Attribute-Efficient PAC Learning of Low-Degree Polynomial Threshold Functions with Nasty Noise

• URL: http://arxiv.org/abs/2306.00673v2
• Date: Tue, 19 Mar 2024 15:01:36 GMT
• Title: Attribute-Efficient PAC Learning of Low-Degree Polynomial Threshold Functions with Nasty Noise
• Authors: Shiwei Zeng, Jie Shen,
• Abstract summary: We study PAC learning of $K$sparse degree-$d$ PTFs on $mathbbRn$. Our main contribution is a new algorithm that runs in time $(nd/epsilon)O(d)$. PAC learns the class up to error rate $epsilon$ with $O(fracK4depsilon2d cdot log5d n)$ even when an $eta leq O(epsilond)
• Score: 10.550885570889527
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The concept class of low-degree polynomial threshold functions (PTFs) plays a fundamental role in machine learning. In this paper, we study PAC learning of $K$-sparse degree-$d$ PTFs on $\mathbb{R}^n$, where any such concept depends only on $K$ out of $n$ attributes of the input. Our main contribution is a new algorithm that runs in time $({nd}/{\epsilon})^{O(d)}$ and under the Gaussian marginal distribution, PAC learns the class up to error rate $\epsilon$ with $O(\frac{K^{4d}}{\epsilon^{2d}} \cdot \log^{5d} n)$ samples even when an $\eta \leq O(\epsilon^d)$ fraction of them are corrupted by the nasty noise of Bshouty et al. (2002), possibly the strongest corruption model. Prior to this work, attribute-efficient robust algorithms are established only for the special case of sparse homogeneous halfspaces. Our key ingredients are: 1) a structural result that translates the attribute sparsity to a sparsity pattern of the Chow vector under the basis of Hermite polynomials, and 2) a novel attribute-efficient robust Chow vector estimation algorithm which uses exclusively a restricted Frobenius norm to either certify a good approximation or to validate a sparsity-induced degree-$2d$ polynomial as a filter to detect corrupted samples.

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