  We study the generation of dependent random numbers in a distributed fashion
in order to enable privatized distributed learning by networked agents. We
propose a method that we refer to as local graph-homomorphic processing; it
relies on the construction of particular noises over the edges to ensure a
certain level of differential privacy. We show that the added noise does not
affect the performance of the learned model. This is a significant improvement
to previous works on differential privacy for distributed algorithms, where the
noise was added in a less structured manner without respecting the graph
topology and has often led to performance deterioration. We illustrate the
theoretical results by considering a linear regression problem over a network
of agents.
