Competing with Automata-based Expert Sequences

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 $∞$-Rényi 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.