Machine Learning

, Volume 28, Issue 1, pp 7–39

A Bayesian/Information Theoretic Model of Learning to Learn via Multiple Task Sampling

  • Jonathan Baxter
Article

Abstract

A Bayesian model of learning to learn by sampling from multiple tasks is presented. The multiple tasks are themselves generated by sampling from a distribution over an environment of related tasks. Such an environment is shown to be naturally modelled within a Bayesian context by the concept of an objective prior distribution. It is argued that for many common machine learning problems, although in general we do not know the true (objective) prior for the problem, we do have some idea of a set of possible priors to which the true prior belongs. It is shown that under these circumstances a learner can use Bayesian inference to learn the true prior by learning sufficiently many tasks from the environment. In addition, bounds are given on the amount of information required to learn a task when it is simultaneously learnt with several other tasks. The bounds show that if the learner has little knowledge of the true prior, but the dimensionality of the true prior is small, then sampling multiple tasks is highly advantageous. The theory is applied to the problem of learning a common feature set or equivalently a low-dimensional-representation (LDR) for an environment of related tasks.

Hierarchichal Bayesian Inference Bias learning Feature Learning Neural Networks Information Theory 

References

  1. Abu-Mostafa, Y.S. (1989). Learning from Hints in Neural Networks. Journal of Complexity, 6:192–198.Google Scholar
  2. Anthony, Martin & Bartlett, Peter. (1995). Function learning from interpolation. In Proceedings of the Second European Conference on Computational Learning Theory, Barcelona. Springer-Verlag.Google Scholar
  3. Barron, Andrew & Clarke, Bertrand. (1994). Jeffreys' Prior is Asymptotically Least Favourable under Entropy Risk. Journal of Statistical Planning and Inference, 41:37–60.CrossRefGoogle Scholar
  4. Bartlett, Peter, Long, Philip & Williamson, Bob. (1994). Fat-Shattering and the Learnability of Real-Valued Functions. In Proccedings of the Seventh ACM Conference on Computational Learning Theory, New York. ACM Press.Google Scholar
  5. Baxter, Jonathan. (1995a). A Model of Bias Learning. Technical Report LSE-MPS-97, London School of Economics, Centre for Discrete and Applicable Mathematics. Submitted for publication.Google Scholar
  6. Baxter, Jonathan. (1995b). Learning Internal Representations. In Proceedings of the Eighth International Conference on Computational Learning Theory, pages 311–320, Santa Cruz, California. ACM Press.Google Scholar
  7. Baxter, Jonathan. (1996a). A Bayesian/Information Theoretic Model of Bias Learning. In Proccedings of the Ninth ACM Conference on Computational Learning Theory, New York. ACM Press.Google Scholar
  8. Baxter, Jonathan. (1996b). Learning Model Bias. In Advances in Neural Information Processing Systems 8, pages 169–175.Google Scholar
  9. Berger, James O. (1985). Statistical Decision Theory and Bayesian Analysis. Springer-Verlag, New York.Google Scholar
  10. Berger, James O. (1986) Multivariate Estimation: Bayes, Empirical Bayes, and Stein Approaches. SIAM.Google Scholar
  11. Bridle, J.S. (1989). Probabilistic interpretation of feedforward classification network outputs, with relationships to statistical pattern recognition. In F Fogelman-Soulie and J Herault, editors, Neurocomputing: Algorithms, Architectures. Springer Verlag, New York.Google Scholar
  12. Caruana, Richard. (1993). Learning Many Related Tasks at the Same Time with Backpropagation. In Advances in Neural Information Processing 5.Google Scholar
  13. Clarke, Bertrand & Barron, Andrew. (1990). Information-Theoretic Asymptotics of Bayes Methods. IEEE Transactions on Information Theory, 36:453–471.CrossRefGoogle Scholar
  14. Cover, T.M. & Thomas, J.A. (1991). Elements of Information Theory. John Wiley & Sons, Inc., New York.Google Scholar
  15. Fefferman, Charles. (1994). Reconstructing a neural network from its output. Rev. Mat. Iberoamericana, 10:507–555.Google Scholar
  16. Good, I.J. (1980). Some History of the Hierarchical Bayesian Methodology. In J M Bernado, M H De Groot, D V Lindley, and A F M Smith, editors, Bayesian Statistics II. University Press, Valencia.Google Scholar
  17. Haussler, David & Opper, Manfred. (1995a). General Bounds on the Mutual Information Between a Parameter and n Conditionally Independent Observations. In Proccedings of the Eighth ACM Conference on Computational Learning Theory, New York. ACM Press.Google Scholar
  18. Haussler, David & Opper, Manfred. (1995b). Mutual Information, Metric Entropy and Risk in Estimation of Probability Distributions. Submitted to Annals of Statistics.Google Scholar
  19. Hornik, K. (1991). Approximation capabilities of multilayer feedforward networks. Neural Networks, 4:251–257.Google Scholar
  20. Mackay, David. (1991). Bayesian Interpolation. Neural Computation, 4:415–447.Google Scholar
  21. Mackay, David. (1991). The Evidence Framework Applied to Classification Networks. Neural Computation, 4:698–714.Google Scholar
  22. Mitchell, Tom M. (1990). The need for biases in learning generalisations. In Tom G Dietterich and Jude Shavlik, editors, Readings in Machine Learning. Morgan Kaufmann.Google Scholar
  23. Mitchell, Tom M. & Thrun, Sebastian. (1994). Learning One More Thing. Technical Report CMU-CS-94-184, CMU.Google Scholar
  24. Pratt, Lori Y. (1992). Discriminability-based transfer between neural networks. In Stephen J Hanson, Jack D Cowan, and C Lee Giles, editors, Advances in Neural Information Processing Systems 5, pages 204–211, San Mateo. Morgan Kaufmann.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Jonathan Baxter
    • 1
    • 2
  1. 1.Department of MathematicsLondon School of EconomicsUK
  2. 2.Department of Computer Science, Royal Holloway CollegeUniversity of LondonUK

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