  Privacy concerns with sensitive data are receiving increasing attention. In
this paper, we study local differential privacy (LDP) in interactive
decentralized optimization. By constructing random local aggregators, we
propose a framework to amplify LDP by a constant. We take Alternating Direction
Method of Multipliers (ADMM), and decentralized gradient descent as two
concrete examples, where experiments support our theory. In an asymptotic view,
we address the following question: Under LDP, is it possible to design a
distributed private minimizer for arbitrary closed convex constraints with
utility loss not explicitly dependent on dimensionality? As an affiliated
result, we also show that with merely linear secret sharing, information
theoretic privacy is achievable for bounded colluding agents.
