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It is well-known that the training of Deep Neural Networks (DNN) can beformalized in the language of optimal control. In this context, this paperleverages classical turnpike properties of optimal control problems to attempta quantifiable answer to the question of how many layers should be consideredin 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 thanthe classical analysis of approximation properties of sigmoidal functions, weprove explicit bounds on the required depths of DNNs based on asymptoticreachability assumptions and a dissipativity-inducing choice of theregularization terms in the training problem. Numerical results obtained forthe two spiral task data set for classification indicate that the proposedestimates can provide non-conservative depth bounds.

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