Skip to main content

Advertisement

Springer Nature Link
Log in

An Analysis of Activation Function Saturation in Particle Swarm Optimization Trained Neural Networks

  • Published:
Neural Processing Letters Aims and scope Submit manuscript

Abstract

The activation functions used in an artificial neural network define how nodes of the network respond to input, directly influence the shape of the error surface and play a role in the difficulty of the neural network training problem. Choice of activation functions is a significant question which must be addressed when applying a neural network to a problem. One issue which must be considered when selecting an activation function is known as activation function saturation. Saturation occurs when a bounded activation function primarily outputs values close to its boundary. Excessive saturation damages the network’s ability to encode information and may prevent successful training. Common functions such as the logistic and hyperbolic tangent functions have been shown to exhibit saturation when the neural network is trained using particle swarm optimization. This study proposes a new measure of activation function saturation, evaluates the saturation behavior of eight common activation functions, and evaluates six measures of controlling activation function saturation in particle swarm optimization based neural network training. Activation functions that result in low levels of saturation are identified. For each activation function recommendations are made regarding which saturation control mechanism is most effective at reducing saturation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Al Hazza MH, Adesta EY (2013) Investigation of the effect of cutting speed on the surface roughness parameters in CNC end milling using artificial neural network. In: IOP conference series: materials science and engineering, vol 53, IOP Publishing. https://doi.org/10.1088/1757-899X/53/1/012089

  2. Bishop C (1995) Neural networks for pattern recognition. Oxford University Press, New York

    MATH  Google Scholar 

  3. Carvalho M, Ludermir R (2006) Particle swarm optimization of feed-forward neural networks with weight decay. In: Proceedings of the international conference on 3D digital imaging and modeling, pp 1–5

  4. Center NGD (2019) Boulder sunspot number data. https://www.sws.bom.gov.au/Educational/2/3/6. Accessed 16 Mar 2019

  5. Dahl G, Sainath T, Hinton G (2013) Improving deep neural networks for LVCSR using rectified linear units and dropout. In: Proceedings of the conference on acoustics, speech and signal processing, pp 8609–8613

  6. Das M, Dulger L (2009) Signature vecification (SV) toolbox: applications of PSO-NN. Eng Appl Artific Intel 22(4):688–694

    Article  Google Scholar 

  7. Dreyfus G (2005) Neural networks: methodology and applications. Springer, Berlin

    MATH  Google Scholar 

  8. Dua D, Graff C (2017) UCI machine learning repository. http://archive.ics.uci.edu/ml. Accessed 16 Mar 2019

  9. Dugas C, Bengio Y, Belisle F, Nadeau C, Garcia R (2001) Incorporating second-order functional knowledge for better option pricing. In: Proceedings of the conference on advances in neural information processing systems, pp 472–478

  10. Eberhart R, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings of the sixth international symposium on micro machine and human science, pp 39–43

  11. Eggensperger K, Lindauer M, Hoos H, Hutter F, Leyton-Brown K (2018) Efficient benchmarking of algorithm configurators via model-based surrogates. Mach Learn 107(1):15–41

    Article  MathSciNet  Google Scholar 

  12. Elliott D (1993) A better activation function for artificial neural networks. Technical report T.R. 93-8, University of Maryland

  13. Engelbrecht A (2012) Particle swarm optimization: velocity initialization. In: Proceedings of the congress on evolutionary computation, pp 1–8

  14. Engelbrecht A (2013) Particle swarm optimization: global best or local best? In: BRICS congress on computational intelligence and 11th Brazilian congress on computational intelligence. IEEE, pp 124–135

  15. Engelbrecht A, Cloete I, Geldenhuys J, Zurada J (1995) Automatic scaling using gamma learning for feedforward neural networks. In: Proceedings of the international workshop on artificial neural networks. Springer, pp 374–381

  16. Fisher R (1936) Iris data set. https://archive.ics.uci.edu/ml/datasets/Iris. Accessed 2 Aug 2018

  17. Forina M et al (1991) Wine data set. https://archive.ics.uci.edu/ml/datasets/Wine. Accessed 2 Aug 2018

  18. Glorot X, Bengio Y (2010) Understanding the difficulty of training deep feedforward neural networks. In: International conference on artificial intelligence and statistics, pp 249–256

  19. Golik P, Doetsch P, Ney H (2013) Cross-entropy vs. squared error training: a theoretical and experimental comparison. Interspeech 13:1756–1760

    Google Scholar 

  20. Gudise V, Venayagamoorthy G (2003) Comparison of particle swarm optimization and backpropagation as training algorithms for neural networks. In: Proceedings of the swarm intelligence symposium, pp 110–117

  21. Harrison K (2018) An analysis of parameter control mechanisms for the particle swarm optimization algorithm. Ph.D. thesis, University of Pretoria

  22. Helwig S, Wanka R (2007) Particle swarm optimization in high dimensional bounded search spaces. In: Proceedings of the swarm intelligence symposium, pp 198–205

  23. Helwig S, Wanka R (2008) Theoretical analysis of initial particle swarm behavior. In: Rudolph G, Jansen T, Beume N, Lucas S, Poloni C (eds) Proceedings of the international conference on parallel problem solving from nature, pp 889–898

  24. Hutter F, Hoos H, Leyton-Brown K (2011) Sequential model-based optimization for general algorithm configuration. In: Coello CA (ed) Learning and intelligent optimization. Springer, Heidelberg, pp 507–523

    Chapter  Google Scholar 

  25. Hutter F, Lücke J, Schmidt-Thieme L (2015) Beyond manual tuning of hyperparameters. KI-Künstliche Intelligenz 29(4):329–337

    Article  Google Scholar 

  26. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of the international conference on neural networks, vol 4, pp 1942–1948

  27. Kennedy J, Mendes R (2002) Population structure and particle swarm performance. In: Proceedings of the international congress on evolutionary computation, vol 2, pp 1671–1676

  28. Lawrence S, Tsoi A, Back A (1996) Function approximation with neural networks and local methods: bias, variance and smoothness. In: Proceedings of the Australian conference on neural networks, australian national university, vol 1621

  29. LeCun Y, Bottou L, Orr G, Müller K (2012) Efficient BackProp. Springer, Berlin, pp 9–48

    Google Scholar 

  30. Maas A, Hannun A, Ng A (2013) Rectifier nonlinearities improve neural network acoustic models. in: Proceedings of the workshop on deep learning for audio, speech, and language processing, vol 30, pp 3–8

  31. Mendes R, Cortez P, Rocha M, Neves J (2002) Particle swarms for feedforward neural network training. In: Proceedings of the international joint conference on neural networks, pp 1895–1899

  32. Moody J, Hanson S, Krogh A, Hertz JA (1995) A simple weight decay can improve generalization. Adv Neural Inf Process Syst 4:950–957

    Google Scholar 

  33. Oldewage E (2018) The perils of particle swarm optimization in high dimensional problem spaces. Master’s thesis, Department of Computer Science, University of Pretoria

  34. Olorunda O, Engelbrecht A (2008) Measuring exploration/exploitation in particle swarms using swarm diversity. In: Proceedings of the international congress on evolutionary computation, pp 1128–1134

  35. Platt J (1991) A resource-allocating network for function interpolation. Neural Comput 3(2):213–225

    Article  MathSciNet  Google Scholar 

  36. Prechelt L (1994) PROBEN1—a set of benchmarks and benchmarking rules for neural network training algorithms. https://github.com/jeffheaton/proben1. Accessed 16 Mar 2019

  37. Rakitianskaia A, Engelbrecht A (2012) Training feedforward neural networks with dynamic particle swarm optimisation. Int J Uncertain Fuzziness Knowl Based Syst 6(3):233–270

    Google Scholar 

  38. Rakitianskaia A, Engelbrecht A (2014) Training high-dimensional neural networks with cooperative particle swarm optimiser. In: Proceedings of the international joint conference on neural networks, pp 4011–4018

  39. Rakitianskaia A, Engelbrecht A (2014) Weight regularisation in particle swarm optimisation neural network training. In: Proceedings of the symposium on swarm intelligence, pp 1–8

  40. Rakitianskaia A, Engelbrecht A (2015) Measuring saturation in neural networks. In: Proceedings of the symposium series on computational intelligence, pp 1423–1430

  41. Rakitianskaia A, Engelbrecht A (2015) Saturation in PSO neural network training: good or evil? In: Proceedings of the international congress on evolutionary computation, pp 125–132

  42. Rini DP, Shamsuddin SM, Yuhaniz SS (2011) Particle swarm optimization: technique, system and challenges. Int J Comput Appl ‘ 14(1):19–27

    Google Scholar 

  43. Röbel A (1994) The dynamic pattern selection algorithm: effective training and controlled generalization of backpropagation neural networks. Technical report, Technische Universität Berlin

  44. Shi Y, Eberhart R (1998) A modified particle swarm optimizer. In: Proceedings of the international congress on evolutionary computation, pp 69–73. https://doi.org/10.1109/ICEC.1998.699146

  45. Stützle T, López-Ibáñez M (2019) Automated design of metaheuristic algorithms. Springer, Cham, pp 541–579

    Google Scholar 

  46. van den Bergh F, Engelbrecht A (2000) Cooperative learning in neural networks using particle swarm optimizers. S Afr Comput J 26:84–90

    Google Scholar 

  47. van Wyk A, Engelbrecht A (2010) Overfitting by PSO trained feedforward neural networks. In: Proceedings of the international congress on evolutionary computation, pp 1–8

  48. van Wyk A, Engelbrecht A (2016) Analysis of activation functions for particle swarm optimized feedforward neural networks. In: Proceedings of the international congress on evolutionary computation, pp 423–430

  49. Volschenk A, Engelbrecht A (2016) An analysis of competitive coevolutionary particle swarm optimizers to train neural network game tree evaluation functions. In: Tan Y, Shi Y, Niu B (eds) Advances in swarm intelligence. Springer, Cham, pp 369–380

    Chapter  Google Scholar 

  50. Werbos PJ (1974) Beyond regression: new tools for prediction and analysis in the behavioural sciences. Ph.D. thesis, Harvard University

  51. Wessels L, Barnard E (1992) Avoiding false local minima by proper initialization of connections. Trans Neural Netw 3(6):899–905

    Article  Google Scholar 

  52. Wolberg W (1990) Breast cancer wisconsin (original) data set. https://archive.ics.uci.edu/ml/datasets/Breast+Cancer+Wisconsin+%28Original%29. Accessed 2 Aug 2018

  53. Wyk A, Engelbrecht A (2011) Lambda-gamma learning with feedforward neural networks using particle swarm optimization. In: Proceedings of the symposium on swarm intelligence, pp 1–8

  54. Xiao X, Wang Z, Li Q, Xia S, Jiang Y (2017) Back-propagation neural network on markov chains from system call sequences: a new approach for detecting android malware with system call sequences. IET Inf Secur 11(1):8–15

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Brock University, St. Catharines, Canada

    Cody Dennis & Beatrice M. Ombuki-Berman

  2. Department of Industrial Engineering and Department of Computer Science, Stellenbosch University, Stellenbosch, South Africa

    Andries P. Engelbrecht

Authors
  1. Cody Dennis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andries P. Engelbrecht
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Beatrice M. Ombuki-Berman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Beatrice M. Ombuki-Berman.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 908 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dennis, C., Engelbrecht, A.P. & Ombuki-Berman, B.M. An Analysis of Activation Function Saturation in Particle Swarm Optimization Trained Neural Networks. Neural Process Lett 52, 1123–1153 (2020). https://doi.org/10.1007/s11063-020-10290-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11063-020-10290-z

Keywords

Search

Navigation