Advertisement

A Chain Rule for the Expected Suprema of Gaussian Processes

  • Andreas Maurer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8776)

Abstract

The expected supremum of a Gaussian process indexed by the image of an index set under a function class is bounded in terms of separate properties of the index set and the function class. The bound is relevant to the estimation of nonlinear transformations or the analysis of learning algorithms whenever hypotheses are chosen from composite classes, as is the case for multi-layer models.

Keywords

Function Class Chain Rule Lipschitz Constant Machine Learn Research Mapping Stage 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bartlett, P.L., Mendelson, S.: Rademacher and Gaussian Complexities: Risk Bounds and Structural Results. Journal of Machine Learning Research 3, 463–482 (2002)MathSciNetGoogle Scholar
  2. 2.
    Baxter, J.: Theoretical Models of Learning to Learn. In: Thrun, S., Pratt, L. (eds.) Learning to Learn. Springer (1998)Google Scholar
  3. 3.
    Baxter, J.: A Model of Inductive Bias Learning. Journal of Artificial Intelligence Research 12, 149–198 (2000)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Boucheron, S., Lugosi, G., Massart, P.: Concentration Inequalities. Oxford University Press (2013)Google Scholar
  5. 5.
    Koltchinskii, V.I., Panchenko, D.: Rademacher processes and bounding the risk of function learning. In: Gine, E., Mason, D., Wellner, J. (eds.) High Dimensional Probability II, pp. 443–459 (2000)Google Scholar
  6. 6.
    Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Springer (1991)Google Scholar
  7. 7.
    Maurer, A.: Bounds for linear multi-task learning. Journal of Machine Learning Research 7, 117–139 (2006)zbMATHGoogle Scholar
  8. 8.
    Maurer, A.: Transfer bounds for linear feature learning. Machine Learning 75(3), 327–350 (2009)CrossRefGoogle Scholar
  9. 9.
    Maurer, A., Pontil, M.: K-dimensional coding schemes in Hilbert spaces. IEEE Transactions on Information Theory 56(11), 5839–5846 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Meir, R., Zhang, T.: Generalization error bounds for Bayesian mixture algorithms. Journal of Machine Learning Research 4, 839–860 (2003)MathSciNetGoogle Scholar
  11. 11.
    Mendelson, S.: l-norm and its application to learning theory. Positivity 5, 177–191 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Shawe-Taylor, J., Cristianini, N.: Kernel Methods for Pattern Analysis. Cambridge University Press (2004)Google Scholar
  13. 13.
    Talagrand, M.: Regularity of Gaussian processes. Acta Mathematica 159, 99–149 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Talagrand, M.: A simple proof of the majorizing measure theorem. Geometric and Functional Analysis 2(1), 118–125 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Talagrand, M.: Majorizing measures without measures. Ann. Probab. 29, 411–417 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Talagrand, M.: The Generic Chaining. Upper and Lower Bounds for Stochastic Processes. Springer, Berlin (2005)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Andreas Maurer
    • 1
  1. 1.MünchenGermany

Personalised recommendations