Provable Robustness of ReLU networks via Maximization of Linear Regions (1810.07481v2)

Abstract: It has been shown that neural network classifiers are not robust. This raises concerns about their usage in safety-critical systems. We propose in this paper a regularization scheme for ReLU networks which provably improves the robustness of the classifier by maximizing the linear regions of the classifier as well as the distance to the decision boundary. Our techniques allow even to find the minimal adversarial perturbation for a fraction of test points for large networks. In the experiments we show that our approach improves upon adversarial training both in terms of lower and upper bounds on the robustness and is comparable or better than the state-of-the-art in terms of test error and robustness.

An Overview of "Provable Robustness of ReLU networks via Maximization of Linear Regions"

The paper "Provable Robustness of ReLU networks via Maximization of Linear Regions" by Francesco Croce, Maksym Andriushchenko, and Matthias Hein addresses the issue of robustness in ReLU networks, particularly in the context of adversarial perturbations. Neural networks, despite their impressive capabilities, have been shown to be vulnerable to small, imperceptible changes in input that can significantly alter their outputs. This vulnerability poses a critical concern for safety-critical applications, such as self-driving cars or medical diagnosis systems.

The authors propose a robust regularization scheme for ReLU networks, which aims to enhance the robustness of classifiers by maximizing the linear regions spanned by the network. Additionally, the proposed method focuses on increasing the distance to the decision boundary. This approach stands in contrast to traditional adversarial training techniques, highlighting its potential to provide better robustness guarantees.

Key Contributions and Findings

1. Regularization for Robustness: The paper introduces a Maximum Margin Regularizer (MMR) for ReLU networks. This regularizer is designed to increase both the margin to the decision boundary and the size of the linear regions in which the classifier operates. The results suggest that this can lead to substantial improvements in the robustness of neural networks without a compromise on classification accuracy.
2. Provable Guarantees: A significant contribution of this work is its ability to provide provable robustness guarantees. By considering the properties of linear regions associated with ReLU networks, the paper shows that, under certain conditions, minimal adversarial perturbations can be effectively approximated or even exactly computed.
3. Comparison with Contemporary Approaches: Experiments conducted in this paper demonstrate that the proposed approach excels in achieving better lower and upper bounds on robustness compared to traditional adversarial training methods. It also competes well with state-of-the-art techniques regarding test error and robust test error bounds. Notably, in several configurations, the authors show that their regularizer achieves provably better results than non-robust techniques.
4. Empirical Validation: The effectiveness of the regularization scheme is validated through experiments on various datasets such as MNIST, Fashion-MNIST, GTS, and CIFAR-10. Across these datasets, the proposed method consistently outperforms or complements existing methods by providing tighter robustness guarantees and demonstrating improved verifiability.

Theoretical and Practical Implications

The paper's proposals and findings have vital implications for both theoretical exploration and practical application in neural network robustness:

• Extension of Robustness Theory: By leveraging linear regions and distance maximization, the work extends the theoretical foundations of neural network robustness, offering new insights into how network architectures can be intrinsically more secure against perturbations.
• Enhanced Safety in Applications: Practically, the ability to guarantee robustness translates to increased safety in deploying neural networks in critical areas, potentially averting harmful consequences induced by adversarial attacks.

Future Directions

The authors indicate potential avenues for advancing this work:

• Scalability: Expanding the applicability to larger networks and more complex architectures can enhance the utility of the MMR approach.
• Integration with Other Techniques: Exploring the integration of this regularization scheme with other robustness strategies could yield synergistic benefits, improving both robustness and efficiency.

In conclusion, this paper makes a substantive contribution to the field of machine learning by proposing a novel method for enhancing the robustness of ReLU networks. Through strategic regularization, the authors provide a framework for achieving provable robustness, offering new potential for deploying neural networks in environments where reliability is paramount.