Curvature-Aware Model Aggregation for Decentralized Federated Learning via Parallel Transport on Discrete Manifolds

Decentralized federated learning (DFL) allows a group of distributed users to train a shared model without relying on a central server. Most existing decentralized aggregation methods, however, still treat the communication network as a flat Euclidean space and overlook the geometric differences that may arise in the model parameter landscape. To better capture these differences, we develop a curvature-aware decentralized federated learning framework, termed CA-DFL, which introduces geometric structure into the aggregation process. The main idea is to view the communication graph from a discrete Riemannian perspective. According to the divergence between neighboring model parameters, each communication edge is associated with one of three constant-curvature geometries: the sphere S², the Euclidean plane R², or the pseudosphere P S². This classification allows model updates to be transferred between tangent spaces through closed-form parallel transport operators, so that the aggregation step respects the local geometry rather than mixing parameters in a purely Euclidean manner. Building on this, we further design a tensor-field-based geometric weighting scheme to assign adaptive aggregation weights using manifold inner products. This weighting scheme only borrows the idea of adaptive neighbor weighting, while the core operation remains curvature-aware transport and aggregation on manifolds. A subtree partition scheme is also introduced to encourage parameter sharing while reducing communication overhead. We establish convergence guarantees for the proposed method under standard assumptions. Experiments on MNIST and CIFAR-10 under ring, exponential, and random communication topologies demonstrate that CA-DFL achieves accuracy competitive with standard decentralized methods while providing adaptive geometric aggregation. A curvature-threshold sensitivity analysis confirms the robustness of the framework to hyperparameter choice.