  Modern large-scale machine learning applications require stochastic
optimization algorithms to be implemented on distributed compute systems. A key
bottleneck of such systems is the communication overhead for exchanging
information across the workers, such as stochastic gradients. Among the many
techniques proposed to remedy this issue, one of the most successful is the
framework of compressed communication with error feedback (EF). EF remains the
only known technique that can deal with the error induced by contractive
compressors which are not unbiased, such as Top-$K$. In this paper, we propose
a new and theoretically and practically better alternative to EF for dealing
with contractive compressors. In particular, we propose a construction which
can transform any contractive compressor into an induced unbiased compressor.
Following this transformation, existing methods able to work with unbiased
compressors can be applied. We show that our approach leads to vast
improvements over EF, including reduced memory requirements, better
communication complexity guarantees and fewer assumptions. We further extend
our results to federated learning with partial participation following an
arbitrary distribution over the nodes, and demonstrate the benefits thereof. We
perform several numerical experiments which validate our theoretical findings.
