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Pushing Stochastic Gradient towards Second-Order Methods – Backpropagation Learning with Transformations in Nonlinearities

  • Tommi Vatanen
  • Tapani Raiko
  • Harri Valpola
  • Yann LeCun
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8226)

Abstract

Recently, we proposed to transform the outputs of each hidden neuron in a multi-layer perceptron network to have zero output and zero slope on average, and use separate shortcut connections to model the linear dependencies instead. We continue the work by firstly introducing a third transformation to normalize the scale of the outputs of each hidden neuron, and secondly by analyzing the connections to second order optimization methods. We show that the transformations make a simple stochastic gradient behave closer to second-order optimization methods and thus speed up learning. This is shown both in theory and with experiments. The experiments on the third transformation show that while it further increases the speed of learning, it can also hurt performance by converging to a worse local optimum, where both the inputs and outputs of many hidden neurons are close to zero.

Keywords

Multi-layer perceptron network deep learning stochastic gradient 

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References

  1. 1.
    Amari, S.: Natural gradient works efficiently in learning. Neural Computation 10(2), 251–276 (1998)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ciresan, D.C., Meier, U., Gambardella, L.M., Schmidhuber, J.: Deep big simple neural nets excel on handwritten digit recognition. CoRR, abs/1003.0358 (2010)Google Scholar
  3. 3.
    Hinton, G.E., Salakhutdinov, R.R.: Reducing the dimensionality of data with neural networks. Science 313(5786), 504–507 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hinton, G.E., Srivastava, N., Krizhevsky, A., Sutskever, I., Salakhutdinov, R.: Improving neural networks by preventing co-adaptation of feature detectors. CoRR, abs/1207.0580 (2012)Google Scholar
  5. 5.
    Krizhevsky, A., Sutskever, I., Hinton, G.E.: Imagenet classification with deep convolutional neural networks (2012)Google Scholar
  6. 6.
    Le Roux, N., Manzagol, P.A., Bengio, Y.: Topmoumoute online natural gradient algorithm. In: Advances in Neural Information Processing Systems 20, NIPS 2007 (2008)Google Scholar
  7. 7.
    LeCun, Y.A., Bottou, L., Orr, G.B., Müller, K.-R.: Efficient backProp. In: Orr, G.B., Müller, K.-R. (eds.) NIPS-WS 1996. LNCS, vol. 1524, pp. 9–48. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  8. 8.
    Martens, J.: Deep learning via Hessian-free optimization. In: Proceedings of the 27th International Conference on Machine Learning, ICML (2010)Google Scholar
  9. 9.
    Raiko, T., Valpola, H., LeCun, Y.: Deep learning made easier by linear transformations in perceptrons. Journal of Machine Learning Research - Proceedings Track 22, 924–932 (2012)Google Scholar
  10. 10.
    Schraudolph, N.N.: Accelerated gradient descent by factor-centering decomposition. Technical Report IDSIA-33-98, Istituto Dalle Molle di Studi sull’Intelligenza Artificiale (1998)Google Scholar
  11. 11.
    Schraudolph, N.N.: Centering neural network gradient factors. In: Orr, G.B., Müller, K.-R. (eds.) NIPS-WS 1996. LNCS, vol. 1524, pp. 207–548. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  12. 12.
    Vatanen, T., Raiko, T., Valpola, H., LeCun, Y.: Pushing stochastic gradient towards second-order methods – backpropagation learning with transformations in nonlinearities (pre-print, 2013), http://arxiv.org/abs/1301.3476

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Tommi Vatanen
    • 1
  • Tapani Raiko
    • 1
  • Harri Valpola
    • 1
  • Yann LeCun
    • 2
  1. 1.Department of Information and Computer ScienceAalto University School of ScienceEspooFinland
  2. 2.New York UniversityNew YorkUSA

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