  Byzantine robustness is an essential feature of algorithms for certain
distributed optimization problems, typically encountered in
collaborative/federated learning. These problems are usually huge-scale,
implying that communication compression is also imperative for their
resolution. These factors have spurred recent algorithmic and theoretical
developments in the literature of Byzantine-robust learning with compression.
In this paper, we contribute to this research area in two main directions.
First, we propose a new Byzantine-robust method with compression -
Byz-DASHA-PAGE - and prove that the new method has better convergence rate (for
non-convex and Polyak-Lojasiewicz smooth optimization problems), smaller
neighborhood size in the heterogeneous case, and tolerates more Byzantine
workers under over-parametrization than the previous method with SOTA
theoretical convergence guarantees (Byz-VR-MARINA). Secondly, we develop the
first Byzantine-robust method with communication compression and error feedback
- Byz-EF21 - along with its bidirectional compression version - Byz-EF21-BC -
and derive the convergence rates for these methods for non-convex and
Polyak-Lojasiewicz smooth case. We test the proposed methods and illustrate our
theoretical findings in the numerical experiments.
