Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funders: National Research Foundation, National Research Foundation Singapore, National Natural Science Foundation of China, Ministry of Education of the People's Republic of China, Government of Jiangsu Province, Natural Science Foundation of Jiangsu Province, Centre for Quantum Technologies, National Science and Technology Major Project, Fundamental Research Funds for the Central Universities, Overseas Expertise Introduction Center for Discipline Innovation of Food Nutrition and Human Health (111 Center)
OpenAlex ID: W7213922009
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