Learning unions of boxes with membership and equivalence queries

Conference · Fri Jul 01 04:00:00 EDT 1994

OSTI ID:10163306

Goldberg, P W ^[1]; Goldman, S A; Mathias, H D ^[2]

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1. Sandia National Laboratories, Albuquerque, NM (United States)
2. Washington Univ., St. Louis, MO (United States)

The authors present two algorithms that use membership and equivalence queries to exactly identify the concepts given by the union of s discretized axis-parallel boxes in d-dimensional discretized Euclidean space where there are n discrete values that each coordinate can have. The first algorithm receives at most sd counterexamples and uses time and membership queries polynomial in s and log n for d any constant. Further, all equivalence queries made can be formulated as the union of O(sd log(s)) axis-parallel boxes. Next, they introduce a new complexity measure that better captures the complexity of a union of boxes than simply the number of boxes and dimensions. Their new measure, {sigma}, is the number of segments in the target polyhedron where a segment is a maximum portion of one of the sides of the polyhedron that lies entirely inside or entirely outside each of the other halfspaces defining the polyhedron. They then present an improvement of their first algorithm that uses time and queries polynomial in {sigma} and log n. The hypothesis class used here is decision trees of height at most 2sd. Further they can show that the time and queries used by this algorithm are polynomial in d and log n for s any constant thus generalizing the exact learnability of DNF formulas with a constant number of terms. In fact, this single algorithm is efficient for either s or d constant.

• Research Organization: Sandia National Labs., Albuquerque, NM (United States)
• Sponsoring Organization: USDOE, Washington, DC (United States); National Science Foundation, Washington, DC (United States)
• DOE Contract Number: AC04-94AL85000
• OSTI ID: 10163306
• Report Number(s): SAND--94-1384C; CONF-940794--2; ON: DE94014662; CNN: Grant CCR-9110108; Grant CCR-9357707
• Country of Publication: United States
• Language: English

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