Fooling Thresholds of Halfspaces

2026-09-18

AI Summary

本文通过设计一种基于Bentkus型平滑器的框架，构建了针对半空间阈值函数的显式伪随机生成器，解决了电路复杂性中的一个关键问题。

Abstract

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.

Problem

• pseudorandom generators
• thresholds of halfspaces
• circuit complexity

Innovation

• pseudorandom generators
• thresholds of halfspaces
• polylogarithmic seed length
• smooth approximation framework
• Boolean anticoncentration theorem