  In this paper, we propose a novel distributed alternating direction method of
multipliers (ADMM) algorithm with synergetic communication and computation,
called SCCD-ADMM, to reduce the total communication and computation cost of the
system. Explicitly, in the proposed algorithm, each node interacts with only
part of its neighboring nodes, the number of which is progressively determined
according to a heuristic searching procedure, which takes into account both the
predicted convergence rate and the communication and computation costs at each
iteration, resulting in a trade-off between communication and computation. Then
the node chooses its neighboring nodes according to an importance sampling
distribution derived theoretically to minimize the variance with the latest
information it locally stores. Finally, the node updates its local information
with a new update rule which adapts to the number of communication nodes. We
prove the convergence of the proposed algorithm and provide an upper bound of
the convergence variance brought by randomness. Extensive simulations validate
the excellent performances of the proposed algorithm in terms of convergence
rate and variance, the overall communication and computation cost, the impact
of network topology as well as the time for evaluation, in comparison with the
traditional counterparts.
