## Neural Network Learning: Theoretical FoundationsThis important work describes recent theoretical advances in the study of artificial neural networks. It explores probabilistic models of supervised learning problems, and addresses the key statistical and computational questions. Chapters survey research on pattern classification with binary-output networks, including a discussion of the relevance of the Vapnik Chervonenkis dimension, and of estimates of the dimension for several neural network models. In addition, Anthony and Bartlett develop a model of classification by real-output networks, and demonstrate the usefulness of classification with a "large margin." The authors explain the role of scale-sensitive versions of the Vapnik Chervonenkis dimension in large margin classification, and in real prediction. Key chapters also discuss the computational complexity of neural network learning, describing a variety of hardness results, and outlining two efficient, constructive learning algorithms. The book is self-contained and accessible to researchers and graduate students in computer science, engineering, and mathematics. |

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### Contents

1 Introduction | 1 |

Pattern Classification with BinaryOutput Neural Networks | 11 |

Pattern Classification with RealOutput Networks | 131 |

Learning RealValued Functions | 229 |

Algorithmics | 297 |

Appendix 1 Useful Results | 357 |

365 | |

379 | |

382 | |

### Common terms and phrases

activation function Adaboost algorithm for H approximate-SEM algorithm Bartlett chapter class F class of functions computation units Computational Learning Theory connected components consider convex combination covering numbers deﬁnition denote e-cover eﬁicient eﬂicient erp(f erp(h erp(L(z estimation error f G F fan-in fatp feed-forward ﬁnite ﬁnite set ﬁrst layer ﬁrst-layer units ﬁxed following result function f functions computed functions deﬁned functions in F functions mapping growth function Haussler Hence implies inequality inﬁnite input labelled examples learnable learning problem Lemma Let F linear threshold networks loss function lower bound neural networks NP-hard output hypothesis output unit Pdim(F polynomial probability at least probability distribution proof of Theorem random variable real number real-valued functions restricted model sample error satisﬁes set components bound set of functions shattered shows sigmoid networks simple perceptron standard sigmoid subset Suppose that F Theorem training sample two-layer networks uniform convergence upper bound VC-dimension VCdim(H weights and thresholds