提出首个高效算法，可在对抗性噪声下学习任意单调单索引模型。
Robustly Learning Monotone Single-Index Models
- 设计新优化框架，利用高斯空间与单调函数性质构造更新向量场。
- 对所有满足二阶矩有界的单调激活函数均实现常数因子近似。
- 适用于连续或间断的激活函数，如带偏置的半平面函数。
我们研究在高斯分布下，存在对抗性标签噪声时，以平方损失学习单索引模型的基本问题。本文主要贡献是首个计算高效的算法，能在所有满足二阶矩有界（阶数为 $2 + ζ$，$ζ > 0$）的单调激活函数类中实现常数因子近似，该类包括所有单调Lipschitz函数，甚至包含可能带偏置的半空间等不连续函数。先前工作在未知激活函数情形下，要么无法达到常数因子近似，要么仅适用于更小的激活函数族。本方法的核心创新在于突破传统梯度方法，通过直接利用问题结构、高斯空间特性及单调函数正则性，构建有效向量场引导算法更新。
原文摘要 · Abstract (English)
We consider the basic problem of learning Single-Index Models with respect to the square loss under the Gaussian distribution in the presence of adversarial label noise. Our main contribution is the first computationally efficient algorithm for this learning task, achieving a constant factor approximation, that succeeds for the class of {\em all} monotone activations with bounded moment of order $2 + ζ,$ for $ζ> 0.$ This class in particular includes all monotone Lipschitz functions and even discontinuous functions like (possibly biased) halfspaces. Prior work for the case of unknown activation either does not attain constant factor approximation or succeeds for a substantially smaller family of activations. The main conceptual novelty of our approach lies in developing an optimization framework that steps outside the boundaries of usual gradient methods and instead identifies a useful vector field to guide the algorithm updates by directly leveraging the problem structure, properties of Gaussian spaces, and regularity of monotone functions.
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