  The alternating direction method of multipliers (ADMM) has been recently
recognized as a promising optimizer for large-scale machine learning models.
However, there are very few results studying ADMM from the aspect of
communication costs, especially jointly with privacy preservation, which are
critical for distributed learning. We investigate the communication efficiency
and privacy-preservation of ADMM by solving the consensus optimization problem
over decentralized networks. Since walk algorithms can reduce communication
load, we first propose incremental ADMM (I-ADMM) based on the walk algorithm,
the updating order of which follows a Hamiltonian cycle instead. However,
I-ADMM cannot guarantee the privacy for agents against external eavesdroppers
even if the randomized initialization is applied. To protect privacy for
agents, we then propose two privacy-preserving incremental ADMM algorithms,
i.e., PI-ADMM1 and PI-ADMM2, where perturbation over step sizes and primal
variables is adopted, respectively. Through theoretical analyses, we prove the
convergence and privacy preservation for PI-ADMM1, which are further supported
by numerical experiments. Besides, simulations demonstrate that the proposed
PI-ADMM1 and PI-ADMM2 algorithms are communication efficient compared with
state-of-the-art methods.
