On the Turnpike to Design of Deep Neural Nets: Explicit Depth Bounds

• URL: http://arxiv.org/abs/2101.03000v1
• Date: Fri, 8 Jan 2021 13:23:37 GMT
• Title: On the Turnpike to Design of Deep Neural Nets: Explicit Depth Bounds
• Authors: Timm Faulwasser and Arne-Jens Hempel and Stefan Streif
• Abstract summary: This paper attempts a quantifiable answer to the question of how many layers should be considered in a Deep Neural Networks (DNN) The underlying assumption is that the number of neurons per layer -- i.e., the width of the DNN -- is kept constant. We prove explicit bounds on the required depths of DNNs based on reachability of assumptions and a dissipativity-inducing choice of the regularization terms in the training problem.
• Score: 0.0
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: 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 estimates can provide non-conservative depth bounds.

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