A Differential Private Method for Distributed Optimization in Directed Networks via State Decomposition

Abstract

In this paper, we study the problem of consensusbased 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.

Discrete-Time Zero-Gradient-Sum Algorithm for Distributed Optimization over Directed Networks

“…• When the objective function is strongly convex and the information-sharing noise is variance-bounded, all agents' estimates converge to the same optimal solution with the convergence rate O k −(1−ϵ) in the mean square sense, where ϵ could be close to 0 infinitely. To the best of our knowledge, this is the first convergence rate result of arriving at the optimal solution, and it is a complement to the convergence rate result of arriving in the optimal solution's neighborhood [17,23,25]. We verify our theoretical results with the numerical example of ridge regression problem.…”

“…Particularly, setting η = β = 1 − 5/8ϵ and α = 1 − 0.25ϵ, the convergence rate of VRA-GT is O 1 k 1−ϵ , where ϵ can close to zero infinitely. Moreover, Theorem 3 may complement the convergence rate result of arriving in the optimal solution's neighborhood [17,23,25].…”

“…To resolve the imperfect communication issue over general directed networks, a series of noise-robust variants of Push-Pull/AB method [21,22] have been developed [15,23,24,24,25]. These methods combat the information-sharing noise through two techniques, i.e.…”

“…Chen et. al [25] propose a modified gradient tracking method by combining the two techniques with a state decomposition mechanism for differentially-private distributed optimization. Similar to R-Push-Pull method, the method proposed in [25] is subject to steady-state errors as the factors adding on weight matrices need to be constant.…”

“…al [25] propose a modified gradient tracking method by combining the two techniques with a state decomposition mechanism for differentially-private distributed optimization. Similar to R-Push-Pull method, the method proposed in [25] is subject to steady-state errors as the factors adding on weight matrices need to be constant. Recently, Wang and Başar [24] propose a new robust gradient-tracking based method by reconstructing the cumulative gradient tracking to accommodate the decaying factors.…”