Critical feature learning in deep neural networks
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913354029727744 |
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| author | Fischer, Kirsten Lindner, Javed Dahmen, David Ringel, Zohar Krämer, Michael Helias, Moritz |
| author_facet | Fischer, Kirsten Lindner, Javed Dahmen, David Ringel, Zohar Krämer, Michael Helias, Moritz |
| contents | 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 $P = αN \rightarrow \infty$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10761 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Critical feature learning in deep neural networks Fischer, Kirsten Lindner, Javed Dahmen, David Ringel, Zohar Krämer, Michael Helias, Moritz Disordered Systems and Neural Networks 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 $P = αN \rightarrow \infty$, 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. |
| title | Critical feature learning in deep neural networks |
| topic | Disordered Systems and Neural Networks |
| url | https://arxiv.org/abs/2405.10761 |