Artificial Constraints and Lipschitz Hints for Unconstrained Online Learning

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