Rademacher Complexity for Adversarially Robust Generalization (1810.11914v4)

Abstract: Many machine learning models are vulnerable to adversarial attacks; for example, adding adversarial perturbations that are imperceptible to humans can often make machine learning models produce wrong predictions with high confidence. Moreover, although we may obtain robust models on the training dataset via adversarial training, in some problems the learned models cannot generalize well to the test data. In this paper, we focus on $\ell_\infty$ attacks, and study the adversarially robust generalization problem through the lens of Rademacher complexity. For binary linear classifiers, we prove tight bounds for the adversarial Rademacher complexity, and show that the adversarial Rademacher complexity is never smaller than its natural counterpart, and it has an unavoidable dimension dependence, unless the weight vector has bounded $\ell_1$ norm. The results also extend to multi-class linear classifiers. For (nonlinear) neural networks, we show that the dimension dependence in the adversarial Rademacher complexity also exists. We further consider a surrogate adversarial loss for one-hidden layer ReLU network and prove margin bounds for this setting. Our results indicate that having $\ell_1$ norm constraints on the weight matrices might be a potential way to improve generalization in the adversarial setting. We demonstrate experimental results that validate our theoretical findings.

• The paper establishes tight upper and lower bounds for adversarial Rademacher complexity in both binary and multi-class classifiers.
• It reveals that adversarial settings introduce inherent dimension dependence in both linear models and neural networks.
• It validates that ℓ1 regularization and surrogate adversarial loss can mitigate generalization errors in high-dimensional adversarial contexts.

An Examination of Rademacher Complexity in Adversarially Robust Machine Learning Models

The paper "Rademacher Complexity for Adversarially Robust Generalization" by Dong Yin, Kannan Ramchandran, and Peter Bartlett provides a rigorous analysis of the adversarially robust generalization issue in machine learning through the lens of Rademacher complexity. This research attempts to address a crucial shortcoming in modern machine learning systems—their vulnerability to adversarial attacks. Such attacks exploit model weaknesses by introducing small, often imperceptible perturbations to inputs, causing erroneous predictions.

Core Contributions

The paper makes several significant contributions, summarized as follows:

1. Binary and Multi-class Linear Classifiers: The authors have successfully derived tight upper and lower bounds for adversarial Rademacher complexity for binary linear classifiers. They establish that the adversarially measured Rademacher complexity is invariably greater than its non-adversarial counterpart. This finding implies that the inherent difficulty of achieving robust generalization is amplified under adversarial conditions. A key insight is that $\ell_\infty$ attack inherently introduces dimension dependence unless constraints such as $\ell_1$ norm are placed on the weight vectors. For multi-class classifiers, the paper extends these results and discusses margin bounds, confirming similar dimension dependencies.
2. Neural Networks: For nonlinear models like neural networks, the paper highlights the persistent dimension dependence in adversarial Rademacher complexity. Interestingly, even with weight norm constraints, an explicit dimensional dependence remains, suggesting inherent challenges in adversarial robustness that are not present in natural settings.
3. Surrogate Adversarial Loss: The researchers also explore the application of a surrogate adversarial loss based on semidefinite programming relaxation. For ReLU networks with a single hidden layer, the paper provides margin bounds and suggests that constraining the $\ell_1$ norms of the weight matrices might offer a feasible pathway to enhance generalization capabilities in adversarial contexts.
4. Experimental Validation: The paper complements its theoretical results with empirical investigations. Experiments conducted on linear classifiers and neural networks on the MNIST dataset validate the hypothesis that $\ell_1$ regularization can mitigate adversarial generalization error. Additionally, they demonstrate that the challenge of adversarial robustness increases with higher dimensionality of the feature space.

Implications and Future Directions

The implications of this research are twofold. Practically, it suggests potential methodologies for improving adversarial robustness in machine learning models. Theoretical insights regarding the dimension dependence of adversarial Rademacher complexity indicate that addressing adversarial challenges might require fundamentally new approaches or significant modifications to current model training paradigms. The paper implicitly suggests the need for further research into alternative norms or constraints that could mitigate adversarial vulnerabilities without exacerbating the generalization challenge.

The findings open several avenues for future work:

• Refining Surrogate Loss Functions: The effectiveness and tightness of surrogate adversarial losses in capturing adversarial robustness merits further examination, potentially leading to improved training techniques for adversarially resilient models.
• Exploration of Norm Constraints: Investigating other norm constraints beyond $\ell_1$, or hybrid approaches that might balance model robustness and computational feasibility remains a fertile area for research.
• Theoretical Exploration of Nonlinear Models: Extending the analysis to deeper and more complex architectures may yield insights directly beneficial for contemporary applications reliant on high-dimensional data.

In conclusion, this paper provides a critical perspective on adversarial robustness in machine learning models through Rademacher complexity, offering both theoretical insights and practical validations. Its contributions lay a foundation for subsequent research aimed at enhancing the resilience of AI systems in adversarial settings.