On the Turnpike to Design of Deep Neural Nets: Explicit Depth Bounds
This work provides a theoretical framework to determine the optimal depth of deep neural networks for practitioners, addressing a fundamental design challenge in network architecture.
This paper formalizes DNN training as an optimal control problem and uses turnpike properties to quantify the required number of layers (depth) for a constant-width DNN. By leveraging asymptotic reachability and dissipativity-inducing regularization, the authors derive explicit depth bounds, which are shown to be non-conservative on the two-spiral classification task.
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.