Attribute-Efficient PAC Learning of Low-Degree Polynomial Threshold Functions with Nasty Noise
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929281734541312 |
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| author | Zeng, Shiwei Shen, Jie |
| author_facet | Zeng, Shiwei Shen, Jie |
| contents | 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 $({nd}/ε)^{O(d)}$ and under the Gaussian marginal distribution, PAC learns the class up to error rate $ε$ with $O(\frac{K^{4d}}{ε^{2d}} \cdot \log^{5d} n)$ samples even when an $η\leq O(ε^d)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_00673 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Attribute-Efficient PAC Learning of Low-Degree Polynomial Threshold Functions with Nasty Noise Zeng, Shiwei Shen, Jie Data Structures and Algorithms Machine Learning 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 $({nd}/ε)^{O(d)}$ and under the Gaussian marginal distribution, PAC learns the class up to error rate $ε$ with $O(\frac{K^{4d}}{ε^{2d}} \cdot \log^{5d} n)$ samples even when an $η\leq O(ε^d)$ 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. |
| title | Attribute-Efficient PAC Learning of Low-Degree Polynomial Threshold Functions with Nasty Noise |
| topic | Data Structures and Algorithms Machine Learning |
| url | https://arxiv.org/abs/2306.00673 |