Incorporating curvature information into on-line learning

Rattray, Magnus and Saad, David (1999). Incorporating curvature information into on-line learning. IN: On-line learning in neural networks. Saad, David (ed.) Publications of the Newton Institute . Isaac Newton Institute, Cambridge: Cambridge University Press.


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.

Publication DOI:
Divisions: Engineering & Applied Sciences > Mathematics
Engineering & Applied Sciences > Systems analytics research institute (SARI)
Additional Information: Copyright of the Institute of Electrical and Electronics Engineers (IEEE)
Event Title: Proceedings of the on-line learning themed week
Event Type: Other
Event Dates: 1999-01-01 - 1999-01-01
Uncontrolled Keywords: multi-layer neural networks,statistical mechanics,optimal asymptotic performance,asymptotic generalization error decay,Matrix momentum
Full Text Link:
Related URLs: http://www.camb ... isbn=0521652634 (Publisher URL)
Published Date: 1999-01
Authors: Rattray, Magnus
Saad, David ( 0000-0001-9821-2623)



Version: Published Version

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