  Gradient compression is a popular technique for improving communication
complexity of stochastic first-order methods in distributed training of machine
learning models. However, the existing works consider only with-replacement
sampling of stochastic gradients. In contrast, it is well-known in practice and
recently confirmed in theory that stochastic methods based on
without-replacement sampling, e.g., Random Reshuffling (RR) method, perform
better than ones that sample the gradients with-replacement. In this work, we
close this gap in the literature and provide the first analysis of methods with
gradient compression and without-replacement sampling. We first develop a
na\"ive combination of random reshuffling with gradient compression (Q-RR).
Perhaps surprisingly, but the theoretical analysis of Q-RR does not show any
benefits of using RR. Our extensive numerical experiments confirm this
phenomenon. This happens due to the additional compression variance. To reveal
the true advantages of RR in the distributed learning with compression, we
propose a new method called DIANA-RR that reduces the compression variance and
has provably better convergence rates than existing counterparts with
with-replacement sampling of stochastic gradients. Next, to have a better fit
to Federated Learning applications, we incorporate local computation, i.e., we
propose and analyze the variants of Q-RR and DIANA-RR -- Q-NASTYA and
DIANA-NASTYA that use local gradient steps and different local and global
stepsizes. Finally, we conducted several numerical experiments to illustrate
our theoretical results.
