  In this paper, we investigate the impact of compression on stochastic
gradient algorithms for machine learning, a technique widely used in
distributed and federated learning. We underline differences in terms of
convergence rates between several unbiased compression operators, that all
satisfy the same condition on their variance, thus going beyond the classical
worst-case analysis. To do so, we focus on the case of least-squares regression
(LSR) and analyze a general stochastic approximation algorithm for minimizing
quadratic functions relying on a random field. We consider weak assumptions on
the random field, tailored to the analysis (specifically, expected H\"older
regularity), and on the noise covariance, enabling the analysis of various
randomizing mechanisms, including compression. We then extend our results to
the case of federated learning.
  More formally, we highlight the impact on the convergence of the covariance
$\mathfrak{C}_{\mathrm{ania}}$ of the additive noise induced by the algorithm.
We demonstrate despite the non-regularity of the stochastic field, that the
limit variance term scales with $\mathrm{Tr}(\mathfrak{C}_{\mathrm{ania}}
H^{-1})/K$ (where $H$ is the Hessian of the optimization problem and $K$ the
number of iterations) generalizing the rate for the vanilla LSR case where it
is $\sigma^2 \mathrm{Tr}(H H^{-1}) / K = \sigma^2 d / K$ (Bach and Moulines,
2013). Then, we analyze the dependency of $\mathfrak{C}_{\mathrm{ania}}$ on the
compression strategy and ultimately its impact on convergence, first in the
centralized case, then in two heterogeneous FL frameworks.
