  We consider a general framework of online learning with expert advice where
regret is defined with respect to sequences of experts accepted by a weighted
automaton. Our framework covers several problems previously studied, including
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 an approximation of the
competitor automaton, in particular n-gram models obtained by minimizing the
\infty-R\'{e}nyi divergence, and present an extensive study of the
approximation properties of such models. Finally, we also extend our algorithms
and results to the framework of sleeping experts.
