Learning in the Presence of Low-dimensional Structure: A Spiked Random Matrix Perspective

Part of Advances in Neural Information Processing Systems 36 (NeurIPS 2023) Main Conference Track

Bibtex Paper


Jimmy Ba, Murat A. Erdogdu, Taiji Suzuki, Zhichao Wang, Denny Wu


We consider the learning of a single-index target function $f_*: \mathbb{R}^d\to\mathbb{R}$ under spiked covariance data: $$f_*(\boldsymbol{x}) = \textstyle\sigma_*(\frac{1}{\sqrt{1+\theta}}\langle\boldsymbol{x},\boldsymbol{\mu}\rangle), ~~ \boldsymbol{x}\overset{\small\mathrm{i.i.d.}}{\sim}\mathcal{N}(0,\boldsymbol{I_d} + \theta\boldsymbol{\mu}\boldsymbol{\mu}^\top), ~~ \theta\asymp d^{\beta} \text{ for } \beta\in[0,1), $$ where the link function $\sigma_*:\mathbb{R}\to\mathbb{R}$ is a degree-$p$ polynomial with information exponent $k$ (defined as the lowest degree in the Hermite expansion of $\sigma_*$), and it depends on the projection of input $\boldsymbol{x}$ onto the spike (signal) direction $\boldsymbol{\mu}\in\mathbb{R}^d$. In the proportional asymptotic limit where the number of training examples $n$ and the dimensionality $d$ jointly diverge: $n,d\to\infty, n/d\to\psi\in(0,\infty)$, we ask the following question: how large should the spike magnitude $\theta$ (i.e., the strength of the low-dimensional component) be, in order for $(i)$ kernel methods, $(ii)$ neural networks optimized by gradient descent, to learn $f_*$? We show that for kernel ridge regression, $\beta\ge 1-\frac{1}{p}$ is both sufficient and necessary. Whereas for two-layer neural networks trained with gradient descent, $\beta>1-\frac{1}{k}$ suffices. Our results demonstrate that both kernel methods and neural networks benefit from low-dimensional structures in the data. Further, since $k\le p$ by definition, neural networks can adapt to such structures more effectively.