  Stochastic gradient descent (SGD) is a prevalent optimization technique for
large-scale distributed machine learning. While SGD computation can be
efficiently divided between multiple machines, communication typically becomes
a bottleneck in the distributed setting. Gradient compression methods can be
used to alleviate this problem, and a recent line of work shows that SGD
augmented with gradient compression converges to an $\varepsilon$-first-order
stationary point. In this paper we extend these results to convergence to an
$\varepsilon$-second-order stationary point ($\varepsilon$-SOSP), which is to
the best of our knowledge the first result of this type. In addition, we show
that, when the stochastic gradient is not Lipschitz, compressed SGD with
RandomK compressor converges to an $\varepsilon$-SOSP with the same number of
iterations as uncompressed SGD [Jin et al.,2021] (JACM), while improving the
total communication by a factor of $\tilde \Theta(\sqrt{d}
\varepsilon^{-3/4})$, where $d$ is the dimension of the optimization problem.
We present additional results for the cases when the compressor is arbitrary
and when the stochastic gradient is Lipschitz.
