37
3

Artificial Constraints and Lipschitz Hints for Unconstrained Online Learning

Abstract

We provide algorithms that guarantee regret RT(u)O~(Gu3+G(u+1)T)R_T(u)\le \tilde O(G\|u\|^3 + G(\|u\|+1)\sqrt{T}) or RT(u)O~(Gu3T1/3+GT1/3+GuT)R_T(u)\le \tilde O(G\|u\|^3T^{1/3} + GT^{1/3}+ G\|u\|\sqrt{T}) for online convex optimization with GG-Lipschitz losses for any comparison point uu without prior knowledge of either GG or u\|u\|. Previous algorithms dispense with the O(u3)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 GuTG\|u\|\sqrt{T} is necessary. Previous penalties were exponential while our bounds are polynomial in all quantities. Further, given a known bound uD\|u\|\le D, our same techniques allow us to design algorithms that adapt optimally to the unknown value of u\|u\| without requiring knowledge of GG.

View on arXiv
Comments on this paper