Robustness via curvature regularization, and vice versa (1811.09716v1)

Abstract: State-of-the-art classifiers have been shown to be largely vulnerable to adversarial perturbations. One of the most effective strategies to improve robustness is adversarial training. In this paper, we investigate the effect of adversarial training on the geometry of the classification landscape and decision boundaries. We show in particular that adversarial training leads to a significant decrease in the curvature of the loss surface with respect to inputs, leading to a drastically more "linear" behaviour of the network. Using a locally quadratic approximation, we provide theoretical evidence on the existence of a strong relation between large robustness and small curvature. To further show the importance of reduced curvature for improving the robustness, we propose a new regularizer that directly minimizes curvature of the loss surface, and leads to adversarial robustness that is on par with adversarial training. Besides being a more efficient and principled alternative to adversarial training, the proposed regularizer confirms our claims on the importance of exhibiting quasi-linear behavior in the vicinity of data points in order to achieve robustness.

• The paper establishes that reducing curvature in the loss landscape leads to enhanced adversarial robustness.
• It employs a locally quadratic approximation to derive theoretical bounds and validates these findings on datasets like CIFAR-10 and SVHN.
• The study introduces CURE, a novel regularization approach that minimizes curvature to offer a more efficient alternative to traditional adversarial training.

Robustness via Curvature Regularization, and Vice Versa

The paper "Robustness via curvature regularization, and vice versa" presents an in-depth investigation of the geometric effects of adversarial training on deep neural networks, particularly focusing on the role of curvature in enhancing model robustness. The authors provide both theoretical insights and empirical validation to argue that reducing curvature in the loss landscape is crucial for achieving adversarial robustness, and they propose an alternative regularization strategy to mimic this effect.

Curvature and Robustness: Empirical Observations

The paper begins with a comprehensive analysis of the impact of adversarial training on the geometry of neural network decision boundaries. It is empirically demonstrated that adversarial training significantly reduces the curvature of the decision boundaries and the loss landscape with respect to the inputs. This reduction in curvature yields a more linear behavior of the network around data points, which is associated with increased robustness to adversarial perturbations. This finding challenges previous notions that attributed the vulnerability of classifiers to excessive linearity.

Theoretical Underpinning: Quadratic Approximation

The authors extend their empirical observations with a theoretical framework using a locally quadratic approximation of the loss surface. They establish that small curvature is conducive to large robustness, formulating upper and lower bounds on the robustness to adversarial perturbations based on curvature. The analysis reveals a direct relationship: as curvature decreases, robustness to adversarial attacks increases.

CURE: An Alternative to Adversarial Training

To further substantiate their claims, the authors introduce Curvature Regularization (CURE), a new regularization approach aimed at directly minimizing curvature. This involves penalizing large curvatures by adding a regularization term to the loss function, effectively encouraging a flattened decision boundary around data points. Evaluations on standard datasets such as CIFAR-10 and SVHN demonstrate that CURE achieves adversarial robustness comparable to that of adversarial training. This result highlights the potential of CURE as a more efficient alternative to traditional adversarial training, reducing computational overhead while maintaining competitive robustness.

Practical and Theoretical Implications

The implications of this work are twofold. Practically, the introduction of CURE offers a computationally attractive method for training robust models without the need for repeated adversarial examples generation, typical of adversarial training. Theoretically, the paper provides strong evidence that regulating curvature, and by extension the linearity of neural networks around data points, is a key factor in achieving robustness. This insight may drive future research on alternative mechanisms for enhancing robustness and could influence new directions in model architecture design.

Future Directions

The exploration of curvature in adversarial training opens several avenues for future work. One potential direction is the development of more sophisticated curvature regularizers that adaptively focus on critical regions of the loss landscape. Moreover, investigating the interplay between curvature and other architectural features such as network depth and topology could yield additional insights into robustness optimization. Lastly, extending the curvature framework to other machine learning domains, beyond image classification, might enhance model reliability across diverse applications.

In conclusion, this paper makes a compelling case for the relevance of curvature in understanding and improving the robustness of neural networks. By shifting focus towards curvature regularization, it challenges existing paradigms and sets the stage for further advancements in the field of adversarial machine learning.