  We study the problem of PAC learning a single neuron in the presence of
Massart noise. Specifically, for a known activation function $f: \mathbb{R} \to
\mathbb{R}$, the learner is given access to labeled examples $(\mathbf{x}, y)
\in \mathbb{R}^d \times \mathbb{R}$, where the marginal distribution of
$\mathbf{x}$ is arbitrary and the corresponding label $y$ is a Massart
corruption of $f(\langle \mathbf{w}, \mathbf{x} \rangle)$. The goal of the
learner is to output a hypothesis $h: \mathbb{R}^d \to \mathbb{R}$ with small
squared loss. For a range of activation functions, including ReLUs, we
establish super-polynomial Statistical Query (SQ) lower bounds for this
learning problem. In more detail, we prove that no efficient SQ algorithm can
approximate the optimal error within any constant factor. Our main technical
contribution is a novel SQ-hard construction for learning $\{ \pm 1\}$-weight
Massart halfspaces on the Boolean hypercube that is interesting on its own
right.
