*Europhys. Lett.*,

**51**(6), pp. 691-697 (2000)

## On-line learning from restricted training sets in multilayer neural networks

^{1}
Department of Mathematics,
King's College - Strand, London WC2R 2LS, UK

^{2}
The Neural Computing Research Group, Aston University -
Birmingham B4 7ET, UK

Received:
7
February
2000

Accepted:
25
July
2000

We analyse the dynamics of on-line learning in multilayer neural networks where training examples are sampled with repetition and where the number of examples scales with the number of network weights. The analysis is based on monitoring a set of macroscopic variables from which the training and generalisation errors can be calculated. A closed set of dynamical equations is derived using the dynamical replica method and is solved numerically. The theoretical results are consistent with those obtained by computer simulations.

PACS: 87.10.+e – Biological and medical physics: General theory and mathematical aspects / 02.50.-r – Probability theory, stochastic processes, and statistics / 05.90.+m – Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems

*© EDP Sciences, 2000*