### Abstract

Halfspaces or linear threshold functions are widely studied in complexity theory, learning theory and algorithm design. In this work we study the natural problem of constructing pseudorandom generators (PRGs) for halfspaces over the sphere, aka spherical caps, which besides being interesting and basic geometric objects, also arise frequently in the analysis of various randomized algorithms (e.g., randomized rounding). We give an explicit PRG which fools spherical caps within error e and has an almost optimal seed-length of O(logn + log(1/∈) ·loglog(1/∈)). For an inverse-polynomially growing error ∈, our generator has a seed-length optimal up to a factor of O(loglog (n)). The most efficient PRG previously known (due to Kane [31]) requires a seed-length of Ω(log^{3/2} (n)) in this setting. We also obtain similar constructions to fool halfspaces with respect to the Gaussian distribution. Our construction and analysis are significantly different from previous works on PRGs for halfspaces and build on the iterative dimension reduction ideas of [45, 9], the classical moment problem from probability theory and explicit constructions of approximate orthogonal designs based on the seminal work of Bourgain and Gamburd [6] on expansion in Lie groups.

Original language | English (US) |
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Title of host publication | STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing |

Publisher | Association for Computing Machinery |

Pages | 247-256 |

Number of pages | 10 |

ISBN (Electronic) | 9781450335362 |

DOIs | |

State | Published - Jun 14 2015 |

Event | 47th Annual ACM Symposium on Theory of Computing, STOC 2015 - Portland, United States Duration: Jun 14 2015 → Jun 17 2015 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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Volume | 14-17-June-2015 |

ISSN (Print) | 0737-8017 |

### Other

Other | 47th Annual ACM Symposium on Theory of Computing, STOC 2015 |
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Country | United States |

City | Portland |

Period | 6/14/15 → 6/17/15 |

### All Science Journal Classification (ASJC) codes

- Software

### Keywords

- Halfspaces
- Orthogonal designs
- Pseudorandom Generators

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## Cite this

*STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing*(pp. 247-256). (Proceedings of the Annual ACM Symposium on Theory of Computing; Vol. 14-17-June-2015). Association for Computing Machinery. https://doi.org/10.1145/2746539.2746611