Are adversarial examples inevitable? (1809.02104v3)

Abstract: A wide range of defenses have been proposed to harden neural networks against adversarial attacks. However, a pattern has emerged in which the majority of adversarial defenses are quickly broken by new attacks. Given the lack of success at generating robust defenses, we are led to ask a fundamental question: Are adversarial attacks inevitable? This paper analyzes adversarial examples from a theoretical perspective, and identifies fundamental bounds on the susceptibility of a classifier to adversarial attacks. We show that, for certain classes of problems, adversarial examples are inescapable. Using experiments, we explore the implications of theoretical guarantees for real-world problems and discuss how factors such as dimensionality and image complexity limit a classifier's robustness against adversarial examples.

• The paper establishes that adversarial examples are inevitable in high-dimensional spaces using isoperimetric inequalities on spheres and hypercubes.
• The paper demonstrates that the density of class distributions and data complexity critically increase the susceptibility of classifiers to adversarial attacks.
• The paper evaluates adversarial vulnerability under various norms and supports its theory with empirical insights from datasets such as MNIST and CIFAR-10.

Analysis of the Inevitability of Adversarial Examples in Neural Networks

The proliferation of adversarial attacks on neural networks has sparked significant interest in determining the robustness and susceptibilities of classifiers. Despite various efforts in developing adversarial defenses, their effectiveness has remained short-lived due to evolving attack strategies. This research paper, "Are adversarial examples inevitable?", investigates the theoretical underpinnings of adversarial examples and provides fundamental bounds on the susceptibility of classifiers to such attacks.

Theoretical Framework and Key Findings

The paper's core premise is to examine whether adversarial examples—slight perturbations introduced to inputs causing misclassification—are unavoidable across various classification challenges. By incorporating sophisticated theoretical analyses, the authors delineate situations wherein adversarial examples cannot be circumvented. The authors leverage results from high-dimensional geometry and isoperimetric inequalities to derive limits on classifier robustness from a theoretical standpoint.

The primary findings can be summarized as:

1. High-Dimensional Sphere and Hypercube Analysis: The paper provides comprehensive proofs demonstrating that for classifiers on spheres and hypercubes, adversarial examples are inevitable for sufficiently large datasets. For instance, by applying isoperimetric inequalities, the paper shows how nearly all points in a sphere may possess adversarial examples when subjected to small perturbations in high dimensional settings.
2. Density Function and Class Susceptibility: The analysis takes into account the density of class distributions. It states that as long as class distributions aren't overly concentrated (having bounded upper values), adversarial examples become more probable with increased dimensionality.
3. Metric-Specific Insights: The research independently evaluates adversarial susceptibility under various norms—most notably the $\ell_2$, $\ell_\infty$, and $\ell_0$ norms—and offers bounds on perturbation sizes which are feasible in reaching adversarial examples. Each norm yields distinct implications on adversarial vulnerability given specific class distributions and dimensional settings.
4. Practical Dimensionality Considerations: The paper includes empirical insights using datasets such as MNIST and CIFAR-10 to assess the practical applicability of the theoretical bounds. These insights suggest that complexity, rather than dimensionality alone, plays a significant role in a classifier's susceptibility to adversarial attacks.

Implications and Prospective Directions

The work holds substantial implications for both theoretical and applied machine learning disciplines. By establishing that adversarial examples are, indeed, inherent to certain classification tasks, the research provides a foundation for future methodologies seeking to improve classifier robustness. It underscores the necessity for adaptable, context-aware adversarial defenses that account for data distribution and dimensionality nuances.

Moreover, the research opens avenues for exploring the trade-offs between classification accuracy and robustness. Understanding the landscape of adversarial vulnerability aids in determining optimal neural architecture design and data handling strategies.

In terms of future developments, understanding the complexity measures of data distributions may yield novel angles to enhance classifier robustness. Additionally, exploring alternative threat models and defense strategies that integrate adversarial training with insight from this theoretical framework could foster advancements in secure AI systems.

Ultimately, this paper delineates critical boundaries within which neural networks operate concerning adversarial examples, providing a structured theoretical basis that reinforces ongoing advancements in adversarial machine learning research.