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The Partial Information Decomposition of Generative Neural Network Models

Tax, Tycho, Mediano, Pedro, Shanahan, Murray·2017·Entropy
DOI 10.3390/e19090474
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Abstract

In this work we study the distributed representations learnt by generative neural network models. In particular, we investigate the properties of redundant and synergistic information that groups of hidden neurons contain about the target variable. To this end, we use an emerging branch of information theory called partial information decomposition (PID) and track the informational properties of the neurons through training. We find two differentiated phases during the training process: a first short phase in which the neurons learn redundant information about the target, and a second phase in which neurons start specialising and each of them learns unique information about the target. We also find that in smaller networks individual neurons learn more specific information about certain features of the input, suggesting that learning pressure can encourage disentangled representations.

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Figure 1 Graphical representation of the discriminative restricted Boltzmann machine (DRBM) and its components. Vectors x and y correspond to the training input and label, respectively, h is the activation of the hidden neurons, and U and W are the weight matrices to be learned. (Adapted from [23]).
Figure 1 Graphical representation of the discriminative restricted Boltzmann machine (DRBM) and its components. Vectors x and y correspond to the training input and label, respectively, h is the activation of the hidden neurons, and U and W are the weight matrices to be learned. (Adapted from [23]).CC-BY · From the PDF
Figure 2 Partial information (PI) diagram for two source variables and a target. The outer ellipse corresponds to the mutual information (MI) between both sources and the target, I ( Y ; { Z 1 , Z 2 } ) , and both inner circles (highlighted in black) to the MI between each source and the target, I ( Y ; Zi ) . Coloured areas represent the PI terms described in the text.
Figure 2 Partial information (PI) diagram for two source variables and a target. The outer ellipse corresponds to the mutual information (MI) between both sources and the target, I ( Y ; { Z 1 , Z 2 } ) , and both inner circles (highlighted in black) to the MI between each source and the target, I ( Y ; Zi ) . Coloured areas represent the PI terms described in the text.CC-BY · From the PDF
Figure 3 Classification error and mutual information between real and predicted labels, I ( ˆ Y , Y ) , calculated through training. Note: X -axis in the rightmost plot is reversed for illustration pusposes, so that training time goes from left to right.
Figure 3 Classification error and mutual information between real and predicted labels, I ( ˆ Y , Y ) , calculated through training. Note: X -axis in the rightmost plot is reversed for illustration pusposes, so that training time goes from left to right.CC-BY · From the PDF
Figure 4 Single-neuron entropy and mutual information follow non-trivial patterns during training. ( a ) Entropy quickly rises up to close to its maximum value of 1 bit. ( b ) Inter-neuron correlation as measured by pairwise MI peaks midway through training. ( c ) Histograms of single-neuron MI (SNMI) split midway through training, implying that some neurons actually lose information. Average SNMI is shown in black projected on the frame box.
Figure 4 Single-neuron entropy and mutual information follow non-trivial patterns during training. ( a ) Entropy quickly rises up to close to its maximum value of 1 bit. ( b ) Inter-neuron correlation as measured by pairwise MI peaks midway through training. ( c ) Histograms of single-neuron MI (SNMI) split midway through training, implying that some neurons actually lose information. Average SNMI is shown in black projected on the frame box.CC-BY · From the PDF
Figure 5 PI terms ( a ) and PI terms normalised by joint mutual information ( b ). Mutual information (MI) in black, redundancy ( R ) in blue, synergy ( S ) in green, and unique information ( U ) in red. MI increases consistently during training, but the PI terms reveal a transition between a redundancy-dominated phase and a unique-information-dominated phase.
Figure 5 PI terms ( a ) and PI terms normalised by joint mutual information ( b ). Mutual information (MI) in black, redundancy ( R ) in blue, synergy ( S ) in green, and unique information ( U ) in red. MI increases consistently during training, but the PI terms reveal a transition between a redundancy-dominated phase and a unique-information-dominated phase.CC-BY · From the PDF
Figure 6 Partial information decomposition (PID) analysis of larger groups of neurons in networks of different sizes. ( a ) Single-neuron MI is consistently smaller in bigger networks, indicating that, although the network as a whole is a better classifier, each individual neuron has a less-efficient encoding; ( b ) Normalised PIK synergy, with network size increasing from left to right and K from top to bottom. Network with maximum synergy for each PIK highlighted with a vertical dashed line. The PI group size with the highest synergy becomes larger in larger networks, indicating that in bigger networks one needs to consider larger groups to capture strong cooperative interactions.
Figure 6 Partial information decomposition (PID) analysis of larger groups of neurons in networks of different sizes. ( a ) Single-neuron MI is consistently smaller in bigger networks, indicating that, although the network as a whole is a better classifier, each individual neuron has a less-efficient encoding; ( b ) Normalised PIK synergy, with network size increasing from left to right and K from top to bottom. Network with maximum synergy for each PIK highlighted with a vertical dashed line. The PI group size with the highest synergy becomes larger in larger networks, indicating that in bigger networks one needs to consider larger groups to capture strong cooperative interactions.CC-BY · From the PDF

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Citations

Cites

Research works printed but not linked: 35

  1. Srivastava, 2014, Dropout: A Simple Way to Prevent Neural Networks from Overfitting, J. Mach. Learn. Res., 15, 1929
  2. Gal, Y., and Ghahramani, Z. (2015). Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning. arXiv.
  3. Bengio, Y., Courville, A., and Vincent, P. (2012). Representation Learning: A Review and New Perspectives. arXiv.
  4. Higgins, I., Matthey, L., Glorot, X., Pal, A., Uria, B., Blundell, C., Mohamed, S., and Lerchner, A. (2016). Early Visual Concept Learning with Unsupervised Deep Learning. arXiv.
  5. Mathieu, M., Zhao, J., Sprechmann, P., Ramesh, A., and LeCun, Y. (2016). Disentangling Factors of Variation in Deep Representations Using Adversarial Training. arXiv.
  6. Siddharth, N., Paige, B., Van de Meent, J.W., Desmaison, A., Wood, F., Goodman, N.D., Kohli, P., and Torr, P.H.S. (2017). Learning Disentangled Representations with Semi-Supervised Deep Generative Models. arXiv.
  7. Garnelo, M., Arulkumaran, K., and Shanahan, M. (2016). Towards Deep Symbolic Reinforcement Learning. arXiv.
  8. Williams, P.L., and Beer, R.D. (2010). Nonnegative Decomposition of Multivariate Information. arXiv.
  9. Rieke, F., Bialek, W., Warland, D., and de Ruyter van Steveninck, R. (1997). Spikes: Exploring the Neural Code, MIT Press.
  10. Le, Q.V., Ranzato, M., Monga, R., Devin, M., Chen, K., Corrado, G.S., Dean, J., and Ng, A.Y. (2011). Building High-Level Features Using Large Scale Unsupervised Learning. arXiv.
  11. Zeiler, M.D., and Fergus, R. (2014, January 6–12). Visualizing and Understanding Convolutional Networks. Proceedings of the European Conference on Computer Vision, Zurich, Switzerland. · DOI · Paper page
  12. Choromanska, A., Henaff, M., Mathieu, M., Arous, G.B., and LeCun, Y. (2014). The Loss Surfaces of Multilayer Networks. arXiv.
  13. Kawaguchi, K. (2016). Deep Learning Without Poor Local Minima. arXiv.
  14. Sørngård, B. (2014). Information Theory for Analyzing Neural Networks. [Master’s Thesis, Norwegian University of Science and Technology].
  15. Schwartz-Ziv, R., and Tishby, N. (2017). Opening the Black Box of Deep Neural Networks via Information. arXiv.
  16. Achille, A., and Soatto, S. (2017). On the Emergence of Invariance and Disentangling in Deep Representations. arXiv. · DOI
  17. Tishby, N., and Zaslavsky, N. (2015). Deep Learning and the Information Bottleneck Principle. arXiv. · DOI
  18. Berglund, 2015, Measuring the Usefulness of Hidden Units in Boltzmann Machines with Mutual Information, Neural Netw., 64, 12, 10.1016/j.neunet.2014.09.004 · DOI
  19. Balduzzi, D., Frean, M., Leary, L., Lewis, J., Ma, K.W.D., and McWilliams, B. (2017). The Shattered Gradients Problem: If Resnets are the Answer, Then What is the Question?. arXiv.
  20. Smolensky, P. (1986). Information Processing in Dynamical Systems: Foundations of Harmony Theory, MIT Press. Technical Report, DTIC Document.
  21. Larochelle, H., and Bengio, Y. (2008, January 5–9). Classification Using Discriminative Restricted Boltzmann Machines. Proceedings of the 25th International Conference on Machine Learning, Helsinki, Finland. · DOI
  22. Hinton, 2006, A Fast Learning Algorithm for Deep Belief Nets, Neural Comput., 18, 1527, 10.1162/neco.2006.18.7.1527 · DOI · Paper page
  23. Tieleman, T. (2008, January 5–9). Training Restricted Boltzmann Machines Using Approximations to the Likelihood Gradient. Proceedings of the 25th International Conference on Machine Learning, Helsinki, Finland. · DOI
  24. Cover, T.M., and Thomas, J.A. (2006). Elements of Information Theory, Wiley. · DOI · Paper page
  25. DeWeese, 1999, How to Measure the Information Gained from one Symbol, Netw. Comput. Neural Syst., 12, 325, 10.1088/0954-898X_10_4_303 · DOI
  26. Ince, R.A.A. (2017). Measuring Multivariate Redundant Information with Pointwise Common Change in Surprisal. Entropy, 19. · DOI
  27. Griffith, 2015, Quantifying Redundant Information in Predicting a Target Random Variable, Entropy, 17, 4644, 10.3390/e17074644 · DOI
  28. Harder, M., Salge, C., and Polani, D. (2013). Bivariate Measure of Redundant Information. Phys. Rev. E, 87. · DOI
  29. Gilbert, T., Kirkilionis, M., and Nicolis, G. Shared Information—New Insights and Problems in Decomposing Information in Complex Systems. Proceedings of the European Conference on Complex Systems 2012.
  30. Williams, P.L. (2011). Information Dynamics: Its Theory and Application to EmbodiedCognitive Systems. [Ph.D. Thesis, Indiana University].
  31. Lizier, J.T. (2010). The Local Information Dynamics of Distributed Computation in Complex Systems, Springer.
  32. Timme, 2014, Synergy, Redundancy, and Multivariate Information Measures: An Experimentalist’s Perspective, J. Comput. Neurosci., 36, 119, 10.1007/s10827-013-0458-4 · DOI
  33. Bertschinger, 2014, Quantifying Unique Information, Entropy, 16, 2161, 10.3390/e16042161 · DOI · Paper page
  34. Ay, 2015, Geometry and Expressive Power of Conditional Restricted Boltzmann Machines, J. Mach. Learn. Res., 16, 2405
  35. Kingma, D.P., and Welling, M. (2013). Auto-Encoding Variational Bayes. arXiv.

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Metadata

Authors
Tax, T., Mediano, P., & Shanahan, M.
Affiliations
  • Corti, Nørrebrogade 45E 2, 2200 Copenhagen N, Denmark
  • Department of Computing, Imperial College London, London SW7 2RH, UK
ORCID
Venue
Entropy
Publisher
MDPI AG
Published
2017-09-06
Volume
19
Issue
9
Pages
474
ISSN
1099-4300
DOI
10.3390/e19090474
Cited by
55
References
2
Language
en
Licence
creativecommons.org/licenses/by/4.0/ (vor)

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