Artificial Constraints and Hints for Unbounded Online Learning

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 $G$.