Robust learning of halfspaces under log-concave marginals

• URL: http://arxiv.org/abs/2505.13708v1
• Date: Mon, 19 May 2025 20:12:16 GMT
• Title: Robust learning of halfspaces under log-concave marginals
• Authors: Jane Lange, Arsen Vasilyan,
• Abstract summary: We give an algorithm that learns linear threshold functions and returns a classifier with boundary volume $O(r+varepsilon)$ at radius perturbation $r$.<n>The time and sample complexity of $dtildeO (1/varepsilon2)$ matches the complexity of Boolean regression.
• Score: 6.852292115526837
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We say that a classifier is \emph{adversarially robust} to perturbations of norm $r$ if, with high probability over a point $x$ drawn from the input distribution, there is no point within distance $\le r$ from $x$ that is classified differently. The \emph{boundary volume} is the probability that a point falls within distance $r$ of a point with a different label. This work studies the task of computationally efficient learning of hypotheses with small boundary volume, where the input is distributed as a subgaussian isotropic log-concave distribution over $\mathbb{R}^d$. Linear threshold functions are adversarially robust; they have boundary volume proportional to $r$. Such concept classes are efficiently learnable by polynomial regression, which produces a polynomial threshold function (PTF), but PTFs in general may have boundary volume $\Omega(1)$, even for $r \ll 1$. We give an algorithm that agnostically learns linear threshold functions and returns a classifier with boundary volume $O(r+\varepsilon)$ at radius of perturbation $r$. The time and sample complexity of $d^{\tilde{O}(1/\varepsilon^2)}$ matches the complexity of polynomial regression. Our algorithm augments the classic approach of polynomial regression with three additional steps: a) performing the $\ell_1$-error regression under noise sensitivity constraints, b) a structured partitioning and rounding step that returns a Boolean classifier with error $\textsf{opt} + O(\varepsilon)$ and noise sensitivity $O(r+\varepsilon)$ simultaneously, and c) a local corrector that ``smooths'' a function with low noise sensitivity into a function that is adversarially robust.

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