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

Puqian Wang, Nikos Zarifis, Ilias Diakonikolas, Jelena Diakonikolas

Keywords: single index models, agnostic learning, adversarial label noise

TL;DR: We provide the first sample and computationally efficient learner for Gaussian Single-Index Models, for link functions with a constant information exponent, in the agnostic model.

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.

Primary Area: Learning theory

Submission Number: 14047