Stronger generalization bounds for deep nets via a compression approach (1802.05296v4)

Abstract: Deep nets generalize well despite having more parameters than the number of training samples. Recent works try to give an explanation using PAC-Bayes and Margin-based analyses, but do not as yet result in sample complexity bounds better than naive parameter counting. The current paper shows generalization bounds that're orders of magnitude better in practice. These rely upon new succinct reparametrizations of the trained net --- a compression that is explicit and efficient. These yield generalization bounds via a simple compression-based framework introduced here. Our results also provide some theoretical justification for widespread empirical success in compressing deep nets. Analysis of correctness of our compression relies upon some newly identified \textquotedblleft noise stability\textquotedblright properties of trained deep nets, which are also experimentally verified. The study of these properties and resulting generalization bounds are also extended to convolutional nets, which had eluded earlier attempts on proving generalization.

• The paper's main contribution is a compression framework that yields significantly improved generalization bounds for deep networks.
• It identifies key noise stability properties, such as layer cushion and activation contraction, that enable effective compression.
• The research validates its findings through both theoretical insights and empirical experiments on convolutional networks like VGG-19 and AlexNet.

Analyzing Generalization Bounds in Deep Networks via Compression

This paper investigates the perplexing ability of deep neural networks to generalize effectively, even when possessing more parameters than training samples. Traditional methods such as PAC-Bayes and margin-based analyses have struggled to provide satisfactory explanations, offering sample complexity bounds that barely surpass naive parameter counting. This work introduces a novel framework based on network compression, proposing significantly improved generalization bounds that align more closely with empirical observations.

Key Contributions

1. Compression Framework: The paper presents a streamlined compression framework to establish generalization bounds for deep networks. This approach simplifies the derivation of bounds compared to prior PAC-Bayes techniques and offers concise proofs of existing results.
2. Noise Stability Properties: The authors identify new noise stability characteristics crucial for deep networks' effective compression. These include layer cushion, interlayer cushion, activation contraction, and interlayer smoothness, crucial for describing each layer's resilience to noise.
3. Theoretical and Empirical Validation: The findings extend to convolutional networks that have traditionally been challenging to analyze theoretically. The research provides algorithms with efficient parameter reductions, achieving generalization bounds better than parameter counting and exhibiting empirical correlations with generalization.

Theoretical Insights

The core theoretical advancement is the realization that deep networks exhibit strong noise stability, allowing for effective compression without significant loss of generalization performance. The framework suggests that networks that generalize well can be significantly compressed while maintaining accuracy, contrary to the previously believed necessity of dense parameterization.

The compression relies on a reparameterization mechanism that maintains network output stability despite the induced noise from approximation, leveraging properties like the interlayer cushion to keep such noise effects minimal.

Empirical Analysis

Experiments on architectures such as VGG-19 and AlexNet, particularly on the CIFAR-10 dataset, demonstrate that trained networks exhibit drastically improved stability properties compared to random initialization. The paper also distinguishes between networks trained on real semantic labels and those trained on random labels, reaffirming the utility of identified properties like layer cushion and activation contraction in determining generalization.

Implications and Future Directions

This research holds substantial implications for understanding and designing deep networks. By connecting compression with generalization, it moves toward a more resource-efficient model deployment, paving the way for innovations in architecture and training methodologies. The approaches can potentially inform practical techniques for model compression in deployment scenarios.

Future investigations may address whether the noise stability properties identified can be enhanced systematically during training, perhaps through modifications to optimization algorithms or architectural choices. Additionally, these techniques could be validated across a broader array of network architectures and tasks to strengthen their general applicability.

Conclusion

The paper offers a transformative view on the generalization of deep networks, aligning theoretical bounds with empirical performance through innovative compression methods. By elucidating the stability properties integral to this process, it opens avenues for further refinement and understanding of deep learning models' inherent capabilities.