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In online convex optimization (OCO), Lipschitz continuity of the functions iscommonly assumed in order to obtain sublinear regret. Moreover, many algorithmshave only logarithmic regret when these functions are also strongly convex.Recently, researchers from convex optimization proposed the notions of"relative Lipschitz continuity" and "relative strong convexity". Both of thenotions are generalizations of their classical counterparts. It has been shownthat subgradient methods in the relative setting have performance analogous totheir performance in the classical setting.  In this work, we consider OCO for relative Lipschitz and relative stronglyconvex functions. We extend the known regret bounds for classical OCOalgorithms to the relative setting. Specifically, we show regret bounds for thefollow the regularized leader algorithms and a variant of online mirrordescent. Due to the generality of these methods, these results yield regretbounds for a wide variety of OCO algorithms. Furthermore, we further extend theresults to algorithms with extra regularization such as regularized dualaveraging.

Regret Bounds without Lipschitz Continuity: Online Learning with Relative-Lipschitz Losses