Critical feature learning in deep neural networks

Kirsten Fischer, Javed Lindner, David Dahmen, Zohar Ringel, Michael Krämer, Moritz Helias
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:13660-13690, 2024.

Abstract

A key property of neural networks driving their success is their ability to learn features from data. Understanding feature learning from a theoretical viewpoint is an emerging field with many open questions. In this work we capture finite-width effects with a systematic theory of network kernels in deep non-linear neural networks. We show that the Bayesian prior of the network can be written in closed form as a superposition of Gaussian processes, whose kernels are distributed with a variance that depends inversely on the network width $N$. A large deviation approach, which is exact in the proportional limit for the number of data points $P=\alpha N\to\infty$, yields a pair of forward-backward equations for the maximum a posteriori kernels in all layers at once. We study their solutions perturbatively, to demonstrate how the backward propagation across layers aligns kernels with the target. An alternative field-theoretic formulation shows that kernel adaptation of the Bayesian posterior at finite-width results from fluctuations in the prior: larger fluctuations correspond to a more flexible network prior and thus enable stronger adaptation to data. We thus find a bridge between the classical edge-of-chaos NNGP theory and feature learning, exposing an intricate interplay between criticality, response functions, and feature scale.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-fischer24a, title = {Critical feature learning in deep neural networks}, author = {Fischer, Kirsten and Lindner, Javed and Dahmen, David and Ringel, Zohar and Kr\"{a}mer, Michael and Helias, Moritz}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {13660--13690}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/fischer24a/fischer24a.pdf}, url = {https://proceedings.mlr.press/v235/fischer24a.html}, abstract = {A key property of neural networks driving their success is their ability to learn features from data. Understanding feature learning from a theoretical viewpoint is an emerging field with many open questions. In this work we capture finite-width effects with a systematic theory of network kernels in deep non-linear neural networks. We show that the Bayesian prior of the network can be written in closed form as a superposition of Gaussian processes, whose kernels are distributed with a variance that depends inversely on the network width $N$. A large deviation approach, which is exact in the proportional limit for the number of data points $P=\alpha N\to\infty$, yields a pair of forward-backward equations for the maximum a posteriori kernels in all layers at once. We study their solutions perturbatively, to demonstrate how the backward propagation across layers aligns kernels with the target. An alternative field-theoretic formulation shows that kernel adaptation of the Bayesian posterior at finite-width results from fluctuations in the prior: larger fluctuations correspond to a more flexible network prior and thus enable stronger adaptation to data. We thus find a bridge between the classical edge-of-chaos NNGP theory and feature learning, exposing an intricate interplay between criticality, response functions, and feature scale.} }
Endnote
%0 Conference Paper %T Critical feature learning in deep neural networks %A Kirsten Fischer %A Javed Lindner %A David Dahmen %A Zohar Ringel %A Michael Krämer %A Moritz Helias %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-fischer24a %I PMLR %P 13660--13690 %U https://proceedings.mlr.press/v235/fischer24a.html %V 235 %X A key property of neural networks driving their success is their ability to learn features from data. Understanding feature learning from a theoretical viewpoint is an emerging field with many open questions. In this work we capture finite-width effects with a systematic theory of network kernels in deep non-linear neural networks. We show that the Bayesian prior of the network can be written in closed form as a superposition of Gaussian processes, whose kernels are distributed with a variance that depends inversely on the network width $N$. A large deviation approach, which is exact in the proportional limit for the number of data points $P=\alpha N\to\infty$, yields a pair of forward-backward equations for the maximum a posteriori kernels in all layers at once. We study their solutions perturbatively, to demonstrate how the backward propagation across layers aligns kernels with the target. An alternative field-theoretic formulation shows that kernel adaptation of the Bayesian posterior at finite-width results from fluctuations in the prior: larger fluctuations correspond to a more flexible network prior and thus enable stronger adaptation to data. We thus find a bridge between the classical edge-of-chaos NNGP theory and feature learning, exposing an intricate interplay between criticality, response functions, and feature scale.
APA
Fischer, K., Lindner, J., Dahmen, D., Ringel, Z., Krämer, M. & Helias, M.. (2024). Critical feature learning in deep neural networks. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:13660-13690 Available from https://proceedings.mlr.press/v235/fischer24a.html.

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