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

TL;DR

This paper establishes near-optimal statistical query lower bounds for smoothed agnostic learning of Boolean halfspaces, demonstrating the complexity of such learning under uniform marginals.

Contribution

It provides tight bounds on the runtime, sample complexity, and SQ lower bounds for smoothed agnostic learning of halfspaces, extending prior work to the discrete setting.

Findings

Polynomial regression achieves near-quadratic runtime in n with respect to epsilon and sigma.

The SQ lower bound nearly matches the polynomial regression upper bound, indicating tight complexity.

Results extend understanding of learning complexity from Gaussian to discrete uniform marginals.

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 $\tilde{O}(n^{O(\log(1/\varepsilon)/\sigma)})$, and prove a nearly matching Statistical Query complexity lower bound of $n^{\Omega(\log(1+\sigma/\varepsilon^2)/\sigma)}$. This complements the recent work of~\cite{DK26}, which established analogous bounds in the continuous setting under Gaussian marginals.