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

Understand

We study the fundamental problems of agnostically learning halfspaces and ReLUs under Gaussian marginals.

• In the former problem, given labeled examples $(\mathbf{x}, y)$ from an unknown distribution on $\mathbb{R}^d \times \{ \pm 1\}$, whose marginal distribution on $\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with 0-1 loss $\mathrm{OPT}+\epsilon$, where $\mathrm{OPT}$ is the 0-1 loss of the best-fitting halfspace.
• In the latter problem, given labeled examples $(\mathbf{x}, y)$ from an unknown distribution on $\mathbb{R}^d \times \mathbb{R}$, whose marginal distribution on $\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with square loss $\mathrm{OPT}+\epsilon$, where $\mathrm{OPT}$ is the square loss of the best-fitting ReLU.
• We prove Statistical Query (SQ) lower bounds of $d^{\mathrm{poly}(1/\epsilon)}$ for both of these problems.