# The sample complexity of pattern classification with neural networks: the size of the weights is more important than the size of the network

Bartlett, P.L. (1998) The sample complexity of pattern classification with neural networks: the size of the weights is more important than the size of the network. *IEEE Transactions on Information Theory*, *44*(2), pp. 525-536.

## Abstract

Sample complexity results from computational learning theory, when applied to neural network learning for pattern classification problems, suggest that for good generalization performance the number of training examples should grow at least linearly with the number of adjustable parameters in the network. Results in this paper show that if a large neural network is used for a pattern classification problem and the learning algorithm finds a network with small weights that has small squared error on the training patterns, then the generalization performance depends on the size of the weights rather than the number of weights. For example, consider a two-layer feedforward network of sigmoid units, in which the sum of the magnitudes of the weights associated with each unit is bounded by A and the input dimension is n. We show that the misclassification probability is no more than a certain error estimate (that is related to squared error on the training set) plus A3 √((log n)/m) (ignoring log A and log m factors), where m is the number of training patterns. This may explain the generalization performance of neural networks, particularly when the number of training examples is considerably smaller than the number of weights. It also supports heuristics (such as weight decay and early stopping) that attempt to keep the weights small during training. The proof techniques appear to be useful for the analysis of other pattern classifiers: when the input domain is a totally bounded metric space, we use the same approach to give upper bounds on misclassification probability for classifiers with decision boundaries that are far from the training examples.

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ID Code: | 43927 |
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Item Type: | Journal Article |

Keywords: | Computer networks , Neural networks , Pattern analysis , Pattern classification , Probability distribution , Statistical learning , Training data , Upper bound , Virtual colonoscopy , Pattern recognition |

DOI: | 10.1109/18.661502 |

Subjects: | Australian and New Zealand Standard Research Classification > INFORMATION AND COMPUTING SCIENCES (080000) > ARTIFICIAL INTELLIGENCE AND IMAGE PROCESSING (080100) Australian and New Zealand Standard Research Classification > ENGINEERING (090000) > ELECTRICAL AND ELECTRONIC ENGINEERING (090600) Australian and New Zealand Standard Research Classification > PSYCHOLOGY AND COGNITIVE SCIENCES (170000) > COGNITIVE SCIENCE (170200) |

Divisions: | Past > QUT Faculties & Divisions > Faculty of Science and Technology Past > Schools > Mathematical Sciences |

Copyright Owner: | IEEE |

Copyright Statement: | Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. |

Deposited On: | 12 Aug 2011 11:07 |

Last Modified: | 12 Aug 2011 11:07 |

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