Sparsity in long-time control of neural ODEs

TL;DR

This paper studies neural ODEs with  penalties, proving optimal controls vanish after a certain time, leading to ordered sparsity in neural network parameters and establishing a turnpike property for long-time control.

Contribution

It introduces a novel analysis of -penalized neural ODE control, showing sparsity patterns and stability estimates without small data assumptions.

Findings

Optimal control vanishes beyond a positive stopping time.

Parameters exhibit ordered sparsity beyond a certain layer.

Empirical risk stability follows a polynomial estimate over time.

Abstract

We consider the neural ODE and optimal control perspective of supervised learning, with $\ell^1$-control penalties, where rather than only minimizing a final cost (the \emph{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 \emph{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 \emph{turnpike property}, for nonsmooth dynamics and functionals with $\ell^1$-penalties, and without any smallness assumptions on the data, both of which are new in the literature.