  This paper develops efficient algorithms for distributed average consensus
with quantized communication using the alternating direction method of
multipliers (ADMM). We first study the effects of probabilistic and
deterministic quantizations on a distributed ADMM algorithm. With probabilistic
quantization, this algorithm yields linear convergence to the desired average
in the mean sense with a bounded variance. When deterministic quantization is
employed, the distributed ADMM either converges to a consensus or cycles with a
finite period after a finite-time iteration. In the cyclic case, local
quantized variables have the same mean over one period and hence each node can
also reach a consensus. We then obtain an upper bound on the consensus error
which depends only on the quantization resolution and the average degree of the
network. Finally, we propose a two-stage algorithm which combines both
probabilistic and deterministic quantizations. Simulations show that the
two-stage algorithm, without picking small algorithm parameter, has consensus
errors that are typically less than one quantization resolution for all
connected networks where agents' data can be of arbitrary magnitudes.
