Distributed Optimization Under Adversarial Nodes

TL;DR

This paper analyzes the vulnerabilities of distributed optimization algorithms to adversarial nodes and proposes a resilient method that guarantees convergence to the convex hull of local minimizers under certain graph conditions.

Contribution

It introduces a resilient distributed optimization algorithm with conditions on graph topology to tolerate adversarial nodes, including necessary and sufficient bounds.

Findings

Resilient algorithm guarantees convergence despite adversaries.

Sufficient graph conditions for adversary tolerance.

NP-hardness of computing maximal F-local sets.

Abstract

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 resilient distributed optimization algorithm that guarantees that the non-adversarial nodes converge to the convex hull of the minimizers of their local functions under certain conditions on the graph topology, regardless of the actions of a certain number of adversarial nodes. In particular, we provide sufficient conditions on the graph topology to tolerate a bounded number of adversaries in the neighborhood of every non-adversarial node, and necessary and sufficient conditions to tolerate a globally bounded number of adversaries. For situations where there are up to F adversaries in the neighborhood of every node, we use the concept of maximal F-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.