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<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.
Federated learning distributes data among n clients, making it vulnerable to malicious attacks and data heterogeneity, which together pose challenges for robust learning. To tackle this issue, centered clipping and Huber aggregators have been exploited for Byzantine robustness. In this paper, we first demonstrate their equivalence via convex conjugate theory, and show that they can yield biased solutions in the presence of outliers, leading to failure under high data heterogeneity and a substantial fraction of outliers. Next, we propose a new robust aggregation rule that utilizes the truncated-quadratic (TQ) loss, effectively mitigating the biases of existing methods, such as centered clipping and Huber aggregators. We show that our aggregator achieves order-optimal Byzantine-robust learning under nonconvex loss functions and heterogeneous data, ultimately enhancing the reliability of federated learning systems. Additionally, we provide a robust deviation estimation strategy for TQ, demonstrating its effectiveness. Furthermore, we show that TQ maintains robustness even when only an estimate of the number of Byzantine clients is available. Finally, experimental results on MNIST, Fashion-MNIST, and CIFAR-10, indicate that our aggregator provides better robustness performance than the competing techniques.
Federated learning enables multiple parties to train a shared model without centralizing raw data with the help of an aggregator, but introduces integrity risks once participants or infrastructure are not fully trustworthy. Two requirements are particularly important: robustness to poisoned or Byzantine client updates, and verifiability of the aggregator so that clients or third parties can audit the reported aggregation without learning individual updates. Existing work has largely treated these goals separately, and efficient public verifiability for robust, outlier-excluding aggregation remains limited. We present a verifiable federated learning protocol that makes a robust aggregation pipeline publicly auditable. Our design combines cryptographic commitments with non-interactive zero-knowledge proofs to certify both (i) cosine-similarity-based outlier exclusion and (ii) aggregation over the selected set, without revealing individual client updates to verifiers. In experiments under representative poisoning attacks, our method maintains high accuracy, with an average accuracy loss below 4% across the evaluated configurations, while keeping verification overhead practical: proof artifacts can be generated and verified within minutes at the scale studied. In summary, our results show that robust outlier exclusion and public verifiability can be jointly achieved in a federated learning setting.
Most existing Byzantine-robust federated learning (FL) methods suffer from slow and unstable convergence. Moreover, when handling a substantial proportion of colluded malicious clients, achieving robustness typically entails compromising model utility. To address these issues, this work introduces FedIDM, which employs distribution matching to construct trustworthy condensed data for identifying and filtering abnormal clients. FedIDM consists of two main components: (1) attack-tolerant condensed data generation, and (2) robust aggregation with negative contribution-based rejection. These components exclude local updates that (1) deviate from the update direction derived from condensed data, or (2) cause a significant loss on the condensed dataset. Comprehensive evaluations on three benchmark datasets demonstrate that FedIDM achieves fast and stable convergence while maintaining acceptable model utility, under multiple state-of-the-art Byzantine attacks involving a large number of malicious clients.