Distributed Optimization Under Adversarial Nodes

• Published in: IEEE transactions on automatic control Vol. 64; no. 3; pp. 1063 - 1076
• Main Authors: Sundaram, Shreyas, Gharesifard, Bahman
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.03.2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Computational geometry; Convexity; Distributed algorithms; fault tolerance; graph theory; Heuristic algorithms; Hulls; Linear programming; Lower bounds; machine learning; multi-agent systems; network security; Nodes; Optimization; Optimization algorithms; Protocols; Safety; Stochastic processes; Topology
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2018.2836919

We investigate the vulnerabilities of consensus-based distributed optimization protocols to nodes that deviate from the prescribed update rule (e.g., due to failures or adversarial attacks). We first characterize certain fundamental limitations on the performance of any distributed optimization algorithm in the presence of adversaries. We then propose a secure distributed optimization algorithm that guarantees that the nonadversarial nodes converge to the convex hull of the minimizers of their local functions under the certain conditions on the graph topology, regardless of the actions of a certain number of the adversarial nodes. In particular, we provide sufficient conditions on the graph topology to tolerate a bounded number of adversaries in the neighborhood of every nonadversarial node, and necessary and sufficient conditions to tolerate a globally bounded number of adversaries. For situations, where there are up to <inline-formula><tex-math notation="LaTeX">F</tex-math></inline-formula> adversaries in the neighborhood of every node, we use the concept of maximal <inline-formula><tex-math notation="LaTeX">F</tex-math></inline-formula>-local sets of graphs to provide lower bounds on the distance-to-optimality of achievable solutions under any algorithm. We show that finding the size of such sets is NP-hard.