Lipschitz regularity of deep neural networks: analysis and efficient estimation (1805.10965v2)

Abstract: Deep neural networks are notorious for being sensitive to small well-chosen perturbations, and estimating the regularity of such architectures is of utmost importance for safe and robust practical applications. In this paper, we investigate one of the key characteristics to assess the regularity of such methods: the Lipschitz constant of deep learning architectures. First, we show that, even for two layer neural networks, the exact computation of this quantity is NP-hard and state-of-art methods may significantly overestimate it. Then, we both extend and improve previous estimation methods by providing AutoLip, the first generic algorithm for upper bounding the Lipschitz constant of any automatically differentiable function. We provide a power method algorithm working with automatic differentiation, allowing efficient computations even on large convolutions. Second, for sequential neural networks, we propose an improved algorithm named SeqLip that takes advantage of the linear computation graph to split the computation per pair of consecutive layers. Third we propose heuristics on SeqLip in order to tackle very large networks. Our experiments show that SeqLip can significantly improve on the existing upper bounds. Finally, we provide an implementation of AutoLip in the PyTorch environment that may be used to better estimate the robustness of a given neural network to small perturbations or regularize it using more precise Lipschitz estimations.

• The paper establishes that even for simple two-layer networks, accurately computing the Lipschitz constant is NP-hard.
• The paper presents AutoLip, an algorithm leveraging automatic differentiation to efficiently upper bound the Lipschitz constant for various network architectures.
• For sequential neural networks, the paper introduces SeqLip, which optimizes Lipschitz estimation and achieves significant improvements over traditional spectral norm approaches.

Lipschitz Regularity of Deep Neural Networks: Analysis and Efficient Estimation

The paper "Lipschitz regularity of deep neural networks: analysis and efficient estimation" authored by Kevin Scaman and Aladin Virmaux explores an important aspect of deep learning models, the Lipschitz continuity, which plays a critical role in understanding the robustness and stability of neural networks. The authors address the challenge of computing the Lipschitz constant for various architectures and propose novel methods to estimate this constant efficiently.

Key Contributions

1. NP-hard Complexity of Lipschitz Constant Computation: The paper begins by establishing the computational complexity of accurately determining the Lipschitz constant in neural networks. Specifically, it demonstrates that even for simple two-layer neural networks, the task is NP-hard. This result underscores the intrinsic difficulty in assessing the robustness of neural networks through exact computation.
2. AutoLip Algorithm: The authors introduce AutoLip, the first general algorithm capable of upper bounding the Lipschitz constant for any automatically differentiable function. This approach leverages automatic differentiation and introduces a systematic way to achieve efficient computation across various network layers.
3. Sequential Neural Network Optimization with SeqLip: For sequential neural networks, another algorithm named SeqLip is developed. It optimizes the estimation of the Lipschitz constant by decomposing calculations across pairs of consecutive layers in the model. Moreover, the authors propose heuristic methods within SeqLip to address scalability issues in large networks.
4. Implementation and Practical Implications: The AutoLip algorithm is implemented using PyTorch, facilitating its use in evaluating and potentially improving the robustness of neural networks against adversarial attacks. This practical implication is particularly relevant for deploying machine learning models in sensitive applications where safety is paramount.

Numerical Results

The experimental section demonstrates that SeqLip provides a significantly tighter upper bound on the Lipschitz constant compared to the more traditional product of spectral norms approach, particularly for deep networks. The experiments conducted on synthetic datasets, as well as well-known architectures like AlexNet, reveal that the proposed methods can achieve up to an eight-fold improvement over existing techniques for certain configurations.

Theoretical and Practical Implications

The theoretical contributions of this paper lie in advancing our understanding of the Lipschitz properties of neural networks and their computational boundaries. In practice, the implications are far-reaching. By providing a more accurate estimation of the Lipschitz constant, these methods contribute to the development of more robust models. This is critical in fields such as computer vision and natural language processing, where adversarial robustness is a key concern.

Future Directions

Building on the findings of this paper, future work may explore:

• Further refinement and optimization of the SeqLip algorithm to handle even larger and more complex network architectures.
• Application of these methods to other types of neural networks, such as recurrent networks, where the sequential dependencies add another layer of complexity.
• Investigation of the relationship between the Lipschitz constant computed by these methods and the empirical robustness observed in real-world applications, to better inform the design of resilient neural networks.

Overall, this paper provides substantial contributions to the field of deep learning, offering both theoretical insights and practical tools to enhance the robustness and safety of neural network models.