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Approachability in unknown games: Online learning meets multi-objective optimization

In the standard setting of approachability there are two players and a target set. The players play repeatedly a known vector-valued game where the first player wants to have the average vector-valued payoff converge to the target set which the other player tries to exclude it from this set. We revisit this setting in the spirit of online learning and do not assume that the first player knows the game structure: she receives an arbitrary vector-valued reward vector at every round. She wishes to approach the smallest ("best") possible set given the observed average payoffs in hindsight. This extension of the standard setting has implications even when the original target set is not approachable and when it is not obvious which expansion of it should be approached instead. We show that it is impossible, in general, to approach the best target set in hindsight and propose achievable though ambitious alternative goals. We further propose a concrete strategy to approach these goals. Our method does not require projection onto a target set and amounts to switching between scalar regret minimization algorithms that are performed in episodes. Applications to global cost minimization and to approachability under sample path constraints are considered.

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