Asymptotic statistical theory of overtraining and cross-validation

Shun Ichi Amari*, Noboru Murata, Klaus Robert Müller, Michael Finke, Howard Hua Yang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

274 Citations (Scopus)


A statistical theory for overtraining is proposed. The analysis treats general realizable stochastic neural networks, trained with Kullback-Leibler divergence in the asymptotic case of a large number of training examples. It is shown that the asymptotic gain in the generalization error is small if we perform early stopping, even if we have access to the optimal stopping time. Considering cross-validation stopping we answer the question: In what ratio the examples should be divided into training and cross-validation sets in order to obtain the optimum performance. Although cross-validated early stopping is useless in the asymptotic region, it surely decreases the generalization error in the nonasymptotic region. Our large scale simulations done on a CM5 are in nice agreement with our analytical findings.

Original languageEnglish
Pages (from-to)985-996
Number of pages12
JournalIEEE Transactions on Neural Networks
Issue number5
Publication statusPublished - 1997
Externally publishedYes


  • Asymptotic analysis
  • Cross-validation
  • Early stopping
  • Generalization
  • Overtraining
  • Stochastic neural networks

ASJC Scopus subject areas

  • Software
  • Computer Science Applications
  • Computer Networks and Communications
  • Artificial Intelligence


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