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Deep Learning Through A Telescoping Lens: A Simple Model Provides Empirical Insights On Grokking, Gradient Boosting & Beyond (2411.00247v1)

Published 31 Oct 2024 in cs.LG, cs.AI, and stat.ML

Abstract: Deep learning sometimes appears to work in unexpected ways. In pursuit of a deeper understanding of its surprising behaviors, we investigate the utility of a simple yet accurate model of a trained neural network consisting of a sequence of first-order approximations telescoping out into a single empirically operational tool for practical analysis. Across three case studies, we illustrate how it can be applied to derive new empirical insights on a diverse range of prominent phenomena in the literature -- including double descent, grokking, linear mode connectivity, and the challenges of applying deep learning on tabular data -- highlighting that this model allows us to construct and extract metrics that help predict and understand the a priori unexpected performance of neural networks. We also demonstrate that this model presents a pedagogical formalism allowing us to isolate components of the training process even in complex contemporary settings, providing a lens to reason about the effects of design choices such as architecture & optimization strategy, and reveals surprising parallels between neural network learning and gradient boosting.

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Authors (3)
  1. Alan Jeffares (9 papers)
  2. Alicia Curth (22 papers)
  3. Mihaela van der Schaar (321 papers)

Summary

Insights from Telescoping Models in Deep Learning Phenomena

The paper "Deep Learning Through A Telescoping Lens" by Jeffares, Curth, and van der Schaar introduces a simplified yet effective model for analyzing neural networks, providing empirical insights into several notable deep learning phenomena. The researchers suggest that viewing neural network training as a series of first-order approximations enables the construction of a telescoping model that not only closely approximates the behavior of trained networks but also serves as a tool for understanding complex behaviors observed in practice.

Telescoping Model Overview

The proposed telescoping model reimagines neural network training through a sequence of linear approximations at every training step, rather than treating the network's final parameters as a monolithic endpoint. This telescopic perspective offers a new way of reasoning about the evolutionary trajectory of a network during training, presenting testable hypotheses about neural architectures, optimization practices, and empirical behaviors that are difficult to detect through traditional means.

Key Phenomena Explored

  1. Double Descent Phenomenon: The paper revisits the double descent behavior in neural networks, a phenomenon wherein test performance improves with model complexity, worsens at certain capacity levels, and then surprisingly improves again with further increases. By applying the telescoping model, the authors leverage a complexity measure ps^0p^{0}_{\hat{\mathbf{s}}} that effectively deconstructs learned complexity into train- and test-time components. This decomposition illustrates quantifiable differences in complexity, thus clarifying the occurrence of benign overfitting in overparameterized networks.
  2. Grokking and Generalization: The model lends significant insight into grokking, where networks achieve perfect training accuracy long before notable improvements in test performance occur. The paper highlights that grokking arises with a notable divergence between effective parameters used for training and test data, indicating a quantifiable measure of benign overfitting. This aspect of the telescoping model points towards understanding the mechanistic process by which networks discover simpler generalizable solutions over extended training periods.
  3. Performance on Tabular Data vs. Gradient Boosting: The research addresses the purported underperformance of deep learning models compared to gradient boosted trees on tabular data and suggests that the intrinsic differences in kernel behavior — implicit to the neural tangent kernels versus the explicit tree kernels — could account for such observations. This line of inquiry reveals that neural networks' behavior can be unexpected under input irregularity, specifying a predictive viewpoint through the behavior of maximum kernel values during testing.
  4. Linear Mode Connectivity (LMC): Within optimization practices, the model aids in understanding linear mode connectivity (LMC), whereby neural networks trained from the same initialization can be averaged to yield equally performant models. The telescoping model frames this transition as associated with a stable regime of the tangent kernel, providing empirical evidence for when and why weight averaging aligns with ensemble predictions.

Implications and Future Directions

The implications of this paper are manifold. The telescoping model encourages a reevaluation of neural network complexity and function that accounts for dynamic training phenomena. The findings suggest potential advancements in optimization strategies, architecture design, and interpretability of neural networks. Future work could expand these insights to broader neural architectures and training regimes, particularly examining telescoping approximations' efficacy in large-scale models and diverse data modalities. This innovative analytical tool bridges an understanding of deep learning as a static entity to a dynamic, comprehensible process, highlighting the need for nuanced approaches in both empirical and theoretical domains of AI.

In conclusion, this paper exemplifies the emergence of novel methodologies in understanding deep learning, fostering deeper insights into established and emerging phenomena that govern model behavior, performance, and generalization. This work sets a robust foundation for future explorations into the surprise-laden behavior of neural networks across varied contexts.

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