Risk Bounds for Levy Processes in the PAC-Learning Framework

Chao Zhang, Dacheng Tao
Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, PMLR 9:948-955, 2010.

Abstract

Levy processes play an important role in the stochastic process theory. However, since samples are non-i.i.d., statistical learning results based on the i.i.d. scenarios cannot be utilized to study the risk bounds for Levy processes. In this paper, we present risk bounds for non-i.i.d. samples drawn from Levy processes in the PAC-learning framework. In particular, by using a concentration inequality for infinitely divisible distributions, we first prove that the function of risk error is Lipschitz continuous with a high probability, and then by using a specific concentration inequality for Levy processes, we obtain the risk bounds for non-i.i.d. samples drawn from Levy processes without Gaussian components. Based on the resulted risk bounds, we analyze the factors that affect the convergence of the risk bounds and then prove the convergence.

Cite this Paper

BibTeX
@InProceedings{pmlr-v9-zhang10a, title = {Risk Bounds for Levy Processes in the PAC-Learning Framework}, author = {Zhang, Chao and Tao, Dacheng}, booktitle = {Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics}, pages = {948--955}, year = {2010}, editor = {Teh, Yee Whye and Titterington, Mike}, volume = {9}, series = {Proceedings of Machine Learning Research}, address = {Chia Laguna Resort, Sardinia, Italy}, month = {13--15 May}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v9/zhang10a/zhang10a.pdf}, url = {https://proceedings.mlr.press/v9/zhang10a.html}, abstract = {Levy processes play an important role in the stochastic process theory. However, since samples are non-i.i.d., statistical learning results based on the i.i.d. scenarios cannot be utilized to study the risk bounds for Levy processes. In this paper, we present risk bounds for non-i.i.d. samples drawn from Levy processes in the PAC-learning framework. In particular, by using a concentration inequality for infinitely divisible distributions, we first prove that the function of risk error is Lipschitz continuous with a high probability, and then by using a specific concentration inequality for Levy processes, we obtain the risk bounds for non-i.i.d. samples drawn from Levy processes without Gaussian components. Based on the resulted risk bounds, we analyze the factors that affect the convergence of the risk bounds and then prove the convergence.} }
Endnote
%0 Conference Paper %T Risk Bounds for Levy Processes in the PAC-Learning Framework %A Chao Zhang %A Dacheng Tao %B Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2010 %E Yee Whye Teh %E Mike Titterington %F pmlr-v9-zhang10a %I PMLR %P 948--955 %U https://proceedings.mlr.press/v9/zhang10a.html %V 9 %X Levy processes play an important role in the stochastic process theory. However, since samples are non-i.i.d., statistical learning results based on the i.i.d. scenarios cannot be utilized to study the risk bounds for Levy processes. In this paper, we present risk bounds for non-i.i.d. samples drawn from Levy processes in the PAC-learning framework. In particular, by using a concentration inequality for infinitely divisible distributions, we first prove that the function of risk error is Lipschitz continuous with a high probability, and then by using a specific concentration inequality for Levy processes, we obtain the risk bounds for non-i.i.d. samples drawn from Levy processes without Gaussian components. Based on the resulted risk bounds, we analyze the factors that affect the convergence of the risk bounds and then prove the convergence.
RIS
TY - CPAPER TI - Risk Bounds for Levy Processes in the PAC-Learning Framework AU - Chao Zhang AU - Dacheng Tao BT - Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics DA - 2010/03/31 ED - Yee Whye Teh ED - Mike Titterington ID - pmlr-v9-zhang10a PB - PMLR DP - Proceedings of Machine Learning Research VL - 9 SP - 948 EP - 955 L1 - http://proceedings.mlr.press/v9/zhang10a/zhang10a.pdf UR - https://proceedings.mlr.press/v9/zhang10a.html AB - Levy processes play an important role in the stochastic process theory. However, since samples are non-i.i.d., statistical learning results based on the i.i.d. scenarios cannot be utilized to study the risk bounds for Levy processes. In this paper, we present risk bounds for non-i.i.d. samples drawn from Levy processes in the PAC-learning framework. In particular, by using a concentration inequality for infinitely divisible distributions, we first prove that the function of risk error is Lipschitz continuous with a high probability, and then by using a specific concentration inequality for Levy processes, we obtain the risk bounds for non-i.i.d. samples drawn from Levy processes without Gaussian components. Based on the resulted risk bounds, we analyze the factors that affect the convergence of the risk bounds and then prove the convergence. ER -
APA
Zhang, C. & Tao, D.. (2010). Risk Bounds for Levy Processes in the PAC-Learning Framework. Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 9:948-955 Available from https://proceedings.mlr.press/v9/zhang10a.html.

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