Interplay between depth and width for interpolation in neural ODEs
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arXiv
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| Format: | Preprint |
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2024
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| author | Álvarez-López, Antonio Slimane, Arselane Hadj Zuazua, Enrique |
| author_facet | Álvarez-López, Antonio Slimane, Arselane Hadj Zuazua, Enrique |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09902 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Interplay between depth and width for interpolation in neural ODEs Álvarez-López, Antonio Slimane, Arselane Hadj Zuazua, Enrique Optimization and Control Machine Learning 34H05, 68T07, 93B05 (Primary) 35Q49 (Secondary) 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. |
| title | Interplay between depth and width for interpolation in neural ODEs |
| topic | Optimization and Control Machine Learning 34H05, 68T07, 93B05 (Primary) 35Q49 (Secondary) |
| url | https://arxiv.org/abs/2401.09902 |