L1 Covering Numbers for Uniformly Bounded Convex Functions

Adityanand Guntuboyina, Bodhisattva Sen
Proceedings of the 25th Annual Conference on Learning Theory, PMLR 23:12.1-12.13, 2012.

Abstract

In this paper we study the covering numbers of the space of convex and uniformly bounded functions in multi-dimension. We find optimal upper and lower bounds for the ε-covering number \emphM(\emphC([\empha, b]^\emphd, \emphB), ε, \emphL_1) in terms of the relevant constants, where \emphd > 1, \empha < \emphb ∈ \emphR, \emphB > 0, and \emphC([\empha, b]^\emphd, \emphB) denotes the set of all convex functions on [\empha, b]^\emphd that are uniformly bounded by \emphB. We summarize previously known results on covering numbers for convex functions and also provide alternate proofs of some known results. Our results have direct implications in the study of rates of convergence of empirical minimization procedures as well as optimal convergence rates in the numerous convexity constrained function estimation problems.

Cite this Paper

BibTeX
@InProceedings{pmlr-v23-guntuboyina12, title = {L1 Covering Numbers for Uniformly Bounded Convex Functions}, author = {Guntuboyina, Adityanand and Sen, Bodhisattva}, booktitle = {Proceedings of the 25th Annual Conference on Learning Theory}, pages = {12.1--12.13}, year = {2012}, editor = {Mannor, Shie and Srebro, Nathan and Williamson, Robert C.}, volume = {23}, series = {Proceedings of Machine Learning Research}, address = {Edinburgh, Scotland}, month = {25--27 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v23/guntuboyina12/guntuboyina12.pdf}, url = {https://proceedings.mlr.press/v23/guntuboyina12.html}, abstract = {In this paper we study the covering numbers of the space of convex and uniformly bounded functions in multi-dimension. We find optimal upper and lower bounds for the ε-covering number \emphM(\emphC([\empha, b]^\emphd, \emphB), ε, \emphL_1) in terms of the relevant constants, where \emphd > 1, \empha < \emphb ∈ \emphR, \emphB > 0, and \emphC([\empha, b]^\emphd, \emphB) denotes the set of all convex functions on [\empha, b]^\emphd that are uniformly bounded by \emphB. We summarize previously known results on covering numbers for convex functions and also provide alternate proofs of some known results. Our results have direct implications in the study of rates of convergence of empirical minimization procedures as well as optimal convergence rates in the numerous convexity constrained function estimation problems.} }
Endnote
%0 Conference Paper %T L1 Covering Numbers for Uniformly Bounded Convex Functions %A Adityanand Guntuboyina %A Bodhisattva Sen %B Proceedings of the 25th Annual Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2012 %E Shie Mannor %E Nathan Srebro %E Robert C. Williamson %F pmlr-v23-guntuboyina12 %I PMLR %P 12.1--12.13 %U https://proceedings.mlr.press/v23/guntuboyina12.html %V 23 %X In this paper we study the covering numbers of the space of convex and uniformly bounded functions in multi-dimension. We find optimal upper and lower bounds for the ε-covering number \emphM(\emphC([\empha, b]^\emphd, \emphB), ε, \emphL_1) in terms of the relevant constants, where \emphd > 1, \empha < \emphb ∈ \emphR, \emphB > 0, and \emphC([\empha, b]^\emphd, \emphB) denotes the set of all convex functions on [\empha, b]^\emphd that are uniformly bounded by \emphB. We summarize previously known results on covering numbers for convex functions and also provide alternate proofs of some known results. Our results have direct implications in the study of rates of convergence of empirical minimization procedures as well as optimal convergence rates in the numerous convexity constrained function estimation problems.
RIS
TY - CPAPER TI - L1 Covering Numbers for Uniformly Bounded Convex Functions AU - Adityanand Guntuboyina AU - Bodhisattva Sen BT - Proceedings of the 25th Annual Conference on Learning Theory DA - 2012/06/16 ED - Shie Mannor ED - Nathan Srebro ED - Robert C. Williamson ID - pmlr-v23-guntuboyina12 PB - PMLR DP - Proceedings of Machine Learning Research VL - 23 SP - 12.1 EP - 12.13 L1 - http://proceedings.mlr.press/v23/guntuboyina12/guntuboyina12.pdf UR - https://proceedings.mlr.press/v23/guntuboyina12.html AB - In this paper we study the covering numbers of the space of convex and uniformly bounded functions in multi-dimension. We find optimal upper and lower bounds for the ε-covering number \emphM(\emphC([\empha, b]^\emphd, \emphB), ε, \emphL_1) in terms of the relevant constants, where \emphd > 1, \empha < \emphb ∈ \emphR, \emphB > 0, and \emphC([\empha, b]^\emphd, \emphB) denotes the set of all convex functions on [\empha, b]^\emphd that are uniformly bounded by \emphB. We summarize previously known results on covering numbers for convex functions and also provide alternate proofs of some known results. Our results have direct implications in the study of rates of convergence of empirical minimization procedures as well as optimal convergence rates in the numerous convexity constrained function estimation problems. ER -
APA
Guntuboyina, A. & Sen, B.. (2012). L1 Covering Numbers for Uniformly Bounded Convex Functions. Proceedings of the 25th Annual Conference on Learning Theory, in Proceedings of Machine Learning Research 23:12.1-12.13 Available from https://proceedings.mlr.press/v23/guntuboyina12.html.

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