On the turnpike to design of deep neural networks : explicit depth bounds

It is well-known that the training of Deep Neural Networks (DNN) can be formalized in the language of optimal control. In this context, this paper leverages classical turnpike properties of optimal control problems to attempt a quantifiable answer to the question of how many layers should be considered in a DNN. The underlying assumption is that the number of neurons per layer—i.e., the width of the DNN—is kept constant. Pursuing a different route than the classical analysis of approximation properties of sigmoidal functions, we prove explicit bounds on the required depths of DNNs based on asymptotic reachability assumptions and a dissipativity-inducing choice of the regularization terms in the training problem. Numerical results obtained for the two spiral task data set for classification indicate that the proposed constructive estimates can provide non-conservative depth bounds.

Subjects

Artificial neural networks

Deep learning

Dissipativity

Machine learning

Turnpike properties

MLE@TUHH

DDC Class

003: Systems Theory

006: Special computer methods

519: Applied Mathematics, Probabilities

Funding(s)

Projekt DEAL

Publication version

publishedVersion

Lizenz

https://creativecommons.org/licenses/by/4.0/

Name

1-s2.0-S2468601824000518-main.pdf

Type

Main Article

Size

753.78 KB

Format

Adobe PDF

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On the turnpike to design of deep neural networks : explicit depth bounds