Artificial Constraints and Hints for Unbounded Online Learning
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Proceedings of the ThirtySecond Conference on Learning Theory, PMLR 99:874894, 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 $G$.
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