## PAC Learning Intersections of Halfspaces with Membership Queries (1998)

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Venue: | ALGORITHMICA |

Citations: | 21 - 1 self |

### BibTeX

@ARTICLE{Kwek98paclearning,

author = {Stephen Kwek and Leonard Pitt},

title = {PAC Learning Intersections of Halfspaces with Membership Queries},

journal = {ALGORITHMICA},

year = {1998},

volume = {22},

pages = {53--75}

}

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### 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=ffl; 1=ffi) 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=ffl, 1=ffi, log(1=d)) time, the probability that Polly fails to output a collection of s halfspaces with classification error at most ffl, is at most ffi . 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 ...