Lipschitz-Margin Training: Scalable Certification of Perturbation Invariance for Deep Neural Networks (1802.04034v3)

Abstract: High sensitivity of neural networks against malicious perturbations on inputs causes security concerns. To take a steady step towards robust classifiers, we aim to create neural network models provably defended from perturbations. Prior certification work requires strong assumptions on network structures and massive computational costs, and thus the range of their applications was limited. From the relationship between the Lipschitz constants and prediction margins, we present a computationally efficient calculation technique to lower-bound the size of adversarial perturbations that can deceive networks, and that is widely applicable to various complicated networks. Moreover, we propose an efficient training procedure that robustifies networks and significantly improves the provably guarded areas around data points. In experimental evaluations, our method showed its ability to provide a non-trivial guarantee and enhance robustness for even large networks.

• The paper proposes connecting Lipschitz constants to prediction margins to scalably certify deep network robustness against adversarial perturbations.
• It introduces Lipschitz-margin training (LMT) which enlarges provably guarded areas around data points by maintaining network smoothness during training.
• Empirical results show LMT significantly improves robustness and provides larger certifiable areas in large networks more effectively than prior methods.

Lipschitz-Margin Training: Scalable Certification of Perturbation Invariance for Deep Neural Networks

The paper addresses the critical issue of adversarial perturbations that deeply affect the robustness and security of neural networks, particularly in high-stakes applications like autonomous driving systems. The authors propose a scalable method that effectively ensures substantial guarded areas around data points in a neural network by connecting the Lipschitz constants to prediction margins.

Key Contributions

1. Lipschitz Constant and Guarded Areas: The paper introduces a method to calculate a lower bound on the size of adversarial perturbations by exploring the relationship between the Lipschitz constants and prediction margins. This relationship facilitates certifying the robustness of neural networks against perturbations in a computationally efficient manner. It devises a method that circumvents the prohibitive computational costs associated with directly calculating local Lipschitz constants for large networks.
2. Lipschitz-margin Training (LMT): A novel training procedure is introduced that prioritizes enlarging the provably guarded areas around data points by leveraging the computed Lipschitz bounds during training. The training approach focuses on maintaining network smoothness, thereby enhancing robustness against adversarial attacks. The Lipschitz-margin training aligns the model’s decision boundaries to ensure the absence of perturbations within certain bounds computed during the training phase.
3. Spectral Bounds and Scalability: The authors provide tighter spectral bounds for components of neural networks and a generalized, fast calculation algorithm for estimating upper bounds of operator norms. The results demonstrate that existing scalability issues in certifying large and complex networks can be resolved without significant computational overhead.

Numerical Results and Implications

Empirical evaluations reveal that Lipschitz-margin training significantly improves the robustness of networks against adversarial attacks and yields increased certifiable guarded areas. For example, the LMT models ensured certifiable invariance for perturbations even in large networks like wide residual networks with 16 layers and width factor 4. The reported median size of provably guarded areas within the L2-norm was non-trivial and exceeded previous computationally affordable methods.

The training algorithm offers improvements over contemporary regularization techniques like weight decay, spectral norm regularization, and Parseval networks in both robustness and execution simplicity. While the authors note that LMT results in moderate accuracy drops on clean data, the gains in adversarial robustness are notable. Furthermore, LMT remains complementary to existing defense techniques, thus widening its potential for integration and cross-utilization in enhancing model robustness.

Future Directions

This research holds promise for extending the existing paradigms in adversarial machine learning and network robustness certifications. Future work could further explore integrating LMT with adversarial training methods and examine applications in more diverse neural network architectures. The paper also hints at the potential for application in other domains sensitive to perturbations, such as GAN training or models prone to noisy labels, thereby suggesting a broad horizon for its theoretical and practical impact.