Artificial Constraints and Lipschitz Hints for Unconstrained Online Learning
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We provide algorithms that guarantee regret R_T(u)<Õ(Gu^3 + G(u+1)√(T)) or R_T(u)<Õ(Gu^3T^1/3 + GT^1/3+ Gu√(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 Gu√(T) is necessary. Previous penalties were exponential while our bounds are polynomial in all quantities. Further, given a known bound u< 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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