Explaining f-Divergence-Based Regularization via Local Curvature and Sharpness-Aware Minimization

• URL: http://arxiv.org/abs/2609.09367v1
• Date: Tue, 08 Sep 2026 19:01:05 GMT
• Title: Explaining f-Divergence-Based Regularization via Local Curvature and Sharpness-Aware Minimization
• Abstract summary: Divergence-based regularization and Sharpness-Aware Minimization (SAM) are two prominent approaches for improving generalization in deep learning.<n>We show that the two methods are locally consistent under parameter-space perturbations.<n>We also show that divergence-based regularization is defined through transformations of the input.
• Score: 7.792020418343022
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Divergence-based regularization and Sharpness-Aware Minimization (SAM) are two prominent approaches for improving generalization in deep learning, both motivated by robustness to perturbations. However, their relationship has remained largely unexplored. Building on classical second-order expansions of $f$-divergences, we show that the two methods are locally consistent under parameter-space perturbations: both induce curvature-sensitive penalties, with divergence regularization yielding a Fisher-weighted quadratic form and SAM penalizing sharpness through the dominant Hessian eigenvalue. For negative log-likelihood objectives with exponential-family output distributions, this correspondence becomes especially transparent, since the Fisher and Gauss-Newton matrices coincide. We further show that the same local geometric perspective extends to input-space perturbations, where divergence-based regularization is defined through transformations of the input. In this setting, the regularizer induces a pullback quadratic form on the input space, providing a more general perturbation framework than standard SAM while preserving the same local sensitivity interpretation. To validate the analysis empirically, we use the asymmetric $α$-skew Jensen-Shannon divergence (JSD) family as a controlled testbed. Its local curvature coefficient scales as $α(1-α)$ and is maximized at the symmetric point $α=\tfrac12$, which recovers the standard JSD. Loss-landscape visualizations in the input-perturbation regime show that stronger induced curvature penalization is associated with flatter local minima. Experiments on four benchmark datasets further demonstrate that both accuracy and negative log-likelihood are consistently best near this regime of maximal curvature penalization.

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