#### Near-Optimal SQ Lower Bounds for Agnostically Learning Halfspaces and ReLUs under Gaussian Marginals

Part of Advances in Neural Information Processing Systems 33 (NeurIPS 2020)

#### Authors

*Ilias Diakonikolas, Daniel Kane, Nikos Zarifis*

#### Abstract

We study the fundamental problems of agnostically learning halfspaces and ReLUs under Gaussian marginals.
In the former problem, given labeled examples $(\bx, y)$ from an unknown distribution on $\R^d \times \{ \pm 1\}$,
whose marginal distribution on $\bx$ is the standard Gaussian and the labels $y$ can be arbitrary,
the goal is to output a hypothesis with 0-1 loss $\opt+\eps$, where $\opt$ is the 0-1 loss of the best-fitting halfspace.
In the latter problem, given labeled examples $(\bx, y)$ from an unknown distribution on $\R^d \times \R$,
whose marginal distribution on $\bx$ is the standard Gaussian and the labels $y$ can be arbitrary,
the goal is to output a hypothesis with square loss $\opt+\eps$, where $\opt$ is the square loss of the best-fitting ReLU.
We prove Statistical Query (SQ) lower bounds of $d^{\poly(1/\eps)}$ for both of these problems.
Our SQ lower bounds provide strong evidence that current upper bounds for these tasks are essentially best possible.