Neural network learns low-dimensional polynomials with SGD near the information-theoretic limit

• URL: http://arxiv.org/abs/2406.01581v2
• Date: Sun, 22 Dec 2024 09:59:09 GMT
• Title: Neural network learns low-dimensional polynomials with SGD near the information-theoretic limit
• Authors: Jason D. Lee, Kazusato Oko, Taiji Suzuki, Denny Wu,
• Abstract summary: We study the problem of gradient descent learning of a single-index target function $f_*(boldsymbolx) = textstylesigma_*left(langleboldsymbolx,boldsymbolthetarangleright)$ We prove that a two-layer neural network optimized by an SGD-based algorithm learns $f_*$ with a complexity that is not governed by information exponents.
• Score: 75.4661041626338
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• Abstract: We study the problem of gradient descent learning of a single-index target function $f_*(\boldsymbol{x}) = \textstyle\sigma_*\left(\langle\boldsymbol{x},\boldsymbol{\theta}\rangle\right)$ under isotropic Gaussian data in $\mathbb{R}^d$, where the unknown link function $\sigma_*:\mathbb{R}\to\mathbb{R}$ has information exponent $p$ (defined as the lowest degree in the Hermite expansion). Prior works showed that gradient-based training of neural networks can learn this target with $n\gtrsim d^{\Theta(p)}$ samples, and such complexity is predicted to be necessary by the correlational statistical query lower bound. Surprisingly, we prove that a two-layer neural network optimized by an SGD-based algorithm (on the squared loss) learns $f_*$ with a complexity that is not governed by the information exponent. Specifically, for arbitrary polynomial single-index models, we establish a sample and runtime complexity of $n \simeq T = \Theta(d\!\cdot\! \mathrm{polylog} d)$, where $\Theta(\cdot)$ hides a constant only depending on the degree of $\sigma_*$; this dimension dependence matches the information theoretic limit up to polylogarithmic factors. More generally, we show that $n\gtrsim d^{(p_*-1)\vee 1}$ samples are sufficient to achieve low generalization error, where $p_* \le p$ is the \textit{generative exponent} of the link function. Core to our analysis is the reuse of minibatch in the gradient computation, which gives rise to higher-order information beyond correlational queries.

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