When Aggregation Hides Directional Structures in Federated Learning

Federated learning (FL) relies on aggregating decentralized client updates into a global update, yet this many-to-one aggregation can obscure important directional structures contained in individual client updates. In particular, distinct client directional configurations may yield similar aggregate updates because opposing directions partially cancel each other, making the underlying directional relationships less observable from the aggregate representation alone. We characterize this phenomenon as Hidden Directional Opposition (HDO), which captures client-level directional opposition that exists before aggregation but becomes partially unobservable after aggregation. To analyze such hidden structures in a shared coordinate system, we construct a geometry-aware tangent-space directional representation that reduces endpoint-dependent interference while retaining relative directional displacement. We further introduce a direction-preserving adaptive optimization mechanism that adapts update magnitudes without introducing additional directional rotation. Experiments on CIFAR-10 and CIFAR-100 show that HDO consistently emerges under heterogeneous federated training, becomes more pronounced with stronger client heterogeneity, and exhibits distinct temporal and layer-wise patterns. These results highlight that federated aggregation determines not only the global update but also which client-level directional structures remain observable at the server.