The Finite–Time Turnpike Property in Machine Learning
Martin Gugat · Preprints.org · 2024
The finite-time turnpike property describes the situation in an optimal control problem where an optimal trajectory reaches the desired state before the end of the time interval and remains there. We consider a machine learning problem with a neural ordinary differential equation that can be seen as a homogenization of a deep ResNet. We show that with appropriate scaling of the quadratic control cost and the non-smooth tracking term the optimal control problem has the finite-time turnpike property, that is the desired state is reached in the interior of the time interval and the optimal state remains there until the terminal time $T$. This property is useful to achieve a compromise between the depth of the network and the size of the optimal system parameters which we hope will be useful to determine optimal depths for neural network architectures in the future.