Neural Computing and Applications

, Volume 31, Issue 12, pp 9241–9260 | Cite as

Evaluating generalization through interval-based neural network inversion

  • Stavros P. AdamEmail author
  • Aristidis C. Likas
  • Michael N. Vrahatis
Original Article


Typically, measuring the generalization ability of a neural network relies on the well-known method of cross-validation which statistically estimates the classification error of a network architecture thus assessing its generalization ability. However, for a number of reasons, cross-validation does not constitute an efficient and unbiased estimator of generalization and cannot be used to assess generalization of neural network after training. In this paper, we introduce a new method for evaluating generalization based on a deterministic approach revealing and exploiting the network’s domain of validity. This is the area of the input space containing all the points for which a class-specific network output provides values higher than a certainty threshold. The proposed approach is a set membership technique which defines the network’s domain of validity by inverting its output activity on the input space. For a trained neural network, the result of this inversion is a set of hyper-boxes which constitute a reliable and \(\varepsilon\)-accurate computation of the domain of validity. Suitably defined metrics on the volume of the domain of validity provide a deterministic estimation of the generalization ability of the trained network not affected by random test set selection as with cross-validation. The effectiveness of the proposed generalization measures is demonstrated on illustrative examples using artificial and real datasets using swallow feed-forward neural networks such as Multi-layer perceptrons.


Neural networks Generalization Inversion Interval analysis Reliable computing 



Highest posterior density


INTerval LABoratory


Interval analysis


Multi-layer perceptron


Off training set


Probability density function


Set computations with subpavings


Set inversion via interval analysis



The authors would like to thank the anonymous reviewers for their valuable suggestions and comments on earlier version of the manuscript that helped to significantly improve the paper at hand.

Compliance with ethical standards

Conflict of interest:

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  • Stavros P. Adam
    • 1
    • 2
    Email author
  • Aristidis C. Likas
    • 3
  • Michael N. Vrahatis
    • 2
  1. 1.Department of Informatics and TelecommunicationsUniversity of IoanninaArtaGreece
  2. 2.Computational Intelligence Laboratory, Department of MathematicsUniversity of PatrasPatrasGreece
  3. 3.Department of Computer Science and EngineeringUniversity of IoanninaIoanninaGreece

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