### Abstract

A randomized learning algorithm POLLY is presented that efficiently learns intersections of s halfspaces in n dimensions, in time polynomial in both s and n. The learning protocol is the PAC (probably approximately correct) model of Valiant, augmented with membership queries. In particular, POLLY receives a set S of m = poly(n, s, 1/ε, 1/δ) randomly generated points from an arbitrary distribution over the unit hypercube, and is told exactly which points are contained in, and which points are not contained in, the convex polyhedron P defined by the halfspaces. POLLY may also obtain the same information about points of its own choosing. It is shown that after poly(n, s, 1/ε, 1/δ, log(1/d)) time, the probability that POLLY fails to output a collection of s halfspaces with classification error at most ε, is at most δ. Here, d is the minimum distance between the boundary of the target and those examples in S that are not lying on the boundary. The parameter log(1/d) can be bounded by the number of bits needed to encode the coefficients of the bounding hyperplanes and the coordinates of the sampled examples S. Moreover, POLLY can be extended to learn unions of k disjoint polyhedra with each polyhedron having at most s facets, in time poly(n, k, s, 1/ε, 1/δ, log(1/d), 1/γ) where γ is the minimum distance between any two distinct polyhedra.

Original language | English (US) |
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Pages (from-to) | 53-75 |

Number of pages | 23 |

Journal | Algorithmica (New York) |

Volume | 22 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 1 1998 |

### Keywords

- Intersections of halfspaces
- Membership queries
- Occam algorithm
- PAC learning
- Unions of polyhedra

### ASJC Scopus subject areas

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics

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## Cite this

*Algorithmica (New York)*,

*22*(1-2), 53-75. https://doi.org/10.1007/PL00013834