Sparsity in long-time control of neural ODEs

We consider the neural ODE and optimal control perspective of supervised learning, with ℓ1-control penalties, where rather than only minimizing a final cost (the empirical risk) for the state, we integrate this cost over the entire time horizon. We prove that any optimal control (for this cost) vani...

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Bibliographic Details
Published inSystems & control letters Vol. 172; p. 105452
Main Authors Esteve-Yagüe, Carlos, Geshkovski, Borjan
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.02.2023
Subjects
Learning
Neural ODEs
Optimal Control
Sparsity
Neural ODEs
Learning
Sparsity
Optimal Control
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Summary:We consider the neural ODE and optimal control perspective of supervised learning, with ℓ1-control penalties, where rather than only minimizing a final cost (the empirical risk) for the state, we integrate this cost over the entire time horizon. We prove that any optimal control (for this cost) vanishes beyond some positive stopping time. When seen in the discrete-time context, this result entails an ordered sparsity pattern for the parameters of the associated residual neural network: ordered in the sense that these parameters are all 0 beyond a certain layer. Furthermore, we provide a polynomial stability estimate for the empirical risk with respect to the time horizon. This can be seen as a turnpike property, for nonsmooth dynamics and functionals with ℓ1 penalties, and without any smallness assumptions on the data, both of which are new in the literature.
ISSN:0167-6911
1872-7956
DOI:10.1016/j.sysconle.2022.105452