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Bounds on the Sample Complexity of Bayesian Learning Using Information Theory and the VC Dimension
 Machine Learning
, 1994
"... In this paper we study a Bayesian or averagecase model of concept learning with a twofold goal: to provide more precise characterizations of learning curve (sample complexity) behavior that depend on properties of both the prior distribution over concepts and the sequence of instances seen by the l ..."
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Cited by 108 (12 self)
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In this paper we study a Bayesian or averagecase model of concept learning with a twofold goal: to provide more precise characterizations of learning curve (sample complexity) behavior that depend on properties of both the prior distribution over concepts and the sequence of instances seen by the learner, and to smoothly unite in a common framework the popular statistical physics and VC dimension theories of learning curves. To achieve this, we undertake a systematic investigation and comparison of two fundamental quantities in learning and information theory: the probability of an incorrect prediction for an optimal learning algorithm, and the Shannon information gain. This study leads to a new understanding of the sample complexity of learning in several existing models. 1 Introduction Consider a simple concept learning model in which the learner attempts to infer an unknown target concept f , chosen from a known concept class F of f0; 1gvalued functions over an instance space X....
Rigorous learning curve bounds from statistical mechanics
 Machine Learning
, 1994
"... Abstract In this paper we introduce and investigate a mathematically rigorous theory of learning curves that is based on ideas from statistical mechanics. The advantage of our theory over the wellestablished VapnikChervonenkis theory is that our bounds can be considerably tighter in many cases, an ..."
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Cited by 53 (9 self)
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Abstract In this paper we introduce and investigate a mathematically rigorous theory of learning curves that is based on ideas from statistical mechanics. The advantage of our theory over the wellestablished VapnikChervonenkis theory is that our bounds can be considerably tighter in many cases, and are also more reflective of the true behavior (functional form) of learning curves. This behavior can often exhibit dramatic properties such as phase transitions, as well as power law asymptotics not explained by the VC theory. The disadvantages of our theory are that its application requires knowledge of the input distribution, and it is limited so far to finite cardinality function classes. We illustrate our results with many concrete examples of learning curve bounds derived from our theory. 1 Introduction According to the VapnikChervonenkis (VC) theory of learning curves [27, 26], minimizing empirical error within a function class F on a random sample of m examples leads to generalization error bounded by ~O(d=m) (in the case that the target function is contained in F) or ~O(pd=m) plus the optimal generalization error achievable within F (in the general case). 1 These bounds are universal: they hold for any class of hypothesis functions F, for any input distribution, and for any target function. The only problemspecific quantity remaining in these bounds is the VC dimension d, a measure of the complexity of the function class F. It has been shown that these bounds are essentially the best distributionindependent bounds possible, in the sense that for any function class, there exists an input distribution for which matching lower bounds on the generalization error can be given [5, 7, 22].
Probably Approximately Correct Learning
 Proceedings of the Eighth National Conference on Artificial Intelligence
, 1990
"... This paper surveys some recent theoretical results on the efficiency of machine learning algorithms. The main tool described is the notion of Probably Approximately Correct (PAC) learning, introduced by Valiant. We define this learning model and then look at some of the results obtained in it. We th ..."
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Cited by 40 (1 self)
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This paper surveys some recent theoretical results on the efficiency of machine learning algorithms. The main tool described is the notion of Probably Approximately Correct (PAC) learning, introduced by Valiant. We define this learning model and then look at some of the results obtained in it. We then consider some criticisms of the PAC model and the extensions proposed to address these criticisms. Finally, we look briefly at other models recently proposed in computational learning theory. 2 Introduction It's a dangerous thing to try to formalize an enterprise as complex and varied as machine learning so that it can be subjected to rigorous mathematical analysis. To be tractable, a formal model must be simple. Thus, inevitably, most people will feel that important aspects of the activity have been left out of the theory. Of course, they will be right. Therefore, it is not advisable to present a theory of machine learning as having reduced the entire field to its bare essentials. All ...
Part 1: Overview of the Probably Approximately Correct (PAC) Learning Framework
, 1995
"... Here we survey some recent theoretical results on the efficiency of machine learning algorithms. The main tool described is the notion of Probably Approximately Correct (PAC) learning, introduced by Valiant. We define this learning model and then look at some of the results obtained in it. We then c ..."
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Cited by 4 (0 self)
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Here we survey some recent theoretical results on the efficiency of machine learning algorithms. The main tool described is the notion of Probably Approximately Correct (PAC) learning, introduced by Valiant. We define this learning model and then look at some of the results obtained in it. We then consider some criticisms of the PAC model and the extensions proposed to address these criticisms. Finally, we look briefly at other models recently proposed in computational learning theory.
Rigorous Learning Curve Bounds from Statistical Mechanics
, 1996
"... . In this paper we introduce and investigate a mathematically rigorous theory of learning curves that is based on ideas from statistical mechanics. The advantage of our theory over the wellestablished VapnikChervonenkis theory is that our bounds can be considerably tighter in many cases, and are al ..."
Abstract
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. In this paper we introduce and investigate a mathematically rigorous theory of learning curves that is based on ideas from statistical mechanics. The advantage of our theory over the wellestablished VapnikChervonenkis theory is that our bounds can be considerably tighter in many cases, and are also more reflective of the true behavior of learning curves. This behavior can often exhibit dramatic properties such as phase transitions, as well as power law asymptotics not explained by the VC theory. The disadvantages of our theory are that its application requires knowledge of the input distribution, and it is limited so far to finite cardinality function classes. We illustrate our results with many concrete examples of learning curve bounds derived from our theory. Keywords: learning curves, statistical mechanics, phase transitions, VC dimension 1. Introduction According to the VapnikChervonenkis (VC) theory of learning curves (Vapnik, 1982; Vapnik & Chervonenkis, 1971), minimizing e...