On the Relationship Between Adversarial Robustness and Decision Region in Deep Neural Network (2207.03400v1)

Abstract: In general, Deep Neural Networks (DNNs) are evaluated by the generalization performance measured on unseen data excluded from the training phase. Along with the development of DNNs, the generalization performance converges to the state-of-the-art and it becomes difficult to evaluate DNNs solely based on this metric. The robustness against adversarial attack has been used as an additional metric to evaluate DNNs by measuring their vulnerability. However, few studies have been performed to analyze the adversarial robustness in terms of the geometry in DNNs. In this work, we perform an empirical study to analyze the internal properties of DNNs that affect model robustness under adversarial attacks. In particular, we propose the novel concept of the Populated Region Set (PRS), where training samples are populated more frequently, to represent the internal properties of DNNs in a practical setting. From systematic experiments with the proposed concept, we provide empirical evidence to validate that a low PRS ratio has a strong relationship with the adversarial robustness of DNNs. We also devise PRS regularizer leveraging the characteristics of PRS to improve the adversarial robustness without adversarial training.

Exploring the Geometric Properties of Deep Neural Networks for Adversarial Robustness

The Concept of Populated Region Set (PRS)

Deep Neural Networks (DNNs) have seen impressive gains in various domains, yet their vulnerability to adversarial attacks poses significant risks, especially in sensitive applications. Recent efforts to improve the adversarial robustness of DNNs have prompted a closer examination of the networks' internal geometrical properties. A novel empirical paper introduces the concept of the Populated Region Set (PRS), which fundamentally represents the internal geometrical configuration of DNNs by focusing on the decision regions populated with training samples. This new viewpoint suggests a strong linkage between the PRS ratio (the ratio of populated regions relative to the training samples) and a model's robustness against adversarial attacks.

PRS and Adversarial Robustness

Through systematic experimentation, the paper demonstrates that models with a lower PRS ratio exhibit enhanced adversarial robustness. Models sharing similar generalization performance diverge in their resilience to adversarial perturbations based on their PRS ratio, where a lower value correlates with heightened robustness. The investigation further reveals that within these robust models, the final layer parameters mapping penultimate features to logits display a higher degree of cosine similarity, suggesting a more parallel configuration that might underpin the observed robustness.

Implications for Test Data

The relationship between PRS and model behavior on unseen test data was also scrutinized. Models with a lower PRS ratio were found to encompass a larger portion of test samples within their trained decision regions, manifesting as higher robustness for these included samples. This implies that models with a streamlined, efficiently populated decision region architecture not only perform well on seen data but also generalize better in the presence of adversarial interventions.

Formulating a PRS-based Regularizer for Robust Learning

Capitalizing on the insights into the PRS concept, the authors propose a PRS regularizer to enhance model robustness without necessitating adversarial training. This novel regularizer encourages models to minimize the PRS ratio by fostering sample populations within major decision regions and ensuring proximity to the major region vector, essentially incentivizing geometric configurations akin to those found in inherently robust models. Empirical validations across different models and datasets confirm that the PRS regularizer effectively balances robustness improvements with the maintenance of generalization performance.

Theoretical and Practical Contributions

From a theoretical standpoint, this research advances our understanding of the geometric properties influencing DNN robustness. It provides an innovative framework to interpret adversarial resilience through the lens of decision regions, a departure from the prevailing focus on decision boundaries. Practically, the PRS regularizer presents a novel methodology for enhancing robustness efficiently, potentially reducing the reliance on computationally expensive adversarial training techniques.

Future Directions

This work opens several avenues for future investigation. The relationship between PRS characteristics and different adversarial attack strategies merits further exploration to understand the bounds of this geometric robustness framework. Additionally, extending the PRS concept to other network architectures, including those with non-linear activation functions beyond ReLU, could broaden the applicability of these insights.

Conclusion

In sum, this paper contributes a significant advance in our understanding of the geometric underpinnings of adversarial robustness in DNNs. By elucidating the role of populated decision regions and introducing a regularizer to leverage this relationship, it offers both theoretical insights and practical tools for enhancing model resilience against adversarial perturbations.