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arXiv · 2310.01770

A simple connection from loss flatness to compressed neural representations

Shirui Chen·Stefano Recanatesi·Eric Shea-Brown
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Abstract

Despite extensive study, the significance of sharpness -- the trace of the loss Hessian at local minima -- remains unclear. We investigate an alternative perspective: how sharpness relates to the geometric structure of neural representations, specifically representation compression, defined as how strongly neural activations concentrate under local input perturbations. We introduce three measures -- Local Volumetric Ratio (LVR), Maximum Local Sensitivity (MLS), and Local Dimensionality -- and derive upper bounds showing these are mathematically constrained by sharpness: flatter minima necessarily limit compression. We extend these bounds to reparametrization-invariant sharpness and introduce network-wide variants (NMLS, NVR) that provide tighter, more stable bounds than prior single-layer analyses. Empirically, we validate consistent positive correlations across feedforward, convolutional, and transformer architectures. Our results suggest that sharpness fundamentally quantifies representation compression, offering a principled resolution to contradictory findings on the sharpness-generalization relationship.

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Shirui Chen, Stefano Recanatesi, Eric Shea-Brown. 2023-10-03. A simple connection from loss flatness to compressed neural representations. https://arxiv.org/abs/2310.01770

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