The Neural Networks Training Problem










      The Neural Networks Training Problem consists in determining the synaptic weights of a neural network to get the desired output for a set of input vectors.



      A neural network (NN in the following) is formed by a set of process units or neurons interconnected. The topology of the NN can be specified by a directed graph where vertices are neurons and arcs are interconnections. Each interconnection has a real value associated with it called its synaptic weight. We can distinguish three types of neurons: input neurons, output neurons and hidden neurons. The outputs of the input neurons are established externally by the environment and constitute the input of the NN. The output of any non-input neuron is computed by applying an activation function to the weighted sum of its inputs. These inputs are computed after the outputs of every incoming neuron weighted by its associtaed synaptic weight. The outputs of the output neurons constitute de final output of the NN. When we present an input vector to the network an output vector is obtained by propagation of the values through the hidden neurons towards the output layer of neurons.

      The pair input vector/desired output vector is called a pattern. Training a NN consists in adjusting the synaptic weights in such a way that the NN learn a set of patterns (i.e., it works out the desired output for every input vector). This problem can be specified as an optimization problem. The goal is to minimize the error between the actual output of the NN and the desired output, computed for all the input vectors. This error can be calculated by means of the Root Medium Square Error (RMSE) whose expression is:








Problem Instances:

      Neural networks can be applied to pattern classification and function approximation. Widely used instances of the pattern classification problem can be obtained from the UCI Machine Learning Repository. However, Prechelt in [PRE94] proposed a benchmark set of instances and benchmarking rules in order to get a standard analysis of results, and thus allow researches to meaningfully compare their results. This benchmark (available for anonymous FTP in includes the following instances:

  • Cancer: Diagnosis of breast cancer. 9 inputs, 2 outputs, 699 examples.
  • Card: Predicts the approval or non-approval of a credit card to a customer. 51 inputs, 2 outputs, 690 examples.
  • Diabetes: Diagnose diabetes of Pima indians. 8 inputs, 2 outputs, 768 examples.
  • Gene: Detects intron/exon boundaries (splice junctions) in nucleotide sequences. 120 inputs, 3 outputs, 3175 examples.
  • Glass: Classifies glass types. 9 inputs, 6 outputs, 214 examples.
  • Heart: Predicts heart disease. 35 inputs, 2 outputs, 920 examples.
  • Horse: Predicts the future health state of a horse that has a colic. 58 inputs, 3 outputs, 364 examples.
  • Mushroom: Discriminates edible from poisonous mushrooms. 125 inputs, 2 outputs, 8124 examples.
  • Soybean: Recognizes 19 different diseasecases of soybeans. 35 inputs, 19 outputs, 683 examples.
  • Thyroid: Diagnoses thyroid hyper- or hypo-function. 21 intputs, 3 outputs, 7200 examples.
  • Building: Prediction of energy consumption in a building.14 inputs, 3 outputs, 4208 examples.
  • Flare: Prediction of solar flares. 24 inputs, 3 outputs, 1066 examples.
  • Hearta: The analog version of the heart disease diagnosis problem.

For the function approximation problem we present here the definition of 15 functions previously used in the literature [FML06]

  • Sexton 1:                                                      
  • Sexton 2:                                                      
  • Sexton 3:                                                      
  • Sexton 4:                                                      
  • Sexton 5:                                                      
  • Branin:                                                         
  • B2:                                                                
  • Easom:                                                         
  • Goldstein:                                                     
  • Shubert:                                                        
  • Beal:                                                             
  • Booth:                                                           
  • Matyas:                                                        
  • SixHumpCamelB:                                        
  • Schwefel:                                                      








Related Papers:

[PRE94] Lutz Prechelt. PROBEN1 - A Set of Neural Network Benchmark Problems and Benchmarking Rules. Technical Report 21, Fakultät für Informatik Universität Karlsruhe, 76128 Karlsruhe, Germany, September 1994.

[SD00] Randall S. Sexton and Robert E. Dorsey. Reliable Classification Using Neural Networks: a Genetic Algorithm and Backpropagation Comparison. Decision Support Systems, 30:11-22, 2000.

[CASL01] C. Cotta, E. Alba, R. Sagarna and P. Larrañaga. Adjusting Weights in Artificial Neural Networks using Evolutionary Algorithms. Estimation of Distribution Algorithms. A New Tool for Evolutionary Computation. Chapter 18, pages 357-373. Kluwer Academic Publishers, P. Larrañaga and J.A. Lozano (eds.), 2001. 

[FML06] El-Fallahi A., Martí, R., and Lasdon, L. (2006), Path Relinking and GRG for Artificial Neural Networks, European Journal of Operational Research, 169, pp. 508-519.








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10/31/06                                                                               For any question or suggestion, click here to contact with us.