Distributed Optimization Under Adversarial Nodes
For distributed systems with potential failures or attacks, this work provides theoretical guarantees and limitations for resilient optimization, though the NP-hardness of key computations limits practical applicability.
This paper identifies fundamental limitations of distributed optimization under adversarial nodes and proposes a resilient algorithm that guarantees convergence of non-adversarial nodes to the convex hull of local minimizers, given certain graph conditions. The algorithm tolerates up to F adversaries per neighborhood, with lower bounds on solution quality derived via maximal F-local sets.
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.