SGD at the Edge of Stability: The Stochastic Sharpness Gap

Abstract

When training neural networks with full-batch gradient descent (GD) and step size $\eta$, the largest eigenvalue of the Hessian -- the sharpness $S(\boldsymbol{\theta})$ -- rises to $2/\eta$ 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(\boldsymbol{\theta})\leq 2/\eta$. For mini-batch stochastic gradient descent (SGD), the sharpness stabilizes below $2/\eta$, 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 $2/\eta$. 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: $\Delta S = \eta \beta \sigma_{\boldsymbol{u}}^{2}/(4\alpha)$, where $\alpha$ is the progressive sharpening rate, $\beta$ is the self-stabilization strength, and $\sigma_{ \boldsymbol{u}}^{2}$ 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.