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

TL;DR

This paper characterizes the asymptotic minimax regret in partial monitoring games with infinite or finite latent spaces, linking it to the generalized information ratio and Rustichini's regret, and provides concrete examples of regret rates.

Contribution

It extends the understanding of minimax regret in partial monitoring to infinite outcome spaces and connects it with the generalized information ratio and Rustichini's regret.

Findings

Minimax regret in infinite partial monitoring games can grow as n^p for p in [1/2,1].

Finite games can have minimax Rustichini regret of order n^{4/7}.

The generalized information ratio determines asymptotic minimax regret under specified conditions.

Abstract

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 $p \in [1/2,1]$ 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.