在高斯分布下高效学习带对抗噪声的单指数模型，误差逼近最优。
Sample and Computationally Efficient Robust Learning of Gaussian Single-Index Models
- 基于赫尔米特系数设计新算法，实现样本与计算双重高效。
- 样本复杂度接近理论下界，误差达最优损失加任意小量ε。
- 适合研究鲁棒学习与统计模型理论的研究者参考。
单指数模型（SIM）形式为 $σ(\mathbf{w}^{ op} \mathbf{x})$，其中 $σ: \mathbb{R} \to \mathbb{R}$ 为已知链接函数，$\mathbf{w}^{ op}$ 为隐藏单位向量。本文研究在高斯分布下，针对 $L^2_2$-损失，在代理（即对抗标签噪声）模型中学习 SIM 的问题。核心结果是提出一种样本与计算均高效的抗噪正确学习器，其 $L^2_2$-误差为 $O(\mathrm{OPT}) + \varepsilon$，其中 $\mathrm{OPT}$ 为最优损失。该算法的样本复杂度为 $\tilde{O}(d^{\lceil k^{\ast}/2\rceil} + d/\varepsilon)$，其中 $k^{\ast}$ 为链接函数 $σ$ 对应的首个非零赫尔米特系数的阶数，称为信息指数。该样本上界几乎逼近已知的 CSQ 低界，即使在可实现情形下亦成立。此前工作多集中于可实现情况或半随机噪声，而现有计算高效鲁棒学习器对链接函数有更强假设。
原文摘要 · Abstract (English)
A single-index model (SIM) is a function of the form $σ(\mathbf{w}^{\ast} \cdot \mathbf{x})$, where $σ: \mathbb{R} \to \mathbb{R}$ is a known link function and $\mathbf{w}^{\ast}$ is a hidden unit vector. We study the task of learning SIMs in the agnostic (a.k.a. adversarial label noise) model with respect to the $L^2_2$-loss under the Gaussian distribution. Our main result is a sample and computationally efficient agnostic proper learner that attains $L^2_2$-error of $O(\mathrm{OPT})+ε$, where $\mathrm{OPT}$ is the optimal loss. The sample complexity of our algorithm is $\tilde{O}(d^{\lceil k^{\ast}/2\rceil}+d/ε)$, where $k^{\ast}$ is the information-exponent of $σ$ corresponding to the degree of its first non-zero Hermite coefficient. This sample bound nearly matches known CSQ lower bounds, even in the realizable setting. Prior algorithmic work in this setting had focused on learning in the realizable case or in the presence of semi-random noise. Prior computationally efficient robust learners required significantly stronger assumptions on the link function.
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