Tensor Programs IIb: Architectural Universality Of Neural Tangent Kernel Training Dynamics

Greg Yang, Etai Littwin
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:11762-11772, 2021.

Abstract

Yang (2020) recently showed that the Neural Tangent Kernel (NTK) at initialization has an infinite-width limit for a large class of architectures including modern staples such as ResNet and Transformers. However, their analysis does not apply to training. Here, we show the same neural networks (in the so-called NTK parametrization) during training follow a kernel gradient descent dynamics in function space, where the kernel is the infinite-width NTK. This completes the proof of the architectural universality of NTK behavior. To achieve this result, we apply the Tensor Programs technique: Write the entire SGD dynamics inside a Tensor Program and analyze it via the Master Theorem. To facilitate this proof, we develop a graphical notation for Tensor Programs, which we believe is also an important contribution toward the pedagogy and exposition of the Tensor Programs technique.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-yang21f, title = {Tensor Programs IIb: Architectural Universality Of Neural Tangent Kernel Training Dynamics}, author = {Yang, Greg and Littwin, Etai}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {11762--11772}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/yang21f/yang21f.pdf}, url = {https://proceedings.mlr.press/v139/yang21f.html}, abstract = {Yang (2020) recently showed that the Neural Tangent Kernel (NTK) at initialization has an infinite-width limit for a large class of architectures including modern staples such as ResNet and Transformers. However, their analysis does not apply to training. Here, we show the same neural networks (in the so-called NTK parametrization) during training follow a kernel gradient descent dynamics in function space, where the kernel is the infinite-width NTK. This completes the proof of the architectural universality of NTK behavior. To achieve this result, we apply the Tensor Programs technique: Write the entire SGD dynamics inside a Tensor Program and analyze it via the Master Theorem. To facilitate this proof, we develop a graphical notation for Tensor Programs, which we believe is also an important contribution toward the pedagogy and exposition of the Tensor Programs technique.} }
Endnote
%0 Conference Paper %T Tensor Programs IIb: Architectural Universality Of Neural Tangent Kernel Training Dynamics %A Greg Yang %A Etai Littwin %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-yang21f %I PMLR %P 11762--11772 %U https://proceedings.mlr.press/v139/yang21f.html %V 139 %X Yang (2020) recently showed that the Neural Tangent Kernel (NTK) at initialization has an infinite-width limit for a large class of architectures including modern staples such as ResNet and Transformers. However, their analysis does not apply to training. Here, we show the same neural networks (in the so-called NTK parametrization) during training follow a kernel gradient descent dynamics in function space, where the kernel is the infinite-width NTK. This completes the proof of the architectural universality of NTK behavior. To achieve this result, we apply the Tensor Programs technique: Write the entire SGD dynamics inside a Tensor Program and analyze it via the Master Theorem. To facilitate this proof, we develop a graphical notation for Tensor Programs, which we believe is also an important contribution toward the pedagogy and exposition of the Tensor Programs technique.
APA
Yang, G. & Littwin, E.. (2021). Tensor Programs IIb: Architectural Universality Of Neural Tangent Kernel Training Dynamics. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:11762-11772 Available from https://proceedings.mlr.press/v139/yang21f.html.

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