## Abstract

We study the problem of agnostically learning halfspaces which is defined by a fixed but unknown distribution D on ℚ^{n} × {±1}. We define Err_{HALF}(D) as the least error of a halfspace classifier for D. A learner who can access D has to return a hypothesis whose error is small compared to Err_{HALF}(D). Using the recently developed method of Daniely, Linial and Shalev-Shwartz we prove hardness of learning results assuming that random K-XOR formulas are hard to (strongly) refute. We show that no efficient learning algorithm has nontrivial worst-case performance even under the guarantees that ErrHALp(D) ≤ η for arbitrarily small constant η > 0, and that D is supported in {±1 }^{n} × {±1}. Namely, even under these favorable conditions, and for every c > 0, it is hard to return a hypothesis with error ≤ 1/2/1/n^{c}. In particular, no efficient algorithm can achieve a constant approximation ratio. Under a stronger version of the assumption (where K can be poly-logarithmic in n), we can take η = 2-^{log1-v(n)} for arbitrarily small ? > 0. These results substantially improve on previously known results, that only show hardness of exact learning.

Original language | English |
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Title of host publication | STOC 2016 - Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Yishay Mansour, Daniel Wichs |

Publisher | Association for Computing Machinery |

Pages | 105-117 |

Number of pages | 13 |

ISBN (Electronic) | 9781450341325 |

DOIs | |

State | Published - 19 Jun 2016 |

Externally published | Yes |

Event | 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016 - Cambridge, United States Duration: 19 Jun 2016 → 21 Jun 2016 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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Volume | 19-21-June-2016 |

ISSN (Print) | 0737-8017 |

### Conference

Conference | 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016 |
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Country/Territory | United States |

City | Cambridge |

Period | 19/06/16 → 21/06/16 |

### Bibliographical note

Publisher Copyright:© 2016 ACM.

## Keywords

- Halfspaces
- Hardness of learning
- Random XOR