A Boundary Tilting Persepective on the Phenomenon of Adversarial Examples (1608.07690v1)

Abstract: Deep neural networks have been shown to suffer from a surprising weakness: their classification outputs can be changed by small, non-random perturbations of their inputs. This adversarial example phenomenon has been explained as originating from deep networks being "too linear" (Goodfellow et al., 2014). We show here that the linear explanation of adversarial examples presents a number of limitations: the formal argument is not convincing, linear classifiers do not always suffer from the phenomenon, and when they do their adversarial examples are different from the ones affecting deep networks. We propose a new perspective on the phenomenon. We argue that adversarial examples exist when the classification boundary lies close to the submanifold of sampled data, and present a mathematical analysis of this new perspective in the linear case. We define the notion of adversarial strength and show that it can be reduced to the deviation angle between the classifier considered and the nearest centroid classifier. Then, we show that the adversarial strength can be made arbitrarily high independently of the classification performance due to a mechanism that we call boundary tilting. This result leads us to defining a new taxonomy of adversarial examples. Finally, we show that the adversarial strength observed in practice is directly dependent on the level of regularisation used and the strongest adversarial examples, symptomatic of overfitting, can be avoided by using a proper level of regularisation.

• The paper introduces a boundary tilting perspective that quantifies adversarial vulnerability using the deviation angle, challenging traditional linear explanations.
• The paper demonstrates that stronger regularization reduces boundary tilting and limits misclassification by mitigating overfitting in deep networks.
• The paper validates its insights through experiments on synthetic and MNIST datasets, emphasizing that overfitting, rather than dimensionality, underlies adversarial examples.

A Critique and Exploration of Adversarial Examples Through Boundary Tilting

The paper "A Boundary Tilting Perspective on the Phenomenon of Adversarial Examples" by Thomas Tanay and Lewis Griffin offers an in-depth analysis of adversarial examples within deep neural networks, proposing an alternative perspective centered around boundary tilting. This essay evaluates the key findings and implications of the paper, providing insights into the foundational concepts and mathematical underpinnings, while contextualizing its contributions within the field of adversarial machine learning.

Challenges and Limitations of Existing Explanations

The phenomenon of adversarial examples challenges the integrity of deep neural networks by illustrating that small perturbations can lead to significant misclassifications. Tanay and Griffin critique the linear explanation posited by Goodfellow et al., which attributes adversarial vulnerability to the high dimensionality and piecewise linear nature of deep network layers. They argue that this explanation is unconvincing, as it fails to universally predict the existence of such examples or their perceptual magnitude. Furthermore, they demonstrate that linear classifiers do not consistently exhibit adversarial susceptibility, highlighting the inadequacy of dimensionality alone as a causal factor.

Introducing the Boundary Tilting Perspective

Tanay and Griffin propose a boundary tilting perspective, suggesting that adversarial examples arise when classification boundaries closely align with the data submanifold. They employ a rigorous mathematical framework to explore this perspective within linear classification. A pivotal contribution is the introduction of the deviation angle—which quantifies the angle between the classifier's decision boundary and the optimal nearest centroid classifier—providing a nuanced measure of adversarial strength. This conceptualization allows for a more precise analysis of how adversarial examples manifest in both linear and potentially non-linear systems.

Impact of Regularization and Overfitting

An important practical insight from the paper is the role of regularization in modulating the adversarial strength of classifiers. The authors demonstrate that the use of higher regularization can mitigate overfitting, thus reducing the emergence of strong adversarial examples. Specifically, they show that as regularization decreases, the decision boundary becomes susceptible to tilting along directions of low variance, leading to overfitting and consequently, robust adversarial examples. This discovery has implications for training robust models, suggesting that careful tuning of regularization parameters is crucial in controlling adversarial behavior.

Empirical Analysis and Theoretical Implications

Through experiments on both synthetic and MNIST datasets, the authors empirically validate their theoretical assertions, showcasing how boundary tilting influences classifier robustness. Their findings prompt a reevaluation of adversarial phenomena, hinting that adversarial vulnerability could be a by-product of overfitting rather than inherent model linearity. This calls for an emphasis on developing regularization techniques that operate directly in pixel space rather than feature space, to prevent unintended adversarial susceptibilities caused by initial preprocessing transformations.

Speculative Future Directions

The paper by Tanay and Griffin opens avenues for future research in adversarial machine learning, particularly in exploring non-linear extensions of boundary tilting. Further investigations could aim to establish whether similar principles apply to more complex architectures like convolutional and recurrent neural networks, offering broader generalization of their results. Additionally, the introduction of new regularization paradigms or architectural adjustments that inherently resist boundary tilting could lead to models with improved adversarial resilience.

In conclusion, this paper enriches the discourse on adversarial examples by providing a compelling alternative perspective that challenges previous explanations and underscores the complexity of designing robust neural networks. Its contributions are pivotal in guiding both theoretical explorations and practical advancements in the continuous effort to enhance the reliability of machine learning models amidst adversarial settings.