The Finite-Time Turnpike Property in Machine Learning

Abstract

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 the 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 within the time interval and the optimal state remains there until the terminal time T. The time t0 where the optimal trajectories reach the desired state can serve as an additional design parameter. Since ResNets can be viewed as discretizations of neural odes, the choice of t0 corresponds to the choice of the number of layers; that is, the depth of the neural network. The choice of t0 allows us 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 the optimal depths for neural network architectures in the future.

Keywords

• ResNet
• neural ODE
• finite-time turnpike property
• turnpike phenomenon
• non-smooth tracking term
• machine learning