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The extreme fragility of deep neural networks, when presented with tinyperturbations in their inputs, was independently discovered by several researchgroups in 2013. However, despite enormous effort, these adversarial examplesremained a counterintuitive phenomenon with no simple testable explanation. Inthis paper, we introduce a new conceptual framework for how the decisionboundary between classes evolves during training, which we call the {\emDimpled Manifold Model}. In particular, we demonstrate that training is dividedinto two distinct phases. The first phase is a (typically fast) clingingprocess in which the initially randomly oriented decision boundary gets veryclose to the low dimensional image manifold, which contains all the trainingexamples. Next, there is a (typically slow) dimpling phase which createsshallow bulges in the decision boundary that move it to the correct side of thetraining examples. This framework provides a simple explanation for whyadversarial examples exist, why their perturbations have such tiny norms, andwhy they look like random noise rather than like the target class. Thisexplanation is also used to show that a network that was adversarially trainedwith incorrectly labeled images might still correctly classify most testimages, and to show that the main effect of adversarial training is just todeepen the generated dimples in the decision boundary. Finally, we discuss anddemonstrate the very different properties of on-manifold and off-manifoldadversarial perturbations. We describe the results of numerous experimentswhich strongly support this new model, using both low dimensional syntheticdatasets and high dimensional natural datasets.

The Dimpled Manifold Model of Adversarial Examples in Machine Learning