Distributed Optimization Under Adversarial Nodes (1606.08939v1)

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.

• The paper demonstrates fundamental limitations of distributed optimization under adversarial nodes and introduces the Local Filtering Dynamics (LF) protocol for resilience.
• LF Dynamics enable nodes to filter out extreme neighbor values, ensuring convergence for regular nodes under specific graph robustness conditions like (2F+1)-robustness.
• The protocol guarantees that non-adversarial nodes converge within the convex hull of their local minimizers, enhancing security for applications in networked systems.

Overview of "Distributed Optimization Under Adversarial Nodes"

The paper "Distributed Optimization Under Adversarial Nodes" authored by Shreyas Sundaram and Bahman Gharesifard explores the robustness of consensus-based distributed optimization protocols in the presence of adversarial nodes. Distributed optimization is pivotal in networked systems where agents (or nodes) aim to reach a global optimum by only interacting with their neighbors, making the study of resilience against malfunction or malicious intent critically important.

Summary of Contributions

The authors commence by illustrating the vulnerabilities inherent in conventional distributed optimization algorithms when certain nodes deviate due to adversarial actions. They identify fundamental performance limitations of any distributed optimization algorithm in adversarial settings. To mitigate these vulnerabilities, a resilient protocol, termed "Local Filtering (LF) Dynamics," is proposed. This protocol ensures convergence within the convex hull of local function minimizers for non-adversarial nodes, despite the presence of adversaries. The convergence proofs rest on conditions related to the topology of the graph interconnecting the nodes.

Key Contributions and Results

1. Fundamental Limitations:
   • The authors demonstrate that no distributed optimization algorithm can be both optimal (in the absence of adversaries) and resilient to adversarial attacks without concessions. Specifically, adversaries can conceal their actions by altering their functions, rendering them indistinguishable from regular nodes, thereby coercing convergence to erroneous results.
2. Local Filtering Dynamics (LF):
   • A novel resilient algorithm is introduced. It allows each node to discard extreme values (highest and lowest) from its neighborhood, presumed to originate from adversarial nodes. This method relies on the graph being sufficiently connected — more precisely, robust.

• The conditions are:
  • For $F$-local adversaries, the graph must be $(2F+1)$-robust.
  • For $F$-total malicious adversaries, the condition relaxes to $(F+1, F+1)$-robustness.
• The LF Dynamics provably ensure that regular nodes reach consensus under these graph conditions.

1. Safety Guarantees:
   • The research guarantees that the final consensus state of non-adversarial nodes lies within the convex hull of their local objective function minimizers. The use of bounded subgradients and appropriately diminishing step-sizes forms the basis for this security against adversarial influence.
2. Graph-Theoretic Considerations:
   • Maximum $F$-local sets in the graph impact the performance of resilient optimization algorithms. Identifying these sets is complex (highlighted as NP-hard), but their structure can profoundly affect the system's distance-to-optimality in adversarial scenarios.

Implications and Future Directions

The implications of this work are twofold: theoretically deepening the understanding of optimization under adversarial conditions by stressing the interplay between graph topology and algorithm design and practically enhancing the robustness of distributed protocols in safety-critical systems. In numerous network applications, such as sensor networks, cyber-physical systems, and data fusion centers, these insights can drastically improve system resilience against failures or attacks.

Future work may explore refining the bounds of achievable performance in distributed algorithms, considering specific classes of function sets, and exploring richer adversarial models. Furthermore, developing efficient heuristics for identifying maximum $F$-local sets and characterizing their influence on optimization performance would bolster the applicability of the proposed methods.

Conclusion

This paper makes significant strides in understanding and enhancing the resilience of distributed optimization algorithms to adversarial behavior. By marrying algorithmic innovation with graph-theoretic insights, it lays down robust foundations for secure distributed computation in networked environments, paving the way for future research and application in resilient algorithm design.