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On the universal consistency of an over-parametrized deep neural network estimate learned by gradient descent

Author

Listed:
  • Selina Drews

    (Technische Universität Darmstadt)

  • Michael Kohler

    (Technische Universität Darmstadt)

Abstract

Estimation of a multivariate regression function from independent and identically distributed data is considered. An estimate is defined which fits a deep neural network consisting of a large number of fully connected neural networks, which are computed in parallel, via gradient descent to the data. The estimate is over-parametrized in the sense that the number of its parameters is much larger than the sample size. It is shown that with a suitable random initialization of the network, a sufficiently small gradient descent step size, and a number of gradient descent steps that slightly exceed the reciprocal of this step size, the estimate is universally consistent. This means that the expected $$L_2$$ L 2 error converges to zero for all distributions of the data where the response variable is square integrable.

Suggested Citation

  • Selina Drews & Michael Kohler, 2024. "On the universal consistency of an over-parametrized deep neural network estimate learned by gradient descent," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 76(3), pages 361-391, June.
    • Handle: RePEc:spr:aistmt:v:76:y:2024:i:3:d:10.1007_s10463-024-00898-6
    DOI: 10.1007/s10463-024-00898-6
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    References listed on IDEAS

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    1. Langer, Sophie, 2021. "Analysis of the rate of convergence of fully connected deep neural network regression estimates with smooth activation function," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
    2. Langer, Sophie, 2021. "Approximating smooth functions by deep neural networks with sigmoid activation function," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
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