Interplay between depth and width for interpolation in neural ODEs

• URL: http://arxiv.org/abs/2401.09902v3
• Date: Tue, 6 Feb 2024 17:05:48 GMT
• Title: Interplay between depth and width for interpolation in neural ODEs
• Authors: Antonio \'Alvarez-L\'opez, Arselane Hadj Slimane, Enrique Zuazua
• Abstract summary: We examine the interplay between their width $p$ and number of layer transitions $L$. In the high-dimensional setting, we demonstrate that $p=O(N)$ neurons are likely sufficient to achieve exact control.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Neural ordinary differential equations (neural ODEs) have emerged as a natural tool for supervised learning from a control perspective, yet a complete understanding of their optimal architecture remains elusive. In this work, we examine the interplay between their width $p$ and number of layer transitions $L$ (effectively the depth $L+1$). Specifically, we assess the model expressivity in terms of its capacity to interpolate either a finite dataset $D$ comprising $N$ pairs of points or two probability measures in $\mathbb{R}^d$ within a Wasserstein error margin $\varepsilon>0$. Our findings reveal a balancing trade-off between $p$ and $L$, with $L$ scaling as $O(1+N/p)$ for dataset interpolation, and $L=O\left(1+(p\varepsilon^d)^{-1}\right)$ for measure interpolation. In the autonomous case, where $L=0$, a separate study is required, which we undertake focusing on dataset interpolation. We address the relaxed problem of $\varepsilon$-approximate controllability and establish an error decay of $\varepsilon\sim O(\log(p)p^{-1/d})$. This decay rate is a consequence of applying a universal approximation theorem to a custom-built Lipschitz vector field that interpolates $D$. In the high-dimensional setting, we further demonstrate that $p=O(N)$ neurons are likely sufficient to achieve exact control.

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