Abstract
In this paper we study the learning performance of regularized least square regression with α-mixing and ϕ-mixing inputs. The capacity independent error bounds and learning rates are derived by means of an integral operator technique. Even for independent samples our learning rates improve those in the literature. The results are sharp in the sense that when the mixing conditions are strong enough the rates are shown to be close to or the same as those for learning with independent samples. They also reveal interesting phenomena of learning with dependent samples: (i) dependent samples contain less information and lead to worse error bounds than independent samples; (ii) the influence of the dependence between samples to the learning process decreases as the smoothness of the target function increases.
Similar content being viewed by others
References
Aronszajn, N.: Theory of reproducing kernels. Trans. Amer. Math. Soc. 68, 337–404 (1950)
Athreya, K.B., Pantula, S.G.: Mixing properties of Harris chains and autoregressive processes. J. Appl. Probab. 23, 880–892 (1986)
Bartlett, P.L., Mendelson, S.: Rademacher and Gaussian complexities: risk bounds and structural results. J. Mach. Learn. Res. 3, 463–482 (2002)
Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)
Bousquet, O., Elisseeff, A.: Stability and generalization. J. Mach. Learn. Res. 2, 499–526 (2002)
Cucker, F., Zhou, D.X.: Learning Theory: An Approximation Theory Viewpoint. Cambridge University Press, Cambridge (2007)
Davydov, Y.A.: The invariance principle for stationary processes. Theory Probab. Appl. 14, 487–498 (1970)
Dehling, H., Philipp, W.: Almost sure invariance principles for weakly dependent vector-valued random variables. Ann. Probab. 10, 689–701 (1982)
Evgeniou, T., Pontil, M., Poggio, T.: Regularization networks and support vector machines. Adv. Comput. Math. 13, 1–50 (2000)
Li, L.Q., Wan, C.G.: Support vector machines with beta-mixing input sequences. In: Wang, J., et al. (eds.) Lecture Notes on Computer Science, vol. 3971, pp. 928–935. Springer, New York (2006)
Modha, D.S.: Minimum complexity regression estimation with weakly dependent observations. IEEE. Trans. Inform. Theory 42, 2133–2145 (1996)
Smale, S., Zhou, D.X.: Shannon sampling and function reconstruction from point values. Bull. Amer. Math. Soc. 41, 279–305 (2004)
Smale, S., Zhou, D.X.: Shannon sampling II: connections to learning theory. Appl. Comput. Harmon. Anal. 19, 285–302 (2005)
Smale, S., Zhou, D.X.: Learning theory estimates via integral operators and their approximations. Constr. Approx. 26, 153–172 (2007)
Vidyasagar, M.: Learning and Generalization with Applications to Neural Networks. Springer, Berlin Heidelberg New York (2003)
Withers, C.S.: Connectionist nonparametric regression: multilayer feedforward networks can learn arbitrary mappings. Neural Netw. 3, 535–549 (2000)
Wu, Q., Ying, Y.M., Zhou, D.X.: Learning rates of least-square regularized regression. Found. Comput. Math. 6, 171–192 (2006)
Xu, Y.L., Chen, D.R.: Learning rates of regularized regression for exponentially strongly mixing sequence. J. Statist. Plann. Inference 138(7), 2180–2189 (2008)
Zhang, T.: Leave-one-out bounds for kernel methods. Neural Comput. 15, 1397–1437 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Yuesheng Xu.
Rights and permissions
About this article
Cite this article
Sun, H., Wu, Q. Regularized least square regression with dependent samples. Adv Comput Math 32, 175–189 (2010). https://doi.org/10.1007/s10444-008-9099-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-008-9099-y
Keywords
- Regularized least square regression
- Integral operator
- Strong mixing condition
- Capacity independent error bounds