  In federated learning, a federator coordinates the training of a model, e.g.,
a neural network, on privately owned data held by several participating
clients. The gradient descent algorithm, a well-known and popular iterative
optimization procedure, is run to train the model. Every client computes
partial gradients based on their local data and sends them to the federator,
which aggregates the results and updates the model. Privacy of the clients'
data is a major concern. In fact, it is shown that observing the partial
gradients can be enough to reveal the clients' data. Existing literature
focuses on private aggregation schemes that tackle the privacy problem in
federated learning in settings where all users are connected to each other and
to the federator. In this paper, we consider a hierarchical wireless system
architecture in which the clients are connected to base stations; the base
stations are connected to the federator either directly or through relays. We
examine settings with and without relays, and derive fundamental limits on the
communication cost under information-theoretic privacy with different collusion
assumptions. We introduce suitable private aggregation schemes tailored for
these settings whose communication costs are multiplicative factors away from
the derived bounds.
