Artificial Constraints and Lipschitz Hints for Unconstrained Online Learning

Abstract: We provide algorithms that guarantee 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 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 $G$.