Approachability, fast and slow

author: Vianney Perchet, Probabilities and Random Models Laboratory, Université Pierre et Marie Curie (UPMC)
published: Aug. 9, 2013,   recorded: June 2013,   views: 3038


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Approachability has become a central tool in the analysis of repeated games and online learning. A player plays a repeated vector-valued game against Nature and her objective is to have her long-term average reward inside some target set. The celebrated results of Blackwell provide a 1/n−√ convergence rate of the expected point-to-set distance if this is achievable, i.e., if the set is approachable. In this paper we provide a characterization for the convergence rates of approachability and show that in some cases a set can be approached with a 1/n rate. Our characterization is solely based on a combination of geometric properties of the set with properties of the repeated game, and not on additional restrictive assumptions on Nature’s behavior.

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