Learning Hierarchical Polynomials of Multiple Nonlinear Features with Three-Layer Networks
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arXiv
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| Format: | Preprint |
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2024
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| author | Fu, Hengyu Wang, Zihao Nichani, Eshaan Lee, Jason D. |
| author_facet | Fu, Hengyu Wang, Zihao Nichani, Eshaan Lee, Jason D. |
| contents | In deep learning theory, a critical question is to understand how neural networks learn hierarchical features. In this work, we study the learning of hierarchical polynomials of \textit{multiple nonlinear features} using three-layer neural networks. We examine a broad class of functions of the form $f^{\star}=g^{\star}\circ \bp$, where $\bp:\mathbb{R}^{d} \rightarrow \mathbb{R}^{r}$ represents multiple quadratic features with $r \ll d$ and $g^{\star}:\mathbb{R}^{r}\rightarrow \mathbb{R}$ is a polynomial of degree $p$. This can be viewed as a nonlinear generalization of the multi-index model \citep{damian2022neural}, and also an expansion upon previous work that focused only on a single nonlinear feature, i.e. $r = 1$ \citep{nichani2023provable,wang2023learning}.
Our primary contribution shows that a three-layer neural network trained via layerwise gradient descent suffices for
\begin{itemize}\item complete recovery of the space spanned by the nonlinear features
\item efficient learning of the target function $f^{\star}=g^{\star}\circ \bp$ or transfer learning of $f=g\circ \bp$ with a different link function
\end{itemize} within $\widetilde{\cO}(d^4)$ samples and polynomial time. For such hierarchical targets, our result substantially improves the sample complexity $Θ(d^{2p})$ of the kernel methods, demonstrating the power of efficient feature learning. It is important to highlight that{ our results leverage novel techniques and thus manage to go beyond all prior settings} such as single-index and multi-index models as well as models depending just on one nonlinear feature, contributing to a more comprehensive understanding of feature learning in deep learning. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17201 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Learning Hierarchical Polynomials of Multiple Nonlinear Features with Three-Layer Networks Fu, Hengyu Wang, Zihao Nichani, Eshaan Lee, Jason D. Machine Learning Artificial Intelligence Statistics Theory In deep learning theory, a critical question is to understand how neural networks learn hierarchical features. In this work, we study the learning of hierarchical polynomials of \textit{multiple nonlinear features} using three-layer neural networks. We examine a broad class of functions of the form $f^{\star}=g^{\star}\circ \bp$, where $\bp:\mathbb{R}^{d} \rightarrow \mathbb{R}^{r}$ represents multiple quadratic features with $r \ll d$ and $g^{\star}:\mathbb{R}^{r}\rightarrow \mathbb{R}$ is a polynomial of degree $p$. This can be viewed as a nonlinear generalization of the multi-index model \citep{damian2022neural}, and also an expansion upon previous work that focused only on a single nonlinear feature, i.e. $r = 1$ \citep{nichani2023provable,wang2023learning}. Our primary contribution shows that a three-layer neural network trained via layerwise gradient descent suffices for \begin{itemize}\item complete recovery of the space spanned by the nonlinear features \item efficient learning of the target function $f^{\star}=g^{\star}\circ \bp$ or transfer learning of $f=g\circ \bp$ with a different link function \end{itemize} within $\widetilde{\cO}(d^4)$ samples and polynomial time. For such hierarchical targets, our result substantially improves the sample complexity $Θ(d^{2p})$ of the kernel methods, demonstrating the power of efficient feature learning. It is important to highlight that{ our results leverage novel techniques and thus manage to go beyond all prior settings} such as single-index and multi-index models as well as models depending just on one nonlinear feature, contributing to a more comprehensive understanding of feature learning in deep learning. |
| title | Learning Hierarchical Polynomials of Multiple Nonlinear Features with Three-Layer Networks |
| topic | Machine Learning Artificial Intelligence Statistics Theory |
| url | https://arxiv.org/abs/2411.17201 |