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Bayesian inference with finitely wide neural networks.

Chi-Ken Lu1

  • 1Department of Mathematics and Computer Science, Rutgers University, Newark, New Jersey 07102, USA.

Physical Review. E
|August 16, 2023
PubMed
Summary

We propose a non-Gaussian distribution to model finite neural network outputs, enabling accurate Bayesian regression. This addresses deviations from Gaussianity in finite-width networks for improved machine learning inference.

Area of Science:

  • Machine Learning
  • Bayesian Inference
  • Deep Learning

Background:

  • Machine learning practitioners often model neural networks as Gaussian processes for analytical benefits.
  • Finite network widths introduce deviations from ideal Gaussianity, complicating inference.
  • Existing methods struggle with accurate modeling of these finite-width effects.

Purpose of the Study:

  • To develop a non-Gaussian distribution for modeling outputs of finite-width random neural networks.
  • To enable accurate Bayesian regression by deriving non-Gaussian posterior distributions.
  • To investigate non-Gaussianity in deep neural networks within a weight space Gaussian process framework.

Main Methods:

  • Utilizing multivariate Edgeworth expansion to derive a differential form for non-Gaussian distributions.

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  • Deriving marginal and conditional properties of the proposed non-Gaussian distribution.
  • Analyzing non-Gaussianity in deep neural networks using marginal kernels and small parameters.

    • Main Results:

    • A novel non-Gaussian distribution is proposed for finite neural network outputs.
    • The method allows for derivation of non-Gaussian posterior distributions in Bayesian regression.
    • Non-Gaussianity in deep Gaussian processes is characterized through specific parameters.

    Conclusions:

    • The proposed non-Gaussian approach enhances Bayesian inference for finite-width neural networks.
    • This work provides a more realistic model for neural network outputs beyond Gaussian assumptions.
    • The findings offer improved analytical tractability for complex deep learning models.