Stability and Generalization of Decentralized Riemannian Stochastic Gradient Descent

Generalization bounds for Euclidean decentralized SGD rely on the nonexpansiveness of averaging. Manifold projection can amplify perturbations caused by replacing a training sample, complicating the extension of these guarantees to manifold constraints. We analyze stochastic decentralized projected Riemannian gradient descent (DPRGD) on compact smooth embedded manifolds without boundary. By bounding projection expansion through consensus error, we derive a stability-based generalization bound whose additional geometric factor is controlled by the sum of squared step sizes. Motivated by this analysis, we propose DPRGD-ATC, an adapt-then-combine variant that averages locally updated models before projection. Under the same assumptions, step sizes, and communication budget, we prove a tighter generalization bound by reducing the terms associated with consensus error. A local curvature construction further shows that DPRGD-ATC can have strictly lower sensitivity to a single-sample replacement. We also establish empirical stationarity guarantees for both methods and compare their network-dependent error terms. Experiments on synthetic PCA across three geometries and on Fashion-MNIST with up to 32 agents illustrate these geometric effects.