PAC-Bayesian inequalities for martingales

IEEE Transactions on Information Theory

Volumn 58, Issue 12, 2012, Pages 7086-7093

  Seldin, Yevgeny ^a,b   Laviolette, François ^c   Cesa Bianchi, Nicolò ^d   Shawe Taylor, John ^b   Auer, Peter ^e  

^a MAX PLANCK INSTITUTE FOR INTELLIGENT SYSTEMS   (Germany)

^b UNIVERSITY COLLEGE LONDON   (United Kingdom)

^c UNIVERSITÉ LAVAL   (Canada)

^d UNIVERSITY OF MILAN   (Italy)

^e UNIVERSITY OF LEOBEN   (Austria)

Author keywords

Bernstein's inequality; Hoeffding Azuma's inequality; martingales; PAC Bayesian bounds

Indexed keywords

BERNSTEIN'S INEQUALITY; CONVEX FUNCTIONS; HOEFFDING-AZUMA'S INEQUALITY; INDEPENDENT BERNOULLI RANDOM VARIABLES; INTERACTIVE LEARNING; LEARNING THEORY; MARTINGALE DIFFERENCE SEQUENCES; MARTINGALES; PAC-BAYESIAN BOUNDS; PROBABILITY THEORY; PROBABLY APPROXIMATELY CORRECT; WEIGHTED AVERAGES;

REINFORCEMENT LEARNING; SAMPLING;

IMPORTANCE SAMPLING;

EID: 84869821119     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2012.2211334     Document Type: Article

References (25)

1. W. Hoeffding, "Probability inequalities for sums of bounded random variables," J. Amer. Statist. Assoc., vol. 58, no. 301, pp. 13-30, 1963.
2. K. Azuma, "Weighted sums of certain dependent random variables," Tôhoku Math. J., vol. 19, no. 3, pp. 357-367, 1967.
3. S. N. Bernstein, Probability Theory (in Russian), 4th ed. Moscow, Russia: Moscow-Leningrad, 1946.
4. Y. Seldin, P. Auer, F. Laviolette, J. Shawe-Taylor, and R. Ortner, "PAC-Bayesian analysis of contextual bandits," in 25th Annu. Conf. Neural Inf. Process. Syst., 2011, pp. 1683-1691.
5. M. D. Donsker and S. S. Varadhan, "Asymptotic evaluation of certain Markov process expectations for large time," Commun. Pure Appl. Math., vol. 28, pp. 1-47, 1975.
6. P. Dupuis and R. S. Ellis, AWeak Convergence Approach to the Theory of Large Deviations. New York: Wiley-Interscience, 1997.
7. R. M. Gray, Entropy and Information Theory, 2nd ed. New York: Springer, 2011.
8. T. M. Cover and J. A. Thomas, Elements of Information Theory. Hoboken, NJ: Wiley, 1991.
9. J. Shawe-Taylor and R. C.Williamson, "A PAC analysis of a Bayesian estimator," in Proc. 10th Annu. Conf. Comput. Learning Theory, 1997, pp. 2-9.
10. J. Shawe-Taylor, P. L. Bartlett, R. C. Williamson, and M. Anthony, "Structural risk minimization over data-dependent hierarchies," IEEE Trans. Inf. Theory, vol. 44, no. 5, pp. 1926-1940, Sep. 1998.
11. D. McAllester, "Some PAC-Bayesian theorems," in Proc. 11th Annu. Conf. Comput. Learning Theory, 1998, pp. 230-234.
12. M. Seeger, "PAC-Bayesian generalization error bounds for Gaussian process classification," J. Mach. Learning Res., vol. 3, pp. 233-269, 2002.
13. L. G. Valiant, "A theory of the learnable," Commun. Assoc. Comput. Mach., vol. 27, no. 11, pp. 1134-1142, 1984.
14. J. Langford and J. Shawe-Taylor, "PAC-Bayes & margins," in Proc. Adv. Neural Inf. Process. Syst., 2002, pp. 439-446.
15. D. McAllester, "PAC-Bayesian stochastic model selection," Mach. Learning, vol. 51, no. 1, pp. 5-21, 2003.
16. P. Germain, A. Lacasse, F. Laviolette, and M. Marchand, "PAC-Bayesian learning of linear classifiers," in Proc. 26th Annu. Int. Conf. Mach. Learning, 2009, pp. 353-360.
17. Y. Seldin and N. Tishby, "PAC-Bayesian analysis of co-clustering and beyond," J. Mach. Learning Res., vol. 11, pp. 3595-3646, 2010.
18. G. Lever, F. Laviolette, and J. Shawe-Taylor, "Distribution- dependent PAC-Bayes priors," presented at the presented at the 21st Int. Conf. Algorithmic Learning Theory, Canberra, Australia, 2010.
19. A. Maurer, A note on the PAC-Bayesian theorem 2004 [Online]. Available: www.arxiv.org
20. D. A. Freedman, "On tail probabilities for martingales," Ann. Probab., vol. 3, no. 1, pp. 100-118, 1975.
21. A. Beygelzimer, J. Langford, L. Li, L. Reyzin, and R. Schapire, "Contextual bandit algorithms with supervised learning guarantees," presented at the presented at the Int. Conf. Artif. Intell. Statist., Fort Lauderdale, FL, 2011.
22. V. Mnih, C. Szepesvári, and J.-Y. Audibert, "Empirical Bernstein stopping," in Proc. 25th Int. Conf. Mach. Learning, 2008, pp. 672-679.
23. J. Y. Audibert, R.Munos, and C. Szepesvári, "Exploration- exploitation trade-off using variance estimates in multi-Armed bandits," J. Theoretical Comput. Sci., vol. 410, pp. 1876-1902, 2009.
24. A. Maurer and M. Pontil, "Empirical Bernstein bounds and sample variance penalization," in Proc. Int. Conf. Comput. Learning Theory, 2009, pp. 1-9.
25. M. Seeger, "Bayesian Gaussian process models: PAC-Bayesian generalization error bounds and sparse approximations," Ph.D. dissertation, University of Edinburgh, Edinburgh, UK, 2003.