Artificial Constraints and Hints for Unbounded Online Learning

Ashok Cutkosky
Proceedings of the Thirty-Second Conference on Learning Theory, PMLR 99:874-894, 2019.

Abstract

We provide algorithms that guarantees regret $R_T(u)\le \tilde O(G\|u\|^3 + G(\|u\|+1)\sqrt{T})$ or $R_T(u)\le \tilde O(G\|u\|^3T^{1/3} + GT^{1/3}+ G\|u\|\sqrt{T})$ for online convex optimization with $G$-Lipschitz losses for any comparison point $u$ without prior knowledge of either $G$ or $\|u\|$. Previous algorithms dispense with the $O(\|u\|^3)$ term at the expense of knowledge of one or both of these parameters, while a lower bound shows that some additional penalty term over $G\|u\|\sqrt{T}$ is necessary. Previous penalties were \emph{exponential} while our bounds are polynomial in all quantities. Further, given a known bound $\|u\|\le D$, our same techniques allow us to design algorithms that adapt optimally to the unknown value of $\|u\|$ without requiring knowledge of $G$.

Cite this Paper

BibTeX
@InProceedings{pmlr-v99-cutkosky19a, title = {Artificial Constraints and Hints for Unbounded Online Learning}, author = {Cutkosky, Ashok}, booktitle = {Proceedings of the Thirty-Second Conference on Learning Theory}, pages = {874--894}, year = {2019}, editor = {Beygelzimer, Alina and Hsu, Daniel}, volume = {99}, series = {Proceedings of Machine Learning Research}, month = {25--28 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v99/cutkosky19a/cutkosky19a.pdf}, url = {https://proceedings.mlr.press/v99/cutkosky19a.html}, abstract = {We provide algorithms that guarantees regret $R_T(u)\le \tilde O(G\|u\|^3 + G(\|u\|+1)\sqrt{T})$ or $R_T(u)\le \tilde O(G\|u\|^3T^{1/3} + GT^{1/3}+ G\|u\|\sqrt{T})$ for online convex optimization with $G$-Lipschitz losses for any comparison point $u$ without prior knowledge of either $G$ or $\|u\|$. Previous algorithms dispense with the $O(\|u\|^3)$ term at the expense of knowledge of one or both of these parameters, while a lower bound shows that some additional penalty term over $G\|u\|\sqrt{T}$ is necessary. Previous penalties were \emph{exponential} while our bounds are polynomial in all quantities. Further, given a known bound $\|u\|\le D$, our same techniques allow us to design algorithms that adapt optimally to the unknown value of $\|u\|$ without requiring knowledge of $G$.} }
Endnote
%0 Conference Paper %T Artificial Constraints and Hints for Unbounded Online Learning %A Ashok Cutkosky %B Proceedings of the Thirty-Second Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2019 %E Alina Beygelzimer %E Daniel Hsu %F pmlr-v99-cutkosky19a %I PMLR %P 874--894 %U https://proceedings.mlr.press/v99/cutkosky19a.html %V 99 %X We provide algorithms that guarantees regret $R_T(u)\le \tilde O(G\|u\|^3 + G(\|u\|+1)\sqrt{T})$ or $R_T(u)\le \tilde O(G\|u\|^3T^{1/3} + GT^{1/3}+ G\|u\|\sqrt{T})$ for online convex optimization with $G$-Lipschitz losses for any comparison point $u$ without prior knowledge of either $G$ or $\|u\|$. Previous algorithms dispense with the $O(\|u\|^3)$ term at the expense of knowledge of one or both of these parameters, while a lower bound shows that some additional penalty term over $G\|u\|\sqrt{T}$ is necessary. Previous penalties were \emph{exponential} while our bounds are polynomial in all quantities. Further, given a known bound $\|u\|\le D$, our same techniques allow us to design algorithms that adapt optimally to the unknown value of $\|u\|$ without requiring knowledge of $G$.
APA
Cutkosky, A.. (2019). Artificial Constraints and Hints for Unbounded Online Learning. Proceedings of the Thirty-Second Conference on Learning Theory, in Proceedings of Machine Learning Research 99:874-894 Available from https://proceedings.mlr.press/v99/cutkosky19a.html.

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