Understanding the Subspace Stabilization of the Hessian and Gradient Covariance Matrix

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.

Full-batch gradient descent on neural networks drives the largest Hessian eigenvalue to the threshold $2/η$, where $η$ is the learning rate. This phenomenon, the Edge of Stability, has resisted a unified explanation: existing accounts establish self-regulation near the edge but do not explain why the trajectory is forced toward $2/η$ from arbitrary initialization. We introduce the edge coupling, a functional on consecutive iterate pairs whose coefficient is uniquely fixed by the gradient-descent update. Differencing its criticality condition yields a step recurrence with stability boundary $2/η$, and a second-order expansion yields a loss-change formula whose telescoping sum forces curvature toward $2/η$. The two formulas involve different Hessian averages, but the mean value theorem localizes each to the true Hessian at an interior point of the step segment, yielding exact forcing of the Hessian eigenvalue with no gap. Setting both gradients of the edge coupling to zero classifies fixed points and period-two orbits; near a fixed point, the problem reduces to a function of the half-amplitude alone, which determines which directions support period-two orbits and on which side of the critical learning rate they appear.

The flatness hypothesis suggests that flatness of the loss landscape, as measured by the eigenvalues of the loss Hessian, correlates with better neural network generalization. While various algorithms reduce these eigenvalues, most focus on procedural design, leaving it unclear how data distributions and NN parameters structurally determine directions toward flat minima. Characterizing these directions analytically is generally intractable. To overcome this mathematical difficulty, recent studies derived the Wolkowicz-Styan (WS) upper bound on the maximum eigenvalue of the cross-entropy loss Hessian in three-layer NNs. Although this upper bound is differentiable, its gradient was not derived. Therefore, we analytically derive the gradient of the WS upper bound to characterize directions leading to flat minima. Based on this, we propose Hessian Spectral Range (HSR) Regularization, which updates parameters along the steepest descent direction of the WS bound. Experiments demonstrate that HSR Regularization narrows the Hessian eigenvalue spectrum, avoids sharp minima and saddle points, and promotes convergence to flat minima. Although the applicability of this method is currently limited to cross-entropy loss and three-layer architectures, to the best of the authors' knowledge, this is the first study to report a closed-form gradient that promotes convergence to flat minima without numerical approximations. Therefore, the theoretical analysis of this gradient is expected to contribute to the further development of NNs.