Study Reveals Superlinear Relationship Between SGD Noise and Loss Landscape Curvature in Deep Neural Networks
Researchers have derived a more general mathematical relationship between stochastic gradient descent (SGD) noise covariance and the curvature of the loss landscape in deep neural networks, challenging a widely used simplifying assumption. Prior work commonly assumed the SGD noise covariance matrix is directly proportional to the Hessian, an equivalence that the authors show holds only under restrictive conditions typically violated in practice. The findings provide a unified, empirically supported characterization of how SGD noise relates to curvature, with implications for understanding why SGD tends to find flat minima.
A preprint posted to arXiv presents theoretical and empirical evidence that the relationship between SGD noise covariance (C) and the loss landscape Hessian (H) in deep neural networks is superlinear rather than simply proportional. The study challenges a common assumption in the field—that the Fisher Information Matrix equals the Hessian for negative log-likelihood losses—showing this equivalence breaks down under conditions typical of modern deep learning. Leveraging a recently introduced framework called Activity–Weight Duality, the authors derive that C is proportional to the expected squared per-sample Hessian, meaning C and H commute approximately but do not coincide exactly. Empirically, the diagonal elements of C and H in fully connected layers follow per-layer power laws of the form C_ii ∝ H_ii^γ, with layer-dependent exponents bounded between 1 and 2. These results were validated across multiple datasets, architectures, and loss functions, lending broad support to the proposed characterization. The work has potential consequences for theoretical analyses of SGD's implicit bias toward flat minima and for practical algorithm design in deep learning.
What's missing
The study focuses exclusively on fully connected layers; it is unclear whether the identified power-law relationships and bounds extend to convolutional, attention, or other layer types common in modern architectures. As a preprint, the work has not yet undergone formal peer review.
What different sources said
- arXiv cs.LGCenter
On the Superlinear Relationship between SGD Noise Covariance and Loss Landscape Curvature
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