The Dimpled Manifold Model of Adversarial Examples in Machine Learning (2106.10151v2)

Abstract: The extreme fragility of deep neural networks, when presented with tiny perturbations in their inputs, was independently discovered by several research groups in 2013. However, despite enormous effort, these adversarial examples remained a counterintuitive phenomenon with no simple testable explanation. In this paper, we introduce a new conceptual framework for how the decision boundary between classes evolves during training, which we call the {\em Dimpled Manifold Model}. In particular, we demonstrate that training is divided into two distinct phases. The first phase is a (typically fast) clinging process in which the initially randomly oriented decision boundary gets very close to the low dimensional image manifold, which contains all the training examples. Next, there is a (typically slow) dimpling phase which creates shallow bulges in the decision boundary that move it to the correct side of the training examples. This framework provides a simple explanation for why adversarial examples exist, why their perturbations have such tiny norms, and why they look like random noise rather than like the target class. This explanation is also used to show that a network that was adversarially trained with incorrectly labeled images might still correctly classify most test images, and to show that the main effect of adversarial training is just to deepen the generated dimples in the decision boundary. Finally, we discuss and demonstrate the very different properties of on-manifold and off-manifold adversarial perturbations. We describe the results of numerous experiments which strongly support this new model, using both low dimensional synthetic datasets and high dimensional natural datasets.

• The paper posits that adversarial examples stem from the interplay between initial decision boundary clinging and subsequent natural dimpling in deep neural networks.
• The study employs both synthetic and real image experiments to demonstrate that adversarial perturbations occur off-manifold, exploiting high-gradient regions near the training data.
• The Dimpled Manifold Model offers a new geometric lens to understand adversarial training, illuminating trade-offs between robustness and accuracy in deep learning.

Understanding Adversarial Examples through the Lens of the Dimpled Manifold Model

Introduction to Adversarial Examples

Adversarial examples pose a significant challenge to the reliability of machine learning systems, particularly deep neural networks. These examples are carefully crafted inputs that cause a model to make incorrect predictions. Despite substantial research, a coherent explanation that is widely accepted and testable for why adversarial examples are effective has remained elusive.

The Dimpled Manifold Hypothesis

A new conceptual framework, termed the Dimpled Manifold Model (DMM), seeks to unravel the mystery behind adversarial examples. According to the DMM, training a deep neural network (DNN) transpires in two stages. The first, known as the clinging phase, occurs quickly and involves the decision boundary of the DNN aligning closely with the manifold representing the data. Subsequently, a slower dimpling phase ensues, during which gentle undulations are formed in the decision boundary to position it correctly with respect to the provided examples. This model suggests that the decision boundaries of networks essentially cling to a low-dimensional manifold that represents the natural images used for training.

The paper posits that adversarial examples exist because of these clinging and dimpling processes. These adversarial perturbations exploit the excessive closeness of the decision boundary to this manifold and the high gradients developed by the networks in the vicinity of their training data.

Empirical Evidence and Experiments

Support for the Dimpled Manifold Model comes from various experiments conducted by the authors. Using both synthetic and natural images as datasets, the properties of adversarial perturbations were examined. It was discovered that adversarial examples tend to be off-manifold perturbations, indicating that they exploit the high-dimensional space that the decision boundary clings to but which does not contain any real images. Essentially, they create pseudo-images that are misclassified by the network despite not resembling any natural images.

Moreover, the DMM also provides a coherent understanding of adversarial training. This process, which aims to enhance the robustness of models by training them with adversarial examples, effectively deepens the decision boundary’s dimples, according to the model. While this makes a network harder to attack in the adversarial sense, it also can lead to a loss in accuracy for standard test images.

Implications and Future Work

The DMM offers an elegant geometric interpretation of adversarial examples and training. It indicates that adversarial perturbations hover very close to the natural image manifold, explaining their small norms and why they don't necessarily resemble the target class visually.

Future research may delve into the aspect of transferability, which refers to the phenomenon where adversarial examples that fool one network can often deceive another, even if they have different architectures or training data. The newfound understanding from the manifold perspective could help clarify why this occurs.

The Dimpled Manifold Model contributes a new layer to our understanding of adversarial examples, moving the field closer to developing robust models that can be trusted in real-world applications. The insights provided by the DMM have implications not only for the ongoing development of defensive techniques against adversarial attacks but also for the foundational principles of how deep learning models perceive and interpret data.