Understanding Deep Learning via Notions of Rank (2408.02111v3)

Abstract: Despite the extreme popularity of deep learning in science and industry, its formal understanding is limited. This thesis puts forth notions of rank as key for developing a theory of deep learning, focusing on the fundamental aspects of generalization and expressiveness. In particular, we establish that gradient-based training can induce an implicit regularization towards low rank for several neural network architectures, and demonstrate empirically that this phenomenon may facilitate an explanation of generalization over natural data (e.g., audio, images, and text). Then, we characterize the ability of graph neural networks to model interactions via a notion of rank, which is commonly used for quantifying entanglement in quantum physics. A central tool underlying these results is a connection between neural networks and tensor factorizations. Practical implications of our theory for designing explicit regularization schemes and data preprocessing algorithms are presented.

• The paper reveals that implicit regularization in neural networks is better explained by low-rank tendencies than by norm minimization.
• It introduces a novel rank metric, using separation rank from quantum physics, to assess Graph Neural Network expressiveness.
• The work develops practical tools such as an edge sparsification algorithm, enhancing performance in models with long-range dependencies.

Understanding Deep Learning via Notions of Rank

The assessment of deep learning systems often involves investigating the underlying principles that govern their effectiveness, with particular emphasis on aspects such as generalization, expressiveness, and implicit regularization. The thesis "Understanding Deep Learning via Notions of Rank" by Noam Razin provides a thorough exploration into these aspects through the lens of rank-related concepts, presenting insights into both the practical utility and theoretical understanding of deep learning models.

Implicit Regularization and Generalization

The work focuses on the implicit regularization observed in neural networks, primarily driven by gradient-based optimizations. It critically evaluates the hypothesis that implicit regularization could be interpreted as the minimization of certain norms, as previously suggested in several studies. Through rigorous mathematical analysis and empirical evidence, the thesis demonstrates that implicit regularization in matrix factorization is better explained by a tendency towards low rank rather than norm minimization. This revelation challenges conventional beliefs and extends the analysis to tensor factorization, which corresponds to non-linear polynomial neural networks. The implications are significant, suggesting that rather than focusing on norm-based measures, understanding and manipulating rank could provide deeper insights into the generalization capabilities of neural networks.

Expressiveness of Graph Neural Networks

Beyond generalization, the thesis explores the expressiveness of Graph Neural Networks (GNNs), which are crucial for modeling dependencies and interactions in data represented as graphs. By formalizing the notion of separation rank, typically used in quantum physics to measure entanglement, the work quantifies the strength of interactions that GNNs can model. The separation rank provides a novel metric for assessing how architectural choices such as depth and width impact a GNN's ability to model complex interactions between different graph regions. This approach offers a quantifiable measure distinct from the typical Weisfeiler-Leman tests, providing a fresh perspective on GNN expressiveness.

Practical Applications

The insights gained from rank considerations are applied to develop practical tools that enhance neural network performance. One such development is a novel explicit regularization scheme aimed at overcoming the limitations observed in convolutional networks dealing with long-range dependencies in input data. Additionally, the thesis introduces an edge sparsification algorithm—Walk Index Sparsification (WIS)—that effectively maintains GNN performance even when edges in input graphs are pruned. This method has demonstrated superiority over alternative approaches, highlighting the potential of rank-based analyses to drive practical advancements in AI.

Future Directions

This thesis opens up several avenues for further research. Extending the established theories to incorporate a broader range of non-linearities beyond polynomial models, such as ReLU, could greatly enhance their applicability. Additionally, further exploration of the WIS algorithm, especially in varying contexts and with different network architectures, would be an important step toward optimizing its implementation. The insights regarding the incremental learning of rank-influenced components offer a framework for continued investigation into the unique dynamics of deep learning models under various optimization constraints.

Conclusion

"Understanding Deep Learning via Notions of Rank" makes significant strides in addressing the complex problem of how neural networks learn and generalize. By shifting the focus from norm-based regularization to rank-based analyses, the thesis marks a pivotal step towards deeper insights into neural network behavior. The combination of theoretical exploration with practical application underscores the transformative potential of this research in shaping future advancements in deep learning.