  While many distributed optimization algorithms have been proposed for solving
smooth or convex problems over the networks, few of them can handle non-convex
and non-smooth problems. Based on a proximal primal-dual approach, this paper
presents a new (stochastic) distributed algorithm with Nesterov momentum for
accelerated optimization of non-convex and non-smooth problems. Theoretically,
we show that the proposed algorithm can achieve an $\epsilon$-stationary
solution under a constant step size with $\mathcal{O}(1/\epsilon^2)$
computation complexity and $\mathcal{O}(1/\epsilon)$ communication complexity.
When compared to the existing gradient tracking based methods, the proposed
algorithm has the same order of computation complexity but lower order of
communication complexity. To the best of our knowledge, the presented result is
the first stochastic algorithm with the $\mathcal{O}(1/\epsilon)$ communication
complexity for non-convex and non-smooth problems. Numerical experiments for a
distributed non-convex regression problem and a deep neural network based
classification problem are presented to illustrate the effectiveness of the
proposed algorithms.
