  While client sampling is a central operation of current state-of-the-art
federated learning (FL) approaches, the impact of this procedure on the
convergence and speed of FL remains under-investigated. In this work, we
provide a general theoretical framework to quantify the impact of a client
sampling scheme and of the clients heterogeneity on the federated optimization.
First, we provide a unified theoretical ground for previously reported sampling
schemes experimental results on the relationship between FL convergence and the
variance of the aggregation weights. Second, we prove for the first time that
the quality of FL convergence is also impacted by the resulting covariance
between aggregation weights. Our theory is general, and is here applied to
Multinomial Distribution (MD) and Uniform sampling, two default unbiased client
sampling schemes of FL, and demonstrated through a series of experiments in
non-iid and unbalanced scenarios. Our results suggest that MD sampling should
be used as default sampling scheme, due to the resilience to the changes in
data ratio during the learning process, while Uniform sampling is superior only
in the special case when clients have the same amount of data.
