We consider a general framework of online learning with expert advice where the regret is defined with respect to a competitor class defined by a weighted automaton over sequences of experts. Our framework covers several problems previously studied, in particular that of competing against k-shifting experts. We give a series of algorithms for this problem, including an automata-based algorithm extending weighted- majority and more efficient algorithms based on the notion of failure transitions. We further present efficient algorithms based on a compact approximation of the competitor automaton, in particular efficient n-gram models obtained by minimizing the Renyi divergence, and present an extensive study of the approximation properties of such models. We also extend our algorithms and results to the framework of sleeping experts. Finally, we describe the extension of our approximation methods to online convex optimization and a general mirror descent setting.
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