On the optimality of sample-based estimates of the expectation of the empirical minimizer

Bartlett, Peter L.; Mendelson, Shahar; Philips, Petra

ESAIM: Probability and Statistics, Tome 14 (2010), pp. 315-337.

We study sample-based estimates of the expectation of the function produced by the empirical minimization algorithm. We investigate the extent to which one can estimate the rate of convergence of the empirical minimizer in a data dependent manner. We establish three main results. First, we provide an algorithm that upper bounds the expectation of the empirical minimizer in a completely data-dependent manner. This bound is based on a structural result due to Bartlett and Mendelson, which relates expectations to sample averages. Second, we show that these structural upper bounds can be loose, compared to previous bounds. In particular, we demonstrate a class for which the expectation of the empirical minimizer decreases as O(1/n) for sample size n, although the upper bound based on structural properties is Ω(1). Third, we show that this looseness of the bound is inevitable: we present an example that shows that a sharp bound cannot be universally recovered from empirical data.

[1] P.L. Bartlett and S. Mendelson, Empirical minimization. Probab. Theory Relat. Fields 135 (2006) 311-334. | Zbl

[2] P.L. Bartlett and M.H. Wegkamp, Classification with a reject option using a hinge loss. J. Machine Learn. Res. 9 (2008) 1823-1840. | Zbl

[3] P.L. Bartlett, O. Bousquet and S. Mendelson, Local Rademacher Complexities. Ann. Statist. 33 (2005) 1497-1537. | Zbl

[4] P.L. Bartlett, M.I. Jordan and J.D. Mcauliffe, Convexity, classification, and risk bounds. J. Am. Statist. Assoc. 101 (2006) 138-156. | Zbl

[5] G. Blanchard, G. Lugosi and N. Vayatis, On the rate of convergence of regularized boosting classifiers. J. Mach. Learn. Res. 4 (2003) 861-894. | Zbl

[6] S. Boucheron, G. Lugosi and P. Massart, Concentration inequalities using the entropy method. Ann. Probab. 31 (2003) 1583-1614. | Zbl

[7] O. Bousquet, Concentration Inequalities and Empirical Processes Theory Applied to the Analysis of Learning Algorithms. Ph.D. thesis, École Polytechnique, 2002.

[8] R.M. Dudley, Uniform Central Limit Theorems, Cambridge University Press (1999). | Zbl

[9] D. Haussler, Sphere Packing Numbers for Subsets of the Boolean n-cube with Bounded Vapnik-Chervonenkis Dimension. J. Combin. Theory Ser. A 69 (1995) 217-232. | Zbl

[10] T. Klein, Une inégalité de concentration gauche pour les processus empiriques. C. R. Math. Acad. Sci. Paris 334 (2002) 501-504. | Zbl

[11] V. Koltchinskii, Local Rademacher Complexities and Oracle Inequalities in Risk Minimization. Ann. Statist. 34 (2006). | Zbl

[12] V. Koltchinskii and D. Panchenko, Rademacher processes and bounding the risk of function learning. High Dimensional Probability, Vol. II (2000) 443-459. | Zbl

[13] M. Ledoux, The Concentration of Measure Phenomenon, volume 89 of Mathematical Surveys and Monographs. American Mathematical Society (2001). | Zbl

[14] W.S. Lee, P.L. Bartlett and R.C. Williamson, The Importance of Convexity in Learning with Squared Loss. IEEE Trans. Informa. Theory 44 (1998) 1974-1980. | Zbl

[15] G. Lugosi and N. Vayatis, On the Bayes-risk consistency of regularized boosting methods (with discussion), Ann. Statist. 32 (2004) 30-55. | Zbl

[16] G. Lugosi and M. Wegkamp, Complexity regularization via localized random penalties. Ann. Statist. 32 (2004) 1679-1697. | Zbl

[17] P. Massart, The constants in Talagrand's concentration inequality for empirical processes. Ann. Probab. 28 (2000) 863-884. | Zbl

[18] P. Massart, Some applications of concentration inequalities to statistics. Ann. Fac. Sci. Toulouse Math. IX (2000) 245-303. | Numdam | Zbl

[19] P. Massart and E. Nédélec, Risk bounds for statistical learning. Ann. Statist. 34 (2006) 2326-2366. | Zbl

[20] S. Mendelson, Improving the sample complexity using global data. IEEE Trans. Inform. Theory 48 (2002) 1977-1991. | Zbl

[21] S. Mendelson, A few notes on Statistical Learning Theory. In Proc. of the Machine Learning Summer School, Canberra 2002, S. Mendelson and A. J. Smola (Eds.), LNCS 2600. Springer (2003). | Zbl

[22] E. Rio, Inégalités de concentration pour les processus empiriques de classes de parties. Probab. Theory Relat. Fields 119 (2001) 163-175. | Zbl

[23] M. Rudelson and R. Vershynin, Combinatorics of random processes and sections of convex bodies. Ann. Math. 164 (2006) 603-648. | Zbl

[24] M. Talagrand, Sharper Bounds for Gaussian and Empirical Processes. Ann. Probab. 22 (1994) 20-76. | Zbl

[25] M. Talagrand, New concentration inequalities in product spaces. Inventiones Mathematicae 126 (1996) 505-563. | Zbl

[26] B. Tarigan and S.A. Van De Geer, Adaptivity of support vector machines with ${\ell }_1$ penalty. Technical Report MI 2004-14, University of Leiden (2004).

[27] A. Tsybakov, Optimal aggregation of classifiers in statistical learning. Ann. Statist. 32 (2004) 135-166. | Zbl

[28] S.A. Van De Geer, A new approach to least squares estimation, with applications. Ann. Statist. 15 (1987) 587-602. | Zbl

[29] S.A. Van De Geer, Empirical Processes in M-Estimation, Cambridge University Press (2000). | Zbl

[30] A. Van Der Vaart and J. Wellner, Weak Convergence and Empirical Processes. Springer (1996). | Zbl

[31] V.N. Vapnik and A.Y. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl. 16 (1971) 264-280. | Zbl