Regularization for Adversarial Robust Learning (2408.09672v2)

Abstract: Despite the growing prevalence of artificial neural networks in real-world applications, their vulnerability to adversarial attacks remains a significant concern, which motivates us to investigate the robustness of machine learning models. While various heuristics aim to optimize the distributionally robust risk using the $\infty$-Wasserstein metric, such a notion of robustness frequently encounters computation intractability. To tackle the computational challenge, we develop a novel approach to adversarial training that integrates $\phi$-divergence regularization into the distributionally robust risk function. This regularization brings a notable improvement in computation compared with the original formulation. We develop stochastic gradient methods with biased oracles to solve this problem efficiently, achieving the near-optimal sample complexity. Moreover, we establish its regularization effects and demonstrate it is asymptotic equivalence to a regularized empirical risk minimization framework, by considering various scaling regimes of the regularization parameter and robustness level. These regimes yield gradient norm regularization, variance regularization, or a smoothed gradient norm regularization that interpolates between these extremes. We numerically validate our proposed method in supervised learning, reinforcement learning, and contextual learning and showcase its state-of-the-art performance against various adversarial attacks.

• The paper introduces φ-divergence regularization into distributionally robust optimization to enable a smoother adversarial training process.
• The paper devises efficient stochastic gradient methods with biased oracles, reducing computational complexity and achieving near-optimal sample efficiency.
• The paper validates the method's robustness across supervised, reinforcement, and contextual learning, demonstrating state-of-the-art performance against adversarial attacks.

Regularization for Adversarial Robust Learning: A Summary

This paper examines the vulnerability of machine learning models, particularly artificial neural networks, to adversarial attacks and proposes a novel approach to enhance their robustness. The key contributions include the integration of $\phi$-divergence regularization into the distributionally robust risk function and the development of efficient stochastic gradient methods, resulting in significant computational improvements.

Key Contributions

1. Novel Regularization Approach: The authors introduce $\phi$-divergence regularization to the existing distributionally robust optimization framework. This allows for a smoother and more computationally tractable adversarial training process.
2. Efficient Stochastic Gradient Methods: To address the intractability of the original problem formulation, stochastic gradient methods with biased oracles are proposed. These methods leverage the strong dual reformulation achieved by the $\phi$-divergence regularization, facilitating near-optimal sample complexity.
3. Unified Regularization Framework:

The paper establishes the asymptotic equivalence of the proposed regularized framework to regularized empirical risk minimization (ERM) under different regimes of the regularization parameter and robustness level. This includes: - Gradient norm regularization - Variance regularization - Smoothed gradient norm regularization

1. Numerical Validation: The proposed method is validated across multiple domains including supervised learning, reinforcement learning, and contextual learning, demonstrating state-of-the-art performance against various adversarial attacks.

Theoretical Insights

The authors provide rigorous theoretical analysis coupled with strong numerical results:

• Strong Duality: The paper derives a strong dual formulation for the $\phi$-divergence regularization framework, providing insights into the worst-case distribution characterization.
• Regularization Effects: The authors show how the proposed framework interpolates between gradient norm and variance regularization depending on the scaling of parameters $\rho$ (robustness level) and $\eta$ (regularization value).
• Generalization Error Bounds: The paper presents generalization error bounds for the proposed adversarial training framework, ensuring that empirical results are robust and generalize well to unseen data.

Practical Implications

1. Computational Efficiency: By reformulating the problem and introducing $\phi$-divergence regularization, the computational burden associated with adversarial training is significantly reduced. The stochastic gradient methods with biased oracles offer near-optimal sample complexity, making the approach scalable to large datasets and complex models.
2. Robustness in Various Domains: The practical utility of the proposed method is demonstrated across diverse applications. In supervised learning, for instance, the algorithm effectively defends against both $\ell_2$ and $\ell_\infty$ norm adversarial attacks. For reinforcement learning, the robust $Q$-learning algorithm showcases superior performance in perturbed environments, illustrating the flexibility and strength of the proposed method.
3. Versatility of the Regularized Framework: The framework's ability to interpolate between different types of regularization provides a deeper understanding of adversarial robustness. This versatility is crucial for devising robust models sensitive to the specific requirements of various applications.

Speculations on Future Developments

Given the robust performance and computational efficiency of the proposed method, several avenues for future research are apparent:

• Extension to Other Divergence Measures: Exploring other types of $\phi$-divergence within the regularization framework could yield further improvements in robustness and computational efficiency.
• Adaptive Regularization Strategies: Developing adaptive algorithms that can dynamically adjust the regularization parameter and robustness level based on the data distribution and the adversarial attack landscape.
• Expansion to Other Learning Paradigms: Extending the regularization approach to other machine learning paradigms like unsupervised and semi-supervised learning could enhance model robustness in broader contexts.

Conclusion

This paper provides a comprehensive solution to enhance the robustness of machine learning models against adversarial attacks through $\phi$-divergence regularization and efficient computational methods. The theoretical contributions and practical validations underscore the significance of this approach in adversarial robust learning. The insights gained here lay the groundwork for future explorations in robust machine learning methodologies.