach to hedgeagainst Byzantine failures due to its ease of implementation and computationalefficiency. However, the MOM estimator has the shortcoming in terms ofstatistical efficiency. The first main contribution of the paper is to proposea variance reduced median-of-means (VRMOM) estimator, which improves thestatistical efficiency over the vanilla MOM estimator and is computationally asefficient as the MOM. Based on the proposed VRMOM estimator, we develop ageneral distributed inference algorithm that is robust against Byzantinefailures. Theoretically, our distributed algorithm achieves a fast convergencerate with only a constant number of rounds of communications. We also providethe asymptotic normality result for the purpose of statistical inference. Tothe best of our knowledge, this is the first normality result in the setting ofByzantine-robust distributed learning. The simulation results are alsopresented to illustrate the effectiveness of our method.

Variance Reduced Median-of-Means Estimator for Byzantine-Robust Distributed Inference