  This paper develops a networked federated learning algorithm to solve
nonsmooth objective functions. To guarantee the confidentiality of the
participants with respect to each other and potential eavesdroppers, we use the
zero-concentrated differential privacy notion (zCDP). Privacy is achieved by
perturbing the outcome of the computation at each client with a
variance-decreasing Gaussian noise. ZCDP allows for better accuracy than the
conventional $(\epsilon, \delta)$-DP and stronger guarantees than the more
recent R\'enyi-DP by assuming adversaries aggregate all the exchanged messages.
The proposed algorithm relies on the distributed Alternating Direction Method
of Multipliers (ADMM) and uses the approximation of the augmented Lagrangian to
handle nonsmooth objective functions. The developed private networked federated
learning algorithm has a competitive privacy accuracy trade-off and handles
nonsmooth and non-strongly convex problems. We provide complete theoretical
proof for the privacy guarantees and the algorithm's convergence to the exact
solution. We also prove under additional assumptions that the algorithm
converges in $O(1/n)$ ADMM iterations. Finally, we observe the performance of
the algorithm in a series of numerical simulations.
