  Alternating direction method of multiplier (ADMM) is a powerful method to
solve decentralized convex optimization problems. In distributed settings, each
node performs computation with its local data and the local results are
exchanged among neighboring nodes in an iterative fashion. During this
iterative process the leakage of data privacy arises and can accumulate
significantly over many iterations, making it difficult to balance the
privacy-accuracy tradeoff. We propose Recycled ADMM (R-ADMM), where a linear
approximation is applied to every even iteration, its solution directly
calculated using only results from the previous, odd iteration. It turns out
that under such a scheme, half of the updates incur no privacy loss and require
much less computation compared to the conventional ADMM. Moreover, R-ADMM can
be further modified (MR-ADMM) such that each node independently determines its
own penalty parameter over iterations. We obtain a sufficient condition for the
convergence of both algorithms and provide the privacy analysis based on
objective perturbation. It can be shown that the privacy-accuracy tradeoff can
be improved significantly compared with conventional ADMM.
