What is it about?
We show that in some situations non-smooth objective functions for optimal control problems with systems that are exactly controllable generate solutions that have the finite-time turnpike property, which means that the desired state is reached exactly in finite time.
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Why is it important?
In machine learning, the training relies on appropriately chosen loss functions. In some situations non-smooth objective functions yields optimal states that have the finite-time turnpike property. This structural properties can be advantageous in certain situations, since it can be used to control the numbers of layers in a deep learning process. The aim is to find an optimal number of layers in a well-defined sense. In the continuous-time framework, this corresponds to a finite time after which the network is inactive. The prize to pay for this situation is the non-smoothness of the loss function.
Perspectives
The choice of the objective functional plays an important role to control learning processes. Therefore it is important to understand how the properties of the objective function influence the properties of the generated optimal state. We think that particular in the context of machine learning, there is a demand for more analytical results that clarify this relation.
Martin Gugat
Friedrich-Alexander-Universitat Erlangen-Nurnberg
Read the Original
This page is a summary of: The Finite-Time Turnpike Property in Machine Learning, Machines, October 2024, MDPI AG,
DOI: 10.3390/machines12100705.
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Resources
Link to the paper in Machines
Here is a link to the paper.
The Exponential Turnpike Property for Neural Differential Equations
This is a paper from MMAR 2025 - 2025 the International Conference on Methods and Models in Automation and Robotics
Optimal control of neural differential equations:The turnpike property
Consider a system that is governed by a neural differential equation.Such systems model deep neural networks with continuous time. For systems of this type, we study an optimal control problem where the objective function consists of a tracking term that is given by the sum of a squared weighted $H^1$-normand a control cost that is given by a squared $L^2$-norm. We prove that a simultaneous exact controllability property of the systemimplies a turnpike property for the optimal statethat depends on the weight in this differentiable objective function.We also show the finite-time turnpike propertyfor the non-differentiable objective function where the tracking term is not squared.Numerical results illustrate our findings.
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