In this paper we study a Bayesian or average-case 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; 1g-valued functions over an instance space X....

: Let V ` f0; 1g n have Vapnik-Chervonenkis dimension d. Let M(k=n;V ) denote the cardinality of the largest W ` V such that any two distinct vectors in W differ on at least k indices. We show that M(k=n;V ) (cn=(k + d)) d for some constant c. This improves on the previous best result of ((cn=k) log(n=k)) d . This new result has applications in the theory of empirical processes. 1 The author gratefully acknowledges the support of the Mathematical Sciences Research Institute at UC Berkeley and ONR grant N00014-91-J-1162. 1 1 Statement of Results Let n be natural number greater than zero. Let V ` f0; 1g n . For a sequence of indices I = (i 1 ; . . . ; i k ), with 1 i j n, let V j I denote the projection of V onto I, i.e. V j I = f(v i 1 ; . . . ; v i k ) : (v 1 ; . . . ; v n ) 2 V g: If V j I = f0; 1g k then we say that V shatters the index sequence I. The Vapnik-Chervonenkis dimension of V is the size of the longest index sequence I that is shattered by V [VC71] (t...

Abstract. In this paper we study a Bayesian or average-case 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.