When Flatness Does (Not) Guarantee Adversarial Robustness
This work provides a nuanced understanding of flatness in adversarial robustness, challenging simplified views and offering insights for researchers in machine learning security.
The paper tackles the hypothesis that flat minima in neural networks guarantee adversarial robustness, showing that flatness only ensures local robustness and not global robustness, with empirical validation across architectures and datasets revealing adversarial examples in flat regions where models are confidently wrong.
Despite their empirical success, neural networks remain vulnerable to small, adversarial perturbations. A longstanding hypothesis suggests that flat minima, regions of low curvature in the loss landscape, offer increased robustness. While intuitive, this connection has remained largely informal and incomplete. By rigorously formalizing the relationship, we show this intuition is only partially correct: flatness implies local but not global adversarial robustness. To arrive at this result, we first derive a closed-form expression for relative flatness in the penultimate layer, and then show we can use this to constrain the variation of the loss in input space. This allows us to formally analyze the adversarial robustness of the entire network. We then show that to maintain robustness beyond a local neighborhood, the loss needs to curve sharply away from the data manifold. We validate our theoretical predictions empirically across architectures and datasets, uncovering the geometric structure that governs adversarial vulnerability, and linking flatness to model confidence: adversarial examples often lie in large, flat regions where the model is confidently wrong. Our results challenge simplified views of flatness and provide a nuanced understanding of its role in robustness.