  In this paper, we study the problem of consensus-based distributed
optimization where a network of agents, abstracted as a directed graph, aims to
minimize the sum of all agents' cost functions collaboratively. In existing
distributed optimization approaches (Push-Pull/AB) for directed graphs, all
agents exchange their states with neighbors to achieve the optimal solution
with a constant stepsize, which may lead to the disclosure of sensitive and
private information. For privacy preservation, we propose a novel
state-decomposition based gradient tracking approach (SD-Push-Pull) for
distributed optimzation over directed networks that preserves differential
privacy, which is a strong notion that protects agents' privacy against an
adversary with arbitrary auxiliary information. The main idea of the proposed
approach is to decompose the gradient state of each agent into two sub-states.
Only one substate is exchanged by the agent with its neighbours over time, and
the other one is kept private. That is to say, only one substate is visible to
an adversary, protecting the privacy from being leaked. It is proved that under
certain decomposition principles, a bound for the sub-optimality of the
proposed algorithm can be derived and the differential privacy is achieved
simultaneously. Moreover, the trade-off between differential privacy and the
optimization accuracy is also characterized. Finally, a numerical simulation is
provided to illustrate the effectiveness of the proposed approach.
