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Byzantine-robustness has been gaining a lot of attention due to the growth ofthe interest in collaborative and federated learning. However, many fruitfuldirections, such as the usage of variance reduction for achieving robustnessand communication compression for reducing communication costs, remain weaklyexplored in the field. This work addresses this gap and proposes Byz-VR-MARINA- a new Byzantine-tolerant method with variance reduction and compression. Akey message of our paper is that variance reduction is key to fightingByzantine workers more effectively. At the same time, communication compressionis a bonus that makes the process more communication efficient. We derivetheoretical convergence guarantees for Byz-VR-MARINA outperforming previousstate-of-the-art for general non-convex and Polyak-Lojasiewicz loss functions.Unlike the concurrent Byzantine-robust methods with variance reduction and/orcompression, our complexity results are tight and do not rely on restrictiveassumptions such as boundedness of the gradients or limited compression.Moreover, we provide the first analysis of a Byzantine-tolerant methodsupporting non-uniform sampling of stochastic gradients. Numerical experimentscorroborate our theoretical findings.

Variance Reduction is an Antidote to Byzantines: Better Rates, Weaker Assumptions and Communication Compression as a Cherry on the Top