Spectrally-normalized margin bounds for neural networks (1706.08498v2)

Abstract: This paper presents a margin-based multiclass generalization bound for neural networks that scales with their margin-normalized "spectral complexity": their Lipschitz constant, meaning the product of the spectral norms of the weight matrices, times a certain correction factor. This bound is empirically investigated for a standard AlexNet network trained with SGD on the mnist and cifar10 datasets, with both original and random labels; the bound, the Lipschitz constants, and the excess risks are all in direct correlation, suggesting both that SGD selects predictors whose complexity scales with the difficulty of the learning task, and secondly that the presented bound is sensitive to this complexity.

• The paper introduces a novel generalization bound based on margin-normalized spectral complexity for neural networks.
• It refines traditional complexity measures by emphasizing the role of spectral norms and Lipschitz constants in controlling excess risk.
• Empirical studies on MNIST and CIFAR-10 validate that margin distributions stabilize despite increasing weight norms under SGD training.

Spectrally-Normalized Margin Bounds for Neural Networks

Introduction

The paper "Spectrally-normalized margin bounds for neural networks" authored by Peter L. Bartlett, Dylan J. Foster, and Matus Telgarsky, offers significant advancements in understanding the generalization capabilities of neural networks. Central to this work is the introduction of a margin-based generalization bound that scales with the margin-normalized spectral complexity and is empirically validated on standard datasets such as MNIST and CIFAR-10. This work diverges from complexity measures based solely on the VC dimension by incorporating the spectral norms of the weight matrices, providing a more nuanced understanding of neural network behavior under Stochastic Gradient Descent (SGD) training.

Key Contributions

This paper's contributions are two-fold:

1. Theoretical Bound: The authors present a rigorous generalization bound for neural networks that scales with the product of the spectral norms of the weight matrices divided by the margins. This bound is free from explicit dependence on combinatorial parameters such as the number of layers or nodes (outside of logarithmic factors) and is applicable to multiclass settings.
2. Empirical Validation: The effectiveness of the proposed bound is evaluated through detailed empirical studies on standard datasets, including CIFAR-10, CIFAR-100, and MNIST.

Theoretical Insights

The paper addresses fundamental questions about why neural networks, despite their capacity to fit arbitrary labels, demonstrate good generalization capabilities. The theoretical bound derived in this paper hinges on the spectral norm of the weight matrices and the margin of the predictor, offering a more refined control over the generalization error compared to traditional methods relying on combinatorial parameters.

Mathematical Form

The primary theorem in the paper provides a bound that scales with the spectral complexity $R_{\mathcal{A}}$ given by: $$R_{\mathcal{A}} := \left( \prod_{i=1}^L \rho_i \|A_i\|_\sigma \right) \left( \sum_{i=1}^L \frac{ \|A_i^{\top} - M_i^{\top}\|_{2,1}^{2/3} }{\|A_i\|_\sigma^{2/3}} \right)^{3/2}$$ where $A_i$ are the weight matrices, $\rho_i$ are Lipschitz constants, and $M_i$ are reference matrices, often chosen as identity maps in the case of ResNet. The bound provided by the theorem is particularly insightful as it demonstrates that the generalization error is primarily controlled by factors that are sensitive to the scale and structure of the neural network, rather than merely its size or the number of its parameters.

Empirical Validation

The authors conduct a thorough empirical paper using the AlexNet architecture trained with SGD on datasets such as CIFAR-10 and MNIST. Key findings from the experiments include:

• Correlation between Lipschitz constant and excess risk: For both original and random labels, the Lipschitz constant of the network correlates tightly with the excess risk, showcasing that the complexity measured scales with the difficulty of the learning task.
• Impact of Margins: Normalizing the Lipschitz constants by the margin results in a decaying curve during training, despite the weight norms continuing to grow, indicating a plateau in the excess risk.
• Dataset Comparisons: Datasets with randomized labels show considerably higher difficulty, reflected in the margin distributions, aligning with the intuition that such tasks are more complex.
• Convergence of Margins: During training, the margin distributions tend to converge even when weight norms continue to increase, suggesting a stabilization in the learned representations.

Further Observations and Open Problems

The paper opens several avenues for further research:

• Adversarial Examples: Low margin points are more susceptible to adversarial noise. Analyzing margin distributions could potentially lead to robust methods against adversarial attacks.
• Regularization: Unlike traditional methods, weight decay regularization appears to have limited effect on improving margins. This highlights the need for new regularization strategies aimed directly at enhancing margins.
• SGD Dynamics: There is a need to understand why SGD tends to select predictors with favorable margin properties naturally and whether this behavior can be systematically exploited or enhanced.

Conclusion

The findings in this paper provide a significant step towards understanding the generalization behavior of neural networks. By introducing a margin-based generalization bound that incorporates the spectral norms of weight matrices, the authors move beyond traditional combinatorial measures, offering insights that align closely with empirical behavior observed under SGD training. Future developments in this line of research could pave the way for more robust and theoretically grounded training methodologies.