ByzaSpline-Fed: Gram-Weighted FunctionalAggregation for Byzantine-Robust FederatedKolmogorov–Arnold Networks in Industrial IoT

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Federated learning enables industrial sites to train shared condition-monitoring models without exchangingraw vibration, acoustic, or sensor data, but its openness admits Byzantine clients that submit corrupted updates. Existingrobust aggregation rules—Krum, trimmed mean, median, and their variants—operate in Euclidean parameter space andimplicitly assume that a malicious update must be far from honest updates to do harm. We show that this assumptionfails for models with Kolmogorov–Arnold (KAN) spline heads: an adversary can hold the spline coefficients close tothe honest consensus in the L2 sense while distorting the functional shape the spline encodes, an attack we termfunctional spline-poisoning. We propose ByzaSpline-Fed, a Byzantine-robust aggregation rule that scores clients byGram-weighted functional B-spline distance on a consensus knot grid and rejects functional outliers. The rule reducesexactly to spline-aware federated averaging under honest participation, admits a breakdown bound of f < K/2 under thestandard honest-majority assumption, and supports optional differential privacy. We evaluate across four industrialbenchmarks spanning classification, unsupervised acoustic anomaly detection, and remaining-useful-life regression—CWRU, Paderborn, MIMII, and C-MAPSS—reporting task-correct metrics under clean and adversarial conditions.Under thirty percent functional poisoning, Euclidean defenses lose up to thirty-four accuracy points while ByzaSplineFed remains within roughly three points of its clean performance. All reported results are averaged over eightindependent random seeds, and the improvements over the strongest competing defense are statistically significantunder the Wilcoxon signed-rank test (p < 0.05).