Online Learning with Feedback Graphs: Beyond Bandits

We study a general class of online learning problems where the feedback is specified by a graph. This class includes online prediction with expert advice and the multiarmed bandit problem, but also several learning problems where the online player does not necessarily observe his own loss. We analyze how the structure of the feedback graph controls the inherent difficulty of the induced T -round learning problem. Specifically, we show that any feedback graph belongs to one of three classes: strongly observable graphs, weakly observable graphs, and unobservable graphs. We prove that the first class induces learning problems with Θ̃(α1/2T 1/2) minimax regret, where α is the independence number of the underlying graph; the second class induces problems with Θ̃(δ1/3T 2/3) minimax regret, where δ is the domination number of a certain portion of the graph; and the third class induces problems with linear minimax regret. Our results subsume much of the previous work on learning with feedback graphs and reveal new connections to partial monitoring games. We also show how the regret is affected if the graphs are allowed to vary with time. Tel Aviv University, Tel Aviv, Israel, and Microsoft Research, Herzliya, Israel, [email protected]. Dipartimento di Informatica, Università degli Studi di Milano, Milan, Italy, [email protected]. Parts of this work were done while the author was at Microsoft Research, Redmond. Microsoft Research, Redmond, Washington; [email protected]. Technion—Israel Institute of Technology, Haifa, Israel, and Microsoft Research, Herzliya, Israel, [email protected]. Parts of this work were done while the author was at Microsoft Research, Redmond.