A Near-optimal SQ Lower Bound for Smoothed Agnostic Learning of Boolean Halfspaces

Abstract

We study the complexity of smoothed agnostic learning of halfspaces on $\{\pm 1{\}}^n$ under uniform marginals in the model of~\cite{KM25}, where each input coordinate is independently flipped with probability $\sigma \in (0,1/2)$. We show that $L^1$ polynomial regression achieves runtime and sample complexity $\tilde{O}(n^{O(\log (1/\epsilon )/\sigma )})$, and prove a nearly matching Statistical Query complexity lower bound of $n^{\Omega (\log (1+\sigma /{\epsilon }^2)/\sigma )}$. This complements the recent work of~\cite{DK26}, which established analogous bounds in the continuous setting under Gaussian marginals.

Fixed several typos and minor proof issues