利用多项式阈值学习噪声半空间的困难性
Hardness of learning noisy halfspaces using polynomial thresholds
摘要：
我们使用多项式阈值函数（PTFs）证明了在对抗性噪声存在下弱学习半空间的硬度。特别是，我们证明了，对于任何常数$d \in \mathbb{Z}^+$和$\varepsilon > 0$，这是NP-难决定：给定一组$\{-1,1\}$标记的点在$\mathbb{R}^n$是否（是的情况下）存在一个半空间，分类$(1-\varepsilon)$ -分数的点正确，或（否情况下）任何程度- $d$ PTF分类最多$(1/2 + \varepsilon)$ -分数的点正确。这将Diakonikolas et al.（2011）使用度- $2$ PTF的学习的NP难度增强到所有恒定度。后一个结果仍然是Feldman et al.（2006）和Guruswami et al.（2006）工作的唯一进展，排除了弱适当学习对抗噪声半空间。
英文摘要：
We prove the hardness of weakly learning halfspaces in the presence of adversarial noise using polynomial threshold functions (PTFs). In particular, we prove that for any constants $d \in \mathbb{Z}^+$ and $\varepsilon > 0$, it is NP-hard to decide: given a set of $\{-1,1\}$-labeled points in $\mathbb{R}^n$ whether (YES Case) there exists a halfspace that classifies $(1-\varepsilon)$-fraction of the points correctly, or (NO Case) any degree-$d$ PTF classifies at most $(1/2 + \varepsilon)$-fraction of the points correctly. This strengthens to all constant degrees the previous NP-hardness of learning using degree-$2$ PTFs shown by Diakonikolas et al. (2011). The latter result had remained the only progress over the works of Feldman et al. (2006) and Guruswami et al. (2006) ruling out weakly proper learning adversarially noisy halfspaces.
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