NDLI: Learning Simple Concepts under Simple Distributions

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Learning Simple Concepts under Simple Distributions

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Li, Ming Vitnyi, Paul M. B.
Copyright Year	1991
Abstract	This paper aims at developing a learning theory where simple concepts are easily learnable. In Valiants learning model, many concepts turn out to be too hard (like NP hard) to learn. Relatively few concept classes were shown to be learnable polynomially. In daily life, it seems that things we care to learn are usually learnable. To model the intuitive notion of learning more closely, it is not required that the learning algorithm learn (polynomially) under all distributions, but only under all simple distributions. A distribution is simple if it is dominated by an enumerable distribution. All distributions with computable parameters that are used in statistics are simple. Simple distributions are complete in the sense that a concept class is learnable under all simple distributions if and only if it is learnable under a fixed universal simple distribution. This holds both for polynomial learning in the discrete case (under a modified model), and for non-time-restricted learning in the continuous case (under the usual model). This completeness result is used to obtain new learning algorithms and several quite general new learnable classes. These include a discrete class that is known to be not polynomial learnable under Valiants model, unless RP = NP, and a continuous class that is not learnable in Valiants model. The results here allow that for each concept class from a wide range of concept classes, for each underlying distribution from a wide range of distributions, the learning algorithm uses a single fixed procedure to draw examples by a single algorithmic process using a random number generator. The universal simple distribution is not computable. To make the theory feasible, a polynomial-time version is developed for it. All results derived for discrete sample spaces hold mutatis mutandis for the polynomial-time versions, including versions of completeness, the new learning algorithms, and the new learnable classes.
Starting Page	911
Ending Page	935
Page Count	25
File Format	PDF
ISSN	00975397
DOI	10.1137/0220056
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	5
Volume Number	20
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-13
Access Restriction	Subscribed
Subject Keyword	universal distribution PAC learning polynomial-time learning algorithms Kolmogorov complexity learning simple concepts completeness discrete and continuous sample spaces enumerable distributions
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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