### Abstract

We show that unless NP = RP, it is hard to (even) weakly PAC-learn intersection of two halfspaces in R^{n} using a hypothesis which is a, function of up to ℓ linear threshold functions for any integer ℓ. Specifically, we show that for every integer ℓ and an arbitrarily small constant ε > 0, unless NP = RP, no polynomial time algorithm can distinguish whether there is an intersection of two halfspaces that correctly classifies a given set of labeled points in R^{n}, or whether any function of ℓ linear threshold functions can correctly classify at most 1/2+ ε fraction of the points.

Original language | English (US) |
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Title of host publication | STOC'08: Proceedings of the 2008 ACM Symposium on Theory of Computing |

Pages | 345-353 |

Number of pages | 9 |

State | Published - 2008 |

Event | 40th Annual ACM Symposium on Theory of Computing, STOC 2008 - Victoria, BC, Canada Duration: May 17 2008 → May 20 2008 |

### Other

Other | 40th Annual ACM Symposium on Theory of Computing, STOC 2008 |
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Country | Canada |

City | Victoria, BC |

Period | 5/17/08 → 5/20/08 |

### Fingerprint

### Keywords

- Approximation
- Halfspaces
- Hardness
- Learning

### ASJC Scopus subject areas

- Software

### Cite this

*STOC'08: Proceedings of the 2008 ACM Symposium on Theory of Computing*(pp. 345-353)

**On hardness of learning intersection of two halfspaces.** / Khot, Subhash; Saket, Rishi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*STOC'08: Proceedings of the 2008 ACM Symposium on Theory of Computing.*pp. 345-353, 40th Annual ACM Symposium on Theory of Computing, STOC 2008, Victoria, BC, Canada, 5/17/08.

}

TY - GEN

T1 - On hardness of learning intersection of two halfspaces

AU - Khot, Subhash

AU - Saket, Rishi

PY - 2008

Y1 - 2008

N2 - We show that unless NP = RP, it is hard to (even) weakly PAC-learn intersection of two halfspaces in Rn using a hypothesis which is a, function of up to ℓ linear threshold functions for any integer ℓ. Specifically, we show that for every integer ℓ and an arbitrarily small constant ε > 0, unless NP = RP, no polynomial time algorithm can distinguish whether there is an intersection of two halfspaces that correctly classifies a given set of labeled points in Rn, or whether any function of ℓ linear threshold functions can correctly classify at most 1/2+ ε fraction of the points.

AB - We show that unless NP = RP, it is hard to (even) weakly PAC-learn intersection of two halfspaces in Rn using a hypothesis which is a, function of up to ℓ linear threshold functions for any integer ℓ. Specifically, we show that for every integer ℓ and an arbitrarily small constant ε > 0, unless NP = RP, no polynomial time algorithm can distinguish whether there is an intersection of two halfspaces that correctly classifies a given set of labeled points in Rn, or whether any function of ℓ linear threshold functions can correctly classify at most 1/2+ ε fraction of the points.

KW - Approximation

KW - Halfspaces

KW - Hardness

KW - Learning

UR - http://www.scopus.com/inward/record.url?scp=57049098727&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=57049098727&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9781605580470

SP - 345

EP - 353

BT - STOC'08: Proceedings of the 2008 ACM Symposium on Theory of Computing

ER -