TY - JOUR

T1 - Learning decision rules for pattern classification under a family of probability measures

AU - Kulkarni, Sanjeev R.

AU - Vidyasagar, Mathukumalli

N1 - Funding Information:
Manuscript received November 1, 1993; revised June 10, 1996. The work of S. R. Kulkami was supported in part by the National Science Foundation under Grant IRI-9209577 and NYI Award RI-9457645 and by the U.S. Army Research Office under Grant DAAL03-92-(3-0320. S. R. Kulkami is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA. M. Vidyasagar is with the Centre for Artificial Intelligence and Robotics, Bangalore 560 001, India. Publisher Item Identifier S 0018-9448(97)00162-4.

PY - 1997

Y1 - 1997

N2 - In this paper, uniformly consistent estimation (learnability) of decision rules for pattern classification under a family of probability measures is investigated. In particular, it is shown that uniform boundedness of the metric entropy of the class of decision rules is both necessary and sufficient for learnability under each of two conditions: i) the family of probability measures is totally bounded, with respect to the total variation metric, and ii) the family of probability measures contains an interior point, when equipped with the same metric. In particular, this shows that insofar as uniform consistency is concerned, when the family of distributions contains a total variation neighborhood, nothing is gained by this knowledge about the distribution. Then two sufficient conditions for learnability are presented. Specifically, it is shown that learnability with respect to each of a finite collection of families of probability measures implies learnability with respect to their union; also, learnability with respect to each of a finite number of measures implies learnability with respect to the convex hull of the corresponding families of uniformly absolutely continuous probability measures.

AB - In this paper, uniformly consistent estimation (learnability) of decision rules for pattern classification under a family of probability measures is investigated. In particular, it is shown that uniform boundedness of the metric entropy of the class of decision rules is both necessary and sufficient for learnability under each of two conditions: i) the family of probability measures is totally bounded, with respect to the total variation metric, and ii) the family of probability measures contains an interior point, when equipped with the same metric. In particular, this shows that insofar as uniform consistency is concerned, when the family of distributions contains a total variation neighborhood, nothing is gained by this knowledge about the distribution. Then two sufficient conditions for learnability are presented. Specifically, it is shown that learnability with respect to each of a finite collection of families of probability measures implies learnability with respect to their union; also, learnability with respect to each of a finite number of measures implies learnability with respect to the convex hull of the corresponding families of uniformly absolutely continuous probability measures.

KW - Class of distributions

KW - Decision rules

KW - Estimation

KW - Metric entropy

KW - Pac learning

KW - Pattern classification

KW - Uniform consistency

KW - Vc dimension

UR - http://www.scopus.com/inward/record.url?scp=0030736447&partnerID=8YFLogxK

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U2 - 10.1109/18.567668

DO - 10.1109/18.567668

M3 - Article

AN - SCOPUS:0030736447

VL - 43

SP - 154

EP - 166

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 1

ER -