Practical Validity Conditions for Byzantine-Tolerant Federated Learning

2026-05-15

AI Summary

This work addresses the limited robustness of existing convex validity conditions in Byzantine-tolerant federated learning under high-dimensional settings and their incompatibility with mainstream aggregation rules. To overcome these limitations, the paper introduces a new geometric validity condition based on the Minimum Enclosing Ball (MEB) and its multiplicatively relaxed variant, c-MEB, thereby establishing a Byzantine-robust aggregation framework suitable for practical systems. Under the standard assumption that honest clients form a majority (n > 2t), this condition is realizable via the optimal MinMax-MEB aggregation rule and provides explicit c-MEB guarantees for widely used aggregators such as the geometric median, medoid, and minimum-diameter averaging. Theoretical analysis demonstrates that c-MEB validity holds whenever c < √2, thus systematically bridging foundational concepts in distributed computing and robust aggregation.

Abstract

Robust aggregation is the core operation in Byzantine-tolerant federated learning. To ensure the quality of aggregation independently of data distribution or attacks, validity conditions are needed. They provide geometric guarantees of where the output of the aggregation must lie. The widespread convex validity requires the output to lie in the convex hull of the honest vectors. Although this guarantee is strong in theory, it is poorly suited to modern federated learning systems, as it has dimension-dependent resilience and excludes many practical aggregation rules. We introduce the minimum enclosing ball (MEB) validity condition for robust aggregation, as well as its multiplicative relaxation, $c$-MEB validity, where $c$ is a constant. We show that exact MEB validity still suffers from limited resilience, while relaxed $c$-MEB validity is achievable if a majority of clients is honest, i.e. $n>2t$. We give an optimal MinMax-MEB rule for the relaxed condition with the bound $c<\sqrt{2}$ and prove explicit relaxed-MEB guarantees for standard aggregators including minimum-diameter averaging, medoid and geometric median. Finally, we relate MEB validity to convex, relaxed-convex and box validity studied in prior literature, thus providing a systematic map of geometric validity conditions for Byzantine-robust aggregation. Our results show that relaxed MEB validity connects validity conditions in distributed computing and Byzantine-tolerant aggregation rules, and offers a practical alternative to convex validity.

Problem

• Byzantine-tolerant federated learning
• robust aggregation
• validity conditions
• minimum enclosing ball
• convex hull

Innovation

• Byzantine-tolerant federated learning
• robust aggregation
• minimum enclosing ball (MEB)
• geometric validity
• MinMax-MEB