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Regret Bounds without Lipschitz Continuity: Online Learning with Relative-Lipschitz Losses
Yihan Zhou · Victor Sanches Portella · Mark Schmidt · Nicholas Harvey

Thu Dec 10 09:00 AM -- 11:00 AM (PST) @ Poster Session 5 #1387

In online convex optimization (OCO), Lipschitz continuity of the functions is commonly assumed in order to obtain sublinear regret. Moreover, many algorithms have only logarithmic regret when these functions are also strongly convex. Recently, researchers from convex optimization proposed the notions of relative Lipschitz continuity'' andrelative strong convexity''. Both of the notions are generalizations of their classical counterparts. It has been shown that subgradient methods in the relative setting have performance analogous to their performance in the classical setting.

In this work, we consider OCO for relative Lipschitz and relative strongly convex functions. We extend the known regret bounds for classical OCO algorithms to the relative setting. Specifically, we show regret bounds for the follow the regularized leader algorithms and a variant of online mirror descent. Due to the generality of these methods, these results yield regret bounds for a wide variety of OCO algorithms. Furthermore, we further extend the results to algorithms with extra regularization such as regularized dual averaging.

Author Information

Yihan Zhou (University of British Columbia)
Victor Sanches Portella (University of British Columbia)
Mark Schmidt (University of British Columbia)
Nicholas Harvey (University of British Columbia)

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