Incorporating curvature information into on-line learning

Magnus Rattray, David Saad

Research output: Chapter in Book/Published conference outputChapter


We analyse the dynamics of a number of second order on-line learning algorithms training multi-layer neural networks, using the methods of statistical mechanics. We first consider on-line Newton's method, which is known to provide optimal asymptotic performance. We determine the asymptotic generalization error decay for a soft committee machine, which is shown to compare favourably with the result for standard gradient descent. Matrix momentum provides a practical approximation to this method by allowing an efficient inversion of the Hessian. We consider an idealized matrix momentum algorithm which requires access to the Hessian and find close correspondence with the dynamics of on-line Newton's method. In practice, the Hessian will not be known on-line and we therefore consider matrix momentum using a single example approximation to the Hessian. In this case good asymptotic performance may still be achieved, but the algorithm is now sensitive to parameter choice because of noise in the Hessian estimate. On-line Newton's method is not appropriate during the transient learning phase, since a suboptimal unstable fixed point of the gradient descent dynamics becomes stable for this algorithm. A principled alternative is to use Amari's natural gradient learning algorithm and we show how this method provides a significant reduction in learning time when compared to gradient descent, while retaining the asymptotic performance of on-line Newton's method.
Original languageEnglish
Title of host publicationOn-line learning in neural networks
EditorsDavid Saad
Place of PublicationIsaac Newton Institute, Cambridge
PublisherCambridge University Press
Number of pages25
ISBN (Print)0521652634
Publication statusPublished - Jan 1999
EventProceedings of the on-line learning themed week -
Duration: 1 Jan 19991 Jan 1999

Publication series

NamePublications of the Newton Institute
PublisherCambridge University Press


OtherProceedings of the on-line learning themed week

Bibliographical note

Copyright of the Institute of Electrical and Electronics Engineers (IEEE)


  • multi-layer neural networks
  • statistical mechanics
  • optimal asymptotic performance
  • asymptotic generalization error decay
  • Matrix momentum


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