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 $σ\in (0, {1}/{2})$. We show that $L^1$ polynomial regression achieves runtime and sample complexity $\tilde{O}(n^{O(\log(1/\varepsilon)/σ)})$, and prove a nearly matching Statistical Query complexity lower bound of $n^{Ω(\log(1+σ/\varepsilon^2)/σ)}$. This complements the recent work of~\cite{DK26}, which established analogous bounds in the continuous setting under Gaussian marginals.