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

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.