Robust Large Margin Deep Neural Networks (1605.08254v3)

Abstract: The generalization error of deep neural networks via their classification margin is studied in this work. Our approach is based on the Jacobian matrix of a deep neural network and can be applied to networks with arbitrary non-linearities and pooling layers, and to networks with different architectures such as feed forward networks and residual networks. Our analysis leads to the conclusion that a bounded spectral norm of the network's Jacobian matrix in the neighbourhood of the training samples is crucial for a deep neural network of arbitrary depth and width to generalize well. This is a significant improvement over the current bounds in the literature, which imply that the generalization error grows with either the width or the depth of the network. Moreover, it shows that the recently proposed batch normalization and weight normalization re-parametrizations enjoy good generalization properties, and leads to a novel network regularizer based on the network's Jacobian matrix. The analysis is supported with experimental results on the MNIST, CIFAR-10, LaRED and ImageNet datasets.

• The paper demonstrates that bounding the spectral norm of the Jacobian near training samples is key to improving DNN generalization, independent of network depth or width.
• The authors derive robust generalization bounds based on classification margins that effectively link network decision boundaries to performance, challenging traditional size-dependent views.
• Empirical validation on datasets like MNIST, CIFAR-10, LaRED, and ImageNet confirms that enforcing spectral norm bounds leads to enhanced model robustness and practical regularization techniques.

Insights on "Robust Large Margin Deep Neural Networks"

The paper "Robust Large Margin Deep Neural Networks" explores the generalization capabilities of deep neural networks (DNNs) by analyzing their classification margins. The authors, Jure Sokolić, Raja Giryes, Guillermo Sapiro, and Miguel R. D. Rodrigues, propose a theoretical framework grounded in the spectral norm of the Jacobian matrix of DNNs, offering a novel perspective on the networks' performance beyond existing literature, which often suggests that generalization error (GE) is constrained by network depth or width.

Summary of Contributions

The paper aims to establish a robust theoretical understanding of DNNs by focusing on:

1. Jacobians and Generality: The authors argue that bounding the spectral norm of a DNN's Jacobian matrix in the vicinity of the training samples is crucial for maintaining a good generalization performance, regardless of the network's depth or width. This approach addresses a significant gap in current literature, which typically implies that GE scales with network dimensions.
2. Generalization Bounds: By introducing bounds based on classification margins that stem from DNNs' decision boundaries, the paper aligns computational robustness and strong classification performance. The authors formalize this through various bounding strategies, including those independent of network size.
3. Empirical Validation: The theoretical insights are supported by empirical results on standard datasets like MNIST, CIFAR-10, LaRED, and ImageNet, confirming that networks adhering to the proposed bounded spectral norms achieve favorable generalization behaviors.

Theoretical and Practical Implications

The work introduces a significant shift in how we understand the underlying factors influencing the GE of DNNs. By decoupling GE from direct dependence on network size and focusing on key structural properties, the research provides:

• Theoretical Deepening: A deeper theoretical framework that emphasizes the relationship between the function learned by DNNs and the properties of the training data.
• Practical Regularization Techniques: Insights into effective regularization strategies, such as those based on spectral norms, weight normalization, and Jacobian regularization, which can be employed to improve model robustness, particularly for weight and batch normalization schemes.
• Data-Driven GE Strategies: A method of assessing GE that considers inherent data structure and complexity as captured by their covering numbers, rather than solely relying on network architecture.

Future Prospects

This paper sets the stage for several future research directions. One avenue lies in exploring the implications of these findings in neural architecture design and the training of deeper networks without increasing complexity or GE. Moreover, additional work may focus on empirically testing the proposed bounds across diverse tasks and neural architectures like Recurrent Neural Networks (RNNs) and convolutional architectures, enhancing robustness across varied scenarios.

The insights might also invigorate the development of optimization techniques that inherently maintain the desired Jacobian properties throughout learning, potentially leading to more autonomous and adaptive learning systems.

In summary, "Robust Large Margin Deep Neural Networks" provides a nuanced understanding of network generalization and offers a concrete path forward for developing more robust DNNs that are efficient at scale, without succumbing to the depth or width constraints as posited by previous research.