On Single-Index Models beyond Gaussian Data

Sparse high-dimensional functions have arisen as a rich framework to study the behavior of gradient-descent methods using shallow neural networks, and showcasing its ability to perform feature learning beyond linear models. Amongst those functions, the simplest are single-index models $f(x) = \phi( x \cdot \theta^*)$, where the labels are generated by an arbitrary non-linear link function $\phi$ of an unknown one-dimensional projection $\theta^*$ of the input data. By focusing on Gaussian data, several recent works have built a remarkable picture, where the so-called information exponent (related to the regularity of the link function) controls the required sample complexity. In essence, these tools exploit the stability and spherical symmetry of Gaussian distributions.In this work, we explore extensions of this picture beyond the Gaussian setting, where both stability or symmetry might be violated. Focusing on the planted setting where $\phi$ is known, our main results establish that Stochastic Gradient Descent recovers the unknown direction $\theta^*$ with constant probability in the high-dimensional regime, under mild assumptions that significantly extend ~[Yehudai and Shamir,20].