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Regret bounds without Lipschitz continuity : online learning with relative-Lipschitz losses Zhou, Yihan


Online convex optimization (OCO) is a powerful algorithmic framework that has extensive applications in different areas. Regret is a commonly-used measurement for the performance of algorithms in this framework. Lipschitz continuity of the cost functions is commonly assumed in order to obtain sublinear regret, that is to say, this condition is usually necessary for theoretical guarantees for good performances of OCO algorithms. Moreover, strong convexity of cost functions can sometimes give even better theoretical performance bounds, more specifically, logarithmic regret. Recently, researchers from convex optimization proposed the notions of “relative Lipschitz continuity” and “relative 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 con- vex 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 extend the results to algorithms with extra regularization such as regularized dual averaging.

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