Sample and Computationally Efficient Robust Learning of Gaussian Single-Index Models

TL;DR

This paper introduces a sample and computationally efficient method for robustly learning single-index models under Gaussian noise, achieving near-optimal error bounds even with adversarial label noise.

Contribution

It presents the first agnostic proper learner for SIMs that nearly matches CSQ lower bounds with minimal assumptions on the link function.

Findings

Achieves $L^2_2$-error of $O(OPT)+ ext{epsilon}$

Sample complexity nearly matches lower bounds

Works under adversarial label noise with minimal assumptions

Abstract

A single-index model (SIM) is a function of the form $\sigma(\mathbf{w}^{\ast} \cdot \mathbf{x})$, where $\sigma: \mathbb{R} \to \mathbb{R}$ is a known link function and $\mathbf{w}^{\ast}$ is a hidden unit vector. We study the task of learning SIMs in the agnostic (a.k.a. adversarial label noise) model with respect to the $L^2_2$-loss under the Gaussian distribution. Our main result is a sample and computationally efficient agnostic proper learner that attains $L^2_2$-error of $O(\mathrm{OPT})+\epsilon$, where $\mathrm{OPT}$ is the optimal loss. The sample complexity of our algorithm is $\tilde{O}(d^{\lceil k^{\ast}/2\rceil}+d/\epsilon)$, where $k^{\ast}$ is the information-exponent of $\sigma$ corresponding to the degree of its first non-zero Hermite coefficient. This sample bound nearly matches known CSQ lower bounds, even in the realizable setting. Prior algorithmic work in this setting had focused on learning in the realizable case or in the presence of semi-random noise. Prior computationally efficient robust learners required significantly stronger assumptions on the link function.