Riemannian Sharpness Explains SGD's Bias Toward Flat Minima

This study introduces Riemannian sharpness, a reparametrization-invariant measure of flatness grounded in Fisher Information Matrix geometry. It proves SGD's stationary distribution concentrates at Riemannian-flat minima and links this geometric bias to generalization via a PAC-Bayes bound. Experiments on MNIST and CIFAR-10 show Riemannian sharpness better tracks generalization than Euclidean sharpness, with scaling consistent with theory.