Issues with Neural Tangent Kernel Approach to Neural Networks (2501.10929v1)

Abstract: Neural tangent kernels (NTKs) have been proposed to study the behavior of trained neural networks from the perspective of Gaussian processes. An important result in this body of work is the theorem of equivalence between a trained neural network and kernel regression with the corresponding NTK. This theorem allows for an interpretation of neural networks as special cases of kernel regression. However, does this theorem of equivalence hold in practice? In this paper, we revisit the derivation of the NTK rigorously and conduct numerical experiments to evaluate this equivalence theorem. We observe that adding a layer to a neural network and the corresponding updated NTK do not yield matching changes in the predictor error. Furthermore, we observe that kernel regression with a Gaussian process kernel in the literature that does not account for neural network training produces prediction errors very close to that of kernel regression with NTKs. These observations suggest the equivalence theorem does not hold well in practice and puts into question whether neural tangent kernels adequately address the training process of neural networks.

• The paper challenges the theoretical equivalence between NTK-based kernel regression and trained neural networks by showing that adding layers benefits networks but not NTKs.
• It rigorously compares NTK performance with Gaussian process and Laplacian kernels, revealing that untrained kernels can sometimes match trained NTK outcomes.
• The study underscores the need to refine NTK assumptions and explore hybrid models to more accurately capture the dynamic learning behaviors of deep neural networks.

Overview of "Issues with Neural Tangent Kernel Approach to Neural Networks"

This paper critically addresses the practical applicability of the neural tangent kernel (NTK) framework in modeling neural network behavior, specifically challenging the notion of equivalence between trained neural networks and kernel regression using NTKs. The authors embark on a rigorous evaluation of this equivalence by exploring both theoretical derivations and empirical experiments.

Neural tangent kernels provide a representation of neural networks' outputs as equivalent to kernel regression under certain conditions, thus offering a lens through which deep learning models could be interpreted as kernel methods. However, foundational assumptions behind this theorem of equivalence, such as convergence under extensive width and the lazy training hypothesis, might not hold true in real-world applications.

Key Insights and Numerical Findings

The authors undertake a comprehensive series of experiments involving NTKs with one and two hidden layers, alongside their neural network counterparts, and contrast these with the performance of both Gaussian process kernels and regular non-neural network-dependent kernels, such as the Laplacian kernel. Their experiments utilize generated data to assess predictive capabilities, with results synthesized over numerous trials to garner statistically significant conclusions.

Experimental results reveal that while NTKs such as NTKB and NTKJ exhibit similar performance metrics in terms of mean root mean squared error (RMSE), these results do not parallel the performance gains observed in neural networks upon adding additional layers. Specifically, adding layers to NTKs does not enhance expressivity in the manner anticipated theoretically, thus deviating from the equivalence principle, as neural networks do demonstrate performance improvement with depth.

Remarkably, empirical findings show the predictions by initial Gaussian process kernels—designed to correspond to untrained networks—being comparably effective as their trained NTK counterparts, which questions the translation of theoretical promises of NTKs into practical advantage. This observation undermines the assumption that NTKs are adequately capturing the iterative learning dynamics of neural networks.

Theoretical and Practical Implications

This investigation prompts a reevaluation of NTKs as a comprehensive tool for understanding and predicting neural networks. While NTKs present a compelling theoretical framework, this paper challenges their practical utility when core assumptions (e.g., infinite width, lazy training) defy realistic conditions. The insights suggest that NTKs, especially when grounded on layered architectures analogous to deeper convolutional neural networks, may not realize expected improvements in function approximation or generalization traits.

The critique extends to caution against over-reliance on NTKs for constructing interpretable models, given the observed lack of congruence between theoretically derived NTK-based equivalency and actual learning patterns of networks. Future explorations might pivot towards refining NTK assumptions, perhaps considering network architectures and hyperparameter settings that align more closely with operating model conditions in applied settings.

Speculation on Future Directions

Considering the limitations highlighted, subsequent research might focus on relaxing certain NTK prerequisites while still striving for even better models of network behavior that maintain desirable interpretative qualities. Another avenue is exploring hybrid models that leverage NTK insights while integrating aspects of network flexibility and adaptive learning through real-time parameter tuning and non-linear transformation strategies.

Continual assessment of NTKs within the broader suite of tools for understanding deep learning models remains vital, as does greater cross-validation of NTK predictions in real-world scenarios. Comprehensive approaches uniting theoretical elegance with practical efficacy could pave the way for refined utilizations of NTKs amid evolving neural network architectures.