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Kernel shape renormalization explains output-output correlations in finite Bayesian one-hidden-layer networks.

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Summary
This summary is machine-generated.

Finite-width neural networks show output correlations absent in infinite-width models. This study explains these correlations using kernel shape renormalization in Bayesian deep learning, validated by numerical experiments.

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Area of Science:

  • Machine Learning
  • Deep Learning Theory
  • Bayesian Inference

Background:

  • Finite-width neural networks exhibit complex output-output correlations, particularly with multiple readout neurons.
  • These correlations disappear in the infinite-width limit, a phenomenon known as lazy-training.
  • Understanding these finite-width effects is crucial for a complete theory of deep learning.

Purpose of the Study:

  • To rationalize the empirical observation of non-trivial output-output correlations in finite-width neural networks.
  • To leverage the proportional limit of Bayesian deep learning to explain these correlations.
  • To validate the theoretical framework with numerical experiments.

Main Methods:

  • Utilizing the proportional limit (P/N finite) of Bayesian deep learning, where P is the training set size and N is network width.
  • Developing a kernel shape renormalization theory for the Neural Network Gaussian Process (NNGP) kernel.
  • Conducting numerical experiments to assess generalization and quantify output-output correlations.

Main Results:

  • Output-output correlations in finite networks are explained by a renormalized NNGP kernel.
  • The proportional limit provides a theoretical framework to understand these finite-width phenomena.
  • Numerical experiments quantitatively match theoretical predictions for correlations.

Conclusions:

  • Kernel shape renormalization is key to understanding correlations in finite Bayesian one-hidden-layer networks.
  • The Bayesian deep learning framework in the proportional limit accurately captures finite-width effects.
  • This work bridges the gap between infinite-width theory and finite-width empirical observations.