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

Tim Sinen

cs.LGMay 4, 2026

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

Authors: Tim Sinen

Organizations: University of Bonn, Germany

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.

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May 26, 2026 · cs.DSJ/K move · Enter open · S save

Proper Agnostic Learning of Functions of Halfspaces under Gaussian Marginals

Sergei Tikhonov, Arsen Vasilyan

UT Austin

We study the problem of computationally efficient proper agnostic learning of multidimensional concept classes under the Gaussian distribution. In this setting, given i.i.d. labeled samples from an unknown distribution over

\mathbb{R}^d \times \{\pm 1\}

whose marginal on

\mathbb{R}^d

is Gaussian, the goal is to output a hypothesis from a target class

\mathcal{F}

whose 0-1 loss is within

ε

of that of the best classifier in

\mathcal{F}

. We give the first efficient proper agnostic learning algorithm for arbitrary Boolean functions of

K

halfspaces under Gaussian marginals. Our algorithm runs in time

d^{O(K^2 \log(1/ε)/ε^2)} + (K/ε)^{O(K^3/ε^{2.5})}

. Prior to our work, the only known algorithm for

K \geq 2

was brute-force search, with run-time exponential in

d

. Moreover, the dependence of our run-time on the dimension

d

matches that of the best known improper learning algorithm, namely

d^{\widetilde{O}(K^2/ε^2)}

. For the special case of a single halfspace (

K=1

), the best previous run-time was

d^{O(1/ε^4)} + (1/ε)^{O(1/ε^6)}

. Our algorithm improves this to

d^{O(1/ε^2)} + (1/ε)^{O(1/ε^{2.5})}

. Once again, the dependence on

d

matches that of the best known improper algorithm, namely

d^{O(1/ε^2)}

. Furthermore, the dependence of our run-time on the dimension

d

is essentially optimal in the statistical query model.

Optimal Sample Complexity Binary Classification Distributional Learning 3D Gaussian Splatting

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Abstract

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