Minimax Regret for Partial Monitoring: Infinite Outcomes and Rustichini's Regret

02/22/2022

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by   Tor Lattimore, et al.

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We show that a version of the generalised information ratio of Lattimore and Gyorgy (2020) determines the asymptotic minimax regret for all finite-action partial monitoring games provided that (a) the standard definition of regret is used but the latent space where the adversary plays is potentially infinite; or (b) the regret introduced by Rustichini (1999) is used and the latent space is finite. Our results are complemented by a number of examples. For any p ∈ [1/2,1] there exists an infinite partial monitoring game for which the minimax regret over n rounds is n^p up to subpolynomial factors and there exist finite games for which the minimax Rustichini regret is n^4/7 up to subpolynomial factors.