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arXiv · 2609.38540

When a Flatness Proxy Is Not a Function: Robustness Certificates and Training Interventions

Vicente Opazo·Jose Calatayud-Mateu·Cristobal Rojas·Cristian Buc Calderon
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Abstract

A valid curvature upper bound need not justify either a robustness certificate or an intervention on an intrinsic predictor property. We demonstrate this distinction for a last-layer relative-flatness proxy used in both settings. First, empirical-risk stationarity does not eliminate pointwise first-order loss terms: at a finite global empirical-risk minimum, the retained certificate expression underestimates a loss increase by over $210\times$. We derive a globally valid, gauge-invariant feature-space repair. Second, common-row softmax shifts preserve predictions and the exact contraction while making the proxy unbounded. Even standard reference-class choices double it on average relative to the centered representation. For a single fixed-feature example with at least three classes, scalar retuning generically cannot align the induced probability updates. Row centering gives the orbit-minimized bound and restores value and full-model gradient invariance under this symmetry. Across 45 paired one-step tests on algorithmic and image models, amplified shifts separate raw-regularized predictors while quotient-regularized predictors remain aligned. Long-horizon CIFAR-10 experiments show substantial, reversible suppression of generalization, while evidence for selective delay after memorization is less consistent. Together, these results show that validity as a curvature upper bound does not by itself justify either inversion into a robustness certificate or differentiation into an intrinsic training intervention.

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Vicente Opazo, Jose Calatayud-Mateu, Cristobal Rojas, Cristian Buc Calderon. 2026-09-29. When a Flatness Proxy Is Not a Function: Robustness Certificates and Training Interventions. https://arxiv.org/abs/2609.38540

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