Abstract
This paper mainly focuses on the least square regularized regression learning algorithm in a setting of unbounded sampling. Our task is to establish learning rates by means of integral operators. By imposing a moment hypothesis on the unbounded sampling outputs and a function space condition associated with marginal distribution ρ X , we derive learning rates which are consistent with those in the bounded sampling setting.
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References
Bennett C., Sharpley R.: Interpolation of Operators. Academic Press, Boston (1988)
Bennett G.: Probability inequalities for the sum of indpendent random variables. J. Am. Stat. Assoc. 57, 33–45 (1962)
Caponnetto A., De Vito E.: Optimal rates for regularized least squares algorithm. Found. Comput. Math. 7, 331–368 (2007)
Cucker F., Smale S.: On the mathmatical foundations of learning. Am. Math. Soc. 39, 1–49 (2001)
Cucker F., Zhou D.X.: Learning Theory: An Approximation Theory Viewpoint. Cambridge University Press, Cambridge (2007)
Evgeniou T., Pontil M., Poggio T.: Reguarization networks and support vector machines. Adv. Comput. Math. 13, 1–50 (2000)
Guo, Z.C., Zhou, D.X.: Concentration estimates for learning with unbounded sampling (preprint, 2010)
Mendelson S., Neeman J.: Regularization in kernel learning. Ann. Stat. 38, 526–565 (2010)
Pinelis I.F., Sakhanenko A.I.: Remarks on inequalities for probabilities of large deviations. Theory Probab. Appl. 30, 143–148 (1985)
Smale S., Zhou D.X.: Shannon sampling II. Connection to learning theory. Appl. Comput. Harmonic Anal. 19, 285–312 (2005)
Smale S., Zhou D.X.: Learning Theory estimates via intergal operators and their approximations. Constr. Approx. 26, 153–172 (2007)
Smale S., Zhou D.X.: Geometry on probability spaces. Constr. Approx. 30, 311–323 (2009)
Smale S., Zhou D.X.: Online learning with Markov sampling. Anal. Appl. 7, 87–113 (2009)
Steinwart, I., Hush, D., Scovel, C.: Optimal rates for regularized least squares regression. In: Dasgupta, S., Klivans, A. (eds.) Proceedings of the 22nd Annual Conference on Learning Theory, pp. 79–93 (2009)
Sun H.W., Wu Q.: Regularized least equare regression with dependent samples. Adv. Comput. Math. 11, 235–249 (2008)
Sun H.W., Wu Q.: A note on application of integral operator in learning theory. Appl. Comput. Harmonic Anal. 26, 416–421 (2009)
Wang C., Zhou D.X.: Optimal learning rates for least square regularized regression with unbounded sampling. J. Complexity 27, 55–67 (2011)
Wu Q., Ying Y.M., Zhou D.X.: Learning rates of least-square regularized regression. Found. Comput. Math. 6, 171–192 (2006)
Ye G.B., Zhou D.X.: SVM learning and L p approximation by Gaussians on Riemannian manifolds. Anal. Appl. 7, 309–339 (2009)
Yurinsky Y.: Sums and Gaussian Vectors. Lecture Notes in Mathematics. Springer, Berlin, Heidelberg (1995)
Zhang T.: Leave-one-out bounds for kernel methods. Neural Comput. 15, 1397–1437 (2003)
Zhou D.X.: Capacity of reproducing kernel spaces in learning theory. IEEE Trans. Inform. Theory 49, 1743–1752 (2003)
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Communicated by L. Littlejohn and J. Stochel.
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Lv, SG., Feng, YL. Integral Operator Approach to Learning Theory with Unbounded Sampling. Complex Anal. Oper. Theory 6, 533–548 (2012). https://doi.org/10.1007/s11785-011-0139-0
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DOI: https://doi.org/10.1007/s11785-011-0139-0
Keywords
- Least square regularized regression
- Reproducing kernel Hilbert spaces
- Integral operator
- Capacity independent error bounds