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Physics  2001 

Supervised Learning with Restricted Training Sets: a Generating Functional Analysis

DOI: 10.1088/0305-4470/34/42/318

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We study the dynamics of supervised on-line learning of realizable tasks in feed-forward neural networks. We focus on the regime where the number of examples used for training is proportional to the number of input channels N. Using generating function techniques from spin glass theory, we are able to average over the composition of the training set and transform the problem for N to infinity to an effective single pattern system, described completely by the student autocovariance, the student-teacher overlap and the student response function, with exact closed equations. Our method applies to arbitrary learning rules, i.e. not necessarily of a gradient-descent type. The resulting exact macroscopic dynamical equations can be integrated without finite-size effects up to any degree of accuracy, but their main value is in providing an exact and simple starting point for analytical approximation schemes. Finally, we show how, in the region of absent anomalous response and using the hypothesis that (as in detailed balance systems) the short-time part of the various operators can be transformed away, one can describe the stationary state of the network succesfully by a set of coupled equations involving only four scalar order parameters.


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