  Distributed optimization methods are often applied to solving huge-scale
problems like training neural networks with millions and even billions of
parameters. In such applications, communicating full vectors, e.g.,
(stochastic) gradients, iterates, is prohibitively expensive, especially when
the number of workers is large. Communication compression is a powerful
approach to alleviating this issue, and, in particular, methods with biased
compression and error compensation are extremely popular due to their practical
efficiency. Sahu et al. (2021) propose a new analysis of Error Compensated SGD
(EC-SGD) for the class of absolute compression operators showing that in a
certain sense, this class contains optimal compressors for EC-SGD. However, the
analysis was conducted only under the so-called $(M,\sigma^2)$-bounded noise
assumption. In this paper, we generalize the analysis of EC-SGD with absolute
compression to the arbitrary sampling strategy and propose the first analysis
of Error Compensated Loopless Stochastic Variance Reduced Gradient method
(EC-LSVRG) with absolute compression for (strongly) convex problems. Our rates
improve upon the previously known ones in this setting. Numerical experiments
corroborate our theoretical findings.
