半空间的阈值欺骗阈值
Fooling Thresholds of Halfspaces
- National University of Singapore（新加坡国立大学）
- Nanjing University（南京大学）
- Hefei National Laboratory（合肥国家实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文为半空间阈值构造显式伪随机生成器，证明多胞体生成器可欺骗此类函数，并给出种子长度界，同时导出噪声敏感性与学习算法。
AI中文摘要：
我们开创了对半空间阈值显式伪随机生成器的研究，其种子长度为半空间数量的多对数。这类函数处于电路复杂性的前沿[CTW26]。我们证明了O'Donnell、Servedio和Tan为多胞体设计的生成器[OST22]也能欺骗这一更广泛的类别。为了分析该生成器，我们开发了一种基于Bentkus型磨光子的阈值特定光滑逼近框架。我们证明了该磨光子的导数界，并通过随机稀疏化论证建立了半空间阈值的布尔反集中定理。这些要素意味着该生成器δ-欺骗每个在{-1,1}^n上的m个半空间的k-out-of-m阈值，种子长度为O~(κ^{6+2ε}log^{6+2ε}m·δ^{-(2+2ε)}log n)，其中ε>0为任意小的常数，κ=min{k,m-k+1}。随机稀疏化论证还产生了半空间阈值的噪声敏感性和高斯表面积界，从而在均匀分布和高斯分布下都得到了学习算法。
英文摘要：
We initiate the study of constructing explicit pseudorandom generators for thresholds of halfspaces with seed length polylogarithmic in the number of halfspaces. This class of functions lies at the frontier of circuit complexity [CTW26]. We show that the generator designed by O'Donnell, Servedio, and Tan for polytopes [OST22] also fools this broader class. To analyze the generator, we develop a threshold-specific smooth approximation framework based on a Bentkus-type mollifier. We prove derivative bounds for this mollifier and also establish a Boolean anticoncentration theorem for thresholds of halfspaces via a random thinning argument. These ingredients imply that the generator $δ$-fools every $k$-out-of-$m$ threshold of $m$ halfspaces over $\{-1,1\}^n$ with seed length $\widetilde{O}(κ^{6+2\varepsilon}\log^{6+2\varepsilon}\!m\cdotδ^{-(2+2\varepsilon)}\log n)$, for any arbitrarily small constant $\varepsilon>0$, where $κ=\min\{k,m-k+1\}$. The random thinning argument also yields bounds on the noise sensitivity and Gaussian surface area for thresholds of halfspaces, leading to learning algorithms under both the uniform and Gaussian distributions.
