Adversarial Risk and Robustness: General Definitions and Implications for the Uniform Distribution (1810.12272v1)

Abstract: We study adversarial perturbations when the instances are uniformly distributed over ${0,1}^n$. We study both "inherent" bounds that apply to any problem and any classifier for such a problem as well as bounds that apply to specific problems and specific hypothesis classes. As the current literature contains multiple definitions of adversarial risk and robustness, we start by giving a taxonomy for these definitions based on their goals, we identify one of them as the one guaranteeing misclassification by pushing the instances to the error region. We then study some classic algorithms for learning monotone conjunctions and compare their adversarial risk and robustness under different definitions by attacking the hypotheses using instances drawn from the uniform distribution. We observe that sometimes these definitions lead to significantly different bounds. Thus, this study advocates for the use of the error-region definition, even though other definitions, in other contexts, may coincide with the error-region definition. Using the error-region definition of adversarial perturbations, we then study inherent bounds on risk and robustness of any classifier for any classification problem whose instances are uniformly distributed over ${0,1}^n$. Using the isoperimetric inequality for the Boolean hypercube, we show that for initial error $0.01$, there always exists an adversarial perturbation that changes $O(\sqrt{n})$ bits of the instances to increase the risk to $0.5$, making classifier's decisions meaningless. Furthermore, by also using the central limit theorem we show that when $n\to \infty$, at most $c \cdot \sqrt{n}$ bits of perturbations, for a universal constant $c< 1.17$, suffice for increasing the risk to $0.5$, and the same $c \cdot \sqrt{n} $ bits of perturbations on average suffice to increase the risk to $1$, hence bounding the robustness by $c \cdot \sqrt{n}$.

• The paper establishes rigorous formulations leveraging binomial coefficient analysis to evaluate ML model robustness under adversarial perturbations.
• It demonstrates how specific ratios and asymptotic behaviors of binomial coefficients bound the impact of adversarial attacks on decision boundaries.
• The study lays the groundwork for developing more resilient AI systems by preemptively addressing vulnerabilities through combinatorial methods.

Revisiting Risk and Robustness under Adversarial Perturbations

This paper explores the mathematical underpinnings of risk and robustness in machine learning models, particularly in the context of adversarial perturbations. The authors present a series of lemmas and theorems that offer a comprehensive analysis of binomial coefficients and their applications in understanding model robustness.

Key Insights from the Paper

The primary focus is on the behavior of models under adversarial conditions and the theoretical formulations that can help evaluate this robustness. For instance, the paper includes detailed derivations and proofs related to the ratios and sums of binomial coefficients, which are critical in understanding how adversarial attacks can optimize their efficacy.

Theoretical Contributions

• Binomial Coefficients Analysis: The paper explores various properties of binomial coefficients, which serve as foundational elements in combinatorial mathematics, applicable to risk assessment in probabilistic models.
• Adversarial Perturbations: It underscores the significance of perturbations and their impact on a model's decision boundaries, through rigorous mathematical formulations.

Notable Lemmas

Several lemmas are noteworthy due to their implications in the broader field of risk analysis:

• Lemma on Useful Ratios: Demonstrates that the ratios of perturbed terms maintain certain bounds, which are essential when evaluating robustness.
• Central Coefficients Lemma: Examines the asymptotic behavior of central coefficients, a critical aspect because it determines the likelihood of model failures under perturbations.

Numerical Implications

While the paper is dense with theoretical constructs, the implications of these findings are profound in terms of practical applications. For example, understanding binomial distributions in the context of adversarial perturbations provides a framework to develop more robust models. These insights allow researchers to anticipate and mitigate potential vulnerabilities within model architectures.

Implications for Future Developments

The paper provides groundwork for future research in AI robustness. By establishing a theoretical lens through which adversarial attacks can be understood, it empowers researchers to:

• Develop models with enhanced resilience against optimization-based adversarial attacks.
• Formulate new algorithms that can preemptively identify and neutralize potential adversarial threats.
• Apply combinatorial mathematics to novel AI applications beyond traditional risk and robustness assessments.

Conclusion

This paper offers a substantial theoretical contribution to the paper of adversarial robustness in machine learning models. By leveraging combinatorial mathematics, it provides a robust framework that experienced researchers can utilize to advance AI systems against adversarial perturbations. The presented lemmas and corollaries not only deepen theoretical understandings but also pave the way for meaningful advancements in AI development and deployment.