Risk bounds for statistical learning

Annals of Statistics

Volumn 34, Issue 5, 2006, Pages 2326-2366

  Massart, Pascal ^a   Nédélec, Élodie ^a  

^a UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

Classification; Concentration inequalities; Empirical processes; Entropy with bracketing; Minimax estimation; Model selection; Pattern recognition; Regression estimation; Statistical learning; Structural minimization of the risk; VC class; VC dimension

Indexed keywords

EID: 33746243474     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053606000000786     Document Type: Article

References (26)

1. BARRON, A. R., BIRGÉ, L. and MASSART, P. (1999). Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 301-413. MR1679028
2. BIRGÉ, L. (2005). A new lower bound for multiple hypothesis testing. IEEE Trans. Inform. Theory 51 1611-1615. MR2241522
3. BIRGÉ, L. and MASSART, P. (1998). Minimum contrast estimators on sieves: Exponential bounds and rates of convergence. Bernoulli 4 329-375. MR1653272
4. BOUSQUET, O. (2002). A Bennett concentration inequality and its application to suprema of empirical processes. C. R. Math. Acad. Sci. Paris 334 495-500. MR1890640
5. DEVROYE, L. and LUGOSI, G. (1995). Lower bounds in pattern recognition and learning. Pattern Recognition 28 1011-1018.
6. DUDLEY, R. M. (1999). Uniform Central Limit Theorems. Cambridge Univ. Press. MR1720712
7. EDELSBRUNNER, H. (1987). Algorithms in Combinatorial Geometry. Springer, Berlin. MR0904271
8. HAUSSLER, D. (1995). Sphere packing numbers for subsets of the Boolean n-cube with bounded Vapnik-Chervonenkis dimension. J. Combin. Theory Ser. A 69 217-232. MR1313896
9. HAUSSLER, D., LITTLESTONE, N. and WARMUTH, M. (1994). Predicting {0,1}-functions on randomly drawn points. Inform. and Comput. 115 248-292. MR1304811
10. KOLTCHINSKII, V. I. (1981). On the central limit theorem for empirical measures. Theor. Probab. Math. Statist. 24 71-82. MR0628431
11. KOROSTELEV, A. P. and TSYBAKOV, A. B. (1993). Minimax Theory of Image Reconstruction. Lecture Notes in Statist. 82. Springer, New York. MR1226450
12. LEDOUX, M. and TALAGRAND, M. (1991). Probability in Banach Spaces. Isoperimetry and Processes. Springer, Berlin. MR1102015
13. LUGOSI, G. (2002). Pattern classification and learning theory. In Principles of Nonparametric Learning (L. Györfi, ed.) 1-56. Springer, Vienna. MR1987656
14. MAMMEN, E. and TSYBAKOV, A. B. (1999). Smooth discrimination analysis. Ann. Statist. 27 1808-1829. MR1765618
15. MASSART, P. (2000). Some applications of concentration inequalities to statistics. Probability theory. Ann. Fac. Sci. Toulouse Math. (6) 9 245-303. MR1813803
16. MASSART, P. (2006). Concentration inequalities and model selection. Lectures on Probability Theory and Statistics. Ecole d'Eté de Probabilités de Saint Flour XXXIII. Lecture Notes in Math. 1896. Springer, Berlin. To appear.
17. MASSART, P. and RIO, E. (1998). A uniform Marcinkiewicz-Zygmund strong law of large numbers for empirical processes. In Festschrift for Miklós Csörgo: Asymptotic Methods in Probability and Statistics (B. Szyszkowicz, ed.) 199-211. North-Holland, Amsterdam. MR1661481
18. MCDIARMID, C. (1989). On the method of bounded differences. In Surveys in Combinatorics 1989 (J. Siemons, ed.) 148-188. Cambridge Univ. Press. MR1036755
19. POLLARD, D. (1982). A central limit theorem for empirical processes. J. Austral. Math. Soc. Ser. A 33 235-248. MR0668445
20. REYNAUD-BOURET, P. (2003). Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities. Probab. Theory Related Fields 126 103-153. MR1981635
21. TALAGRAND, M. (1996). New concentration inequalities in product spaces. Invent. Math. 126 505-563. MR1419006
22. TSYBAKOV, A. B. (2004). Optimal aggregation of classifiers in statistical learning. Ann. Statist. 32 135-166. MR2051002
23. VAPNIK, V. N. (1982). Estimation of Dependences Based on Empirical Data. Springer, New York. MR0672244
24. VAPNIK, V. N. and CHERVONENKIS, A. YA. (1974). Theory of Pattern Recognition. Nauka, Moscow. (In Russian.) MR0474638
25. YANG, Y. and BARRON, A. R. (1999). Information- theoretic determination of minimax rates of convergence. Ann. Statist. 27 1564-1599. MR1742500
26. YU, B. (1997). Assouad, Fano, and Le Cam. In Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics (D. Pollard, E. Torgersen and G. L. Yang, eds.) 423-435. Springer, New York. MR1462963