Mini-batch Noise Lowers Sharpness via Dominant-Subspace Fluctuations

When training neural networks with full-batch gradient descent (GD) and step size $η$, the largest eigenvalue of the Hessian -- the sharpness $S(\boldsymbolθ)$ -- rises to $2/η$ and hovers there, a phenomenon termed the Edge of Stability (EoS). \citet{damian2023selfstab} showed that this behavior is explained by a self-stabilization mechanism driven by third-order structure of the loss, and that GD implicitly follows projected gradient descent (PGD) on the constraint $S(\boldsymbolθ)\leq 2/η$. For mini-batch stochastic gradient descent (SGD), the sharpness stabilizes below $2/η$, with the gap widening as the batch size decreases; yet no theoretical explanation exists for this suppression. We introduce stochastic self-stabilization, extending the self-stabilization framework to SGD. Our key insight is that gradient noise injects variance into the oscillatory dynamics along the top Hessian eigenvector, strengthening the cubic sharpness-reducing force and shifting the equilibrium below $2/η$. Following the approach of \citet{damian2023selfstab}, we define stochastic predicted dynamics relative to a moving projected gradient descent trajectory and prove a stochastic coupling theorem that bounds the deviation of SGD from these predictions. We derive a closed-form equilibrium sharpness gap: $ΔS = ηβσ_{\boldsymbol{u}}^{2}/(4α)$, where $α$ is the progressive sharpening rate, $β$ is the self-stabilization strength, and $σ_{ \boldsymbol{u}}^{2}$ is the gradient noise variance projected onto the top eigenvector. This formula predicts that smaller batch sizes yield flatter solutions and recovers GD when the batch equals the full dataset.

Deep neural network training often exhibits highly anisotropic loss geometry, where a few sharp dominant Hessian directions coexist with a large flatter bulk. Gradients tend to align disproportionately with these dominant directions, although stable progress often requires movement through flatter bulk directions. Estimating the dominant subspace is therefore useful but costly with direct Hessian-based methods. We show that standard Local SGD exposes this geometry through worker disagreement. We theoretically show that the worker-average gap covariance is shaped by stochastic-gradient noise and Hessian curvature, causing workers to disagree along sharp, curvature-sensitive directions. Thus, worker-average gaps provide a cheap Hessian-free estimator of the dominant subspace. Experiments on MLPs, CNNs, and Transformers show that subspaces formed by worker-average gaps capture a substantial fraction of the gradient component lying in the dominant Hessian eigenspace.

A widely held intuition in deep learning is that stochastic gradient descent (SGD) implicitly favors flat minima and that flat minima generalize better, but standard Euclidean measures of flatness such as the trace or maximum eigenvalue of the loss Hessian are not invariant under reparametrizations that preserve the network function, which undermines the theoretical foundations of this narrative. In this study we resolve this issue by grounding flatness in the Riemannian geometry of the statistical manifold induced by the Fisher Information Matrix (FIM). We define Riemannian sharpness mathematically and prove that it is invariant under smooth, function-preserving reparametrizations, which directly addresses the critique of Dinh et al. in the paper ``Sharp minima can generalize for deep nets''.We note that this invariance is a property of the true FIM; the diagonal empirical estimator used in practice (and in all experiments below) inherits invariance only approximately, and exact invariance under arbitrary reparametrizations would require structured estimators such as K-FAC. We formalize the gradient noise of mini-batch SGD as having a covariance structure proportional to the FIM, derive the stationary distribution of the resulting stochastic differential equation, and then show that the probability mass is exponentially concentrated at Riemannian-flat minima. A PAC-Bayes generalization bound controlled explicitly by SR formally links this geometric bias to test performance. Our experiments on MNIST and CIFAR-10 confirm that SR reliably tracks generalization in ways that Euclidean sharpness does not, and that its scaling with $η/B$ matches the theoretical predictions. Together these results provide a rigorous, reparametrization-invariant account of why flat minima generalize.