Neural Networks Learn Generic Multi-Index Models Near Information-Theoretic Limit

• URL: http://arxiv.org/abs/2511.15120v1
• Date: Wed, 19 Nov 2025 04:46:47 GMT
• Title: Neural Networks Learn Generic Multi-Index Models Near Information-Theoretic Limit
• Authors: Bohan Zhang, Zihao Wang, Hengyu Fu, Jason D. Lee,
• Abstract summary: We study the gradient descent learning of a general Gaussian Multi-index model $f(boldsymbolx)=g(boldsymbolUboldsymbolx)$ with hidden subspace $boldsymbolUin mathbbRrtimes d$.<n>We prove that under generic non-degenerate assumptions on the link function, a standard two-layer neural network trained via layer-wise gradient descent can agnostically learn the target with $o_d(1)$ test error.
• Score: 66.20349460098275
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In deep learning, a central issue is to understand how neural networks efficiently learn high-dimensional features. To this end, we explore the gradient descent learning of a general Gaussian Multi-index model $f(\boldsymbol{x})=g(\boldsymbol{U}\boldsymbol{x})$ with hidden subspace $\boldsymbol{U}\in \mathbb{R}^{r\times d}$, which is the canonical setup to study representation learning. We prove that under generic non-degenerate assumptions on the link function, a standard two-layer neural network trained via layer-wise gradient descent can agnostically learn the target with $o_d(1)$ test error using $\widetilde{\mathcal{O}}(d)$ samples and $\widetilde{\mathcal{O}}(d^2)$ time. The sample and time complexity both align with the information-theoretic limit up to leading order and are therefore optimal. During the first stage of gradient descent learning, the proof proceeds via showing that the inner weights can perform a power-iteration process. This process implicitly mimics a spectral start for the whole span of the hidden subspace and eventually eliminates finite-sample noise and recovers this span. It surprisingly indicates that optimal results can only be achieved if the first layer is trained for more than $\mathcal{O}(1)$ steps. This work demonstrates the ability of neural networks to effectively learn hierarchical functions with respect to both sample and time efficiency.

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