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Classification with non-i.i.d. sampling. (English) Zbl 1228.62074
Summary: We study learning algorithms for classification generated by regularization schemes in reproducing kernel Hilbert spaces associated with a general convex loss function in a non-i.i.d. process. Error analysis is studied and our main purpose is to provide elaborate capacity dependent error bounds by applying concentration techniques involving the $\ell ^{2}$-empirical covering numbers.

62H30Classification and discrimination; cluster analysis (statistics)
68T99Artificial intelligence
46N30Applications of functional analysis in probability theory and statistics
Full Text: DOI
[1] Evgeniou, T.; Pontil, M.; Poggio, T.: Regularization networks and suport vector machine, Advances in computational mathematics 10, 51-80 (1999)
[2] Blanchard, G.; Bousquet, O.; Massart, P.: Statistical performance of support vector machines, Annals of statistics 36, 489-531 (2008) · Zbl 1133.62044
[3] Bartlett, P. L.; Jordan, M. I.; Mcauliffe, J. D.: Convexity, classification, and risk bounds, Journal of the American statistical association 101, 138-156 (2006) · Zbl 1118.62330
[4] Chen, D. R.; Wu, Q.; Ying, Y.; Zhou, D. X.: Support vector machine soft margin classifiers: error analysis, Journal of machine learning research 5, 1143-1175 (2004) · Zbl 1222.68167
[5] Steinwart, I.; Scovel, C.: Fast rates for suppport vector machines using Gaussian kernels, Annals of statistics 35, 575-607 (2007) · Zbl 1127.68091
[6] Wu, Q.; Ying, Y.; Zhou, D. X.: Multi-kernel regularized classifiers, Journal of complexity 23, 108-134 (2007) · Zbl 1171.65043
[7] Zhang, T.: Statistical behavior and consistency of classification methods based on convex risk minimization, Annals of statistics 32, 56-85 (2004) · Zbl 1105.62323
[8] Smale, S.; Zhou, D. X.: Online learning with Markov sampling, Analysis and applications 7, 87-113 (2009) · Zbl 1170.68022
[9] Xiao, Q. W.; Pan, Z. W.: Learning from non-identical sampling for classification, Advances in computational mathematics 33, 97-112 (2010) · Zbl 1213.68511
[10] Yu, B.: Rates of convergence for emipircal processes of stationary mixing sequence, The annals of probability 22, 94-116 (1994)
[11] Bradley, R. C.: Basic properties of strong mixing conditions. A survey and some open questions, Probability surveys 2, 107-144 (2005) · Zbl 1189.60077
[12] Vidyasagar, M.: Learning and generalization: with applications to neural networks, (2003) · Zbl 1008.68102
[13] Modha, D. S.; Masry, E.: Minimum complexity regression estimation with weakly dependent observations, IEEE transactions on information theory 42, 2133-2145 (2002) · Zbl 0868.62015
[14] Mohri, M.; Rostamizadeh, A.: Rademacher complexity bounds for non-i.i.d. Processes, Advances in neural information processing systems 21, 1097-1104 (2009)
[15] Mohri, M.; Rostamizadeh, A.: Stability bounds for stationary $\phi $-mixing and ${\beta}$-mixing processes, Journal of machine learning research 11, 789-814 (2010) · Zbl 1242.68238
[16] Pan, Z. W.; Xiao, Q. W.: Least-squares regularized regression with non-i.i.d. Sampling, Journal of statistical planning and inference 139, 3579-3587 (2009) · Zbl 1176.68163
[17] Steinwart, I.; Hush, D.; Scovel, C.: Learning from dependent observations, Journal of multivariate analysis 100, 175-194 (2008) · Zbl 1158.68040
[18] Sun, H.; Wu, Q.: Regularized least squares regression with dependent samples, Advances in computational mathematics 32, 1019-7168 (2010)
[19] Xu, Y. L.; Chen, D. R.: Learning rates of regularized regression for exponentially strongly mixing sequence, Journal of statistical planning and inference 138, 2180-2189 (2008) · Zbl 1134.62050
[20] Smale, S.; Zhou, D. X.: Learning theory estimates via integral operators and their approximations, Constructive approximation 26, 153-172 (2007) · Zbl 1127.68088
[21] Cucker, F.; Zhou, D. X.: Learning theory: an approximation theory viewpoint, (2007) · Zbl 1274.41001
[22] Zhou, D. X.: The covering number in learning theory, Journal of complexity 18, 739-767 (2002) · Zbl 1016.68044
[23] Xiang, D. H.: Classification with gaussians and convex loss II: Improving error bounds by noise conditions, Science China mathematics 53, 1-7 (2010)
[24] Van Der Vaart, A. W.; Wellner, J. A.: Weak convergence and emprical processes, (1996) · Zbl 0862.60002
[25] Zhou, D. X.: Capacity of reproducing kernel spaces in learning theory, IEEE transactions on information theory 49, 1743-1752 (2003) · Zbl 1290.62033
[26] Bousquet, O.: A bennett concentration inequality and its application to surprema of emprical processes, Comptes rendus mathematique 334, 495-500 (2002) · Zbl 1001.60021
[27] Mendelson, S.: Improving the sample complexity using global data, IEEE transactions on information theory 7, 1977-1991 (2002) · Zbl 1061.68128