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

Selina Drews () and Michael Kohler ()
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Selina Drews: Technische Universität Darmstadt
Michael Kohler: Technische Universität Darmstadt

Annals of the Institute of Statistical Mathematics, 2024, vol. 76, issue 3, No 1, 391 pages

Abstract: 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.

Keywords: Neural networks; Nonparametric regression; Over-parametrization; Universal consistency (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10463-024-00898-6

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