NDLI: Exact Learning of Discretized Geometric Concepts

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Mathias, H. David Goldberg, Paul W. Bshouty, Nader H. Goldman, Sally A.
Copyright Year	1998
Abstract	We first present an algorithm that uses membership and equivalence queries to exactly identify a discretized geometric concept defined by the union of m axis-parallel boxes in d-dimensional discretized Euclidean space where each coordinate can have n discrete values. This algorithm receives at most md counterexamples and uses time and membership queries polynomial in m and log n for any constant d. Furthermore, all equivalence queries can be formulated as the union of O(md log m) axis-parallel boxes.Next, we show how to extend our algorithm to efficiently learn, from only equivalence queries, any discretized geometric concept generated from any number of halfspaces with any number of known (to the learner) slopes in a constant dimensional space. In particular, our algorithm exactly learns (from equivalence queries only) unions of discretized axis-parallel boxes in constant dimensional space in polynomial time. Furthermore, this equivalence query only algorithm can be modified to handle a polynomial number of lies in the counterexamples provided by the environment.Finally, we introduce a new complexity measure that better captures the complexity of the union of m boxes than simply the number of boxes and the dimension. Our new measure, $\sigma$, is the number of segments in the target, where a segment is a maximum portion of one of the sides of the target that lies entirely inside or entirely outside each of the other halfspaces defining the target. We present a modification of our first algorithm that uses time and queries polynomial in $\sigma$ and log n. In fact, the time and queries (both membership and equivalence) used by this single algorithm are polynomial for eitherm or d constant.
Starting Page	674
Ending Page	699
Page Count	26
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539794274246
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	2
Volume Number	28
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-28
Access Restriction	Subscribed
Subject Keyword	Learning and adaptive systems Analysis of algorithms and problem complexity geometric concepts computational learning exact learning membership and equivalence queries
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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