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.

翻译：低次多项式阈值函数（PTFs）的概念类在机器学习中扮演着基础性角色。本文研究$\mathbb{R}^n$上$K$-稀疏度-$d$ PTF的PAC学习问题，其中每个此类概念仅依赖于输入$n$个属性中的$K$个。我们的主要贡献是一种新算法，运行时间为$({nd}/{\epsilon})^{O(d)}$，在高斯边际分布下，即使当$\eta \leq O(\epsilon^d)$比例的训练样本被Bshouty等人（2002）提出的恶性噪声（可能最强的破坏模型）污染时，该算法仍能以$O(\frac{K^{4d}}{\epsilon^{2d}} \cdot \log^{5d} n)$个样本实现误差率$\epsilon$的PAC学习。在此工作之前，属性高效的鲁棒算法仅针对稀疏齐次半空间的特殊情况建立。我们的关键要素包括：1）一个结构结果，将属性稀疏性转化为埃尔米特多项式基下Chow向量的稀疏模式；2）一种新颖的属性高效鲁棒Chow向量估计算法，该算法专门使用受限Frobenius范数来认证良好的近似，或验证作为滤波器检测污染样本的稀疏诱导的$2d$次多项式。