Fast swap regret minimization and applications to approximate correlated equilibria

• URL: http://arxiv.org/abs/2310.19647v2
• Date: Tue, 14 Nov 2023 17:39:27 GMT
• Title: Fast swap regret minimization and applications to approximate correlated equilibria
• Authors: Binghui Peng and Aviad Rubinstein
• Abstract summary: For any constant $varepsilon>0$, our algorithm obtains $varepsilon T$-swap regret within only $T = mathsfpolylog(n)$ rounds. Our algorithm has an exponential dependence on $varepsilon$, but we prove a new, matching lower bound.
• Score: 20.047624953965965
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We give a simple and computationally efficient algorithm that, for any constant $\varepsilon>0$, obtains $\varepsilon T$-swap regret within only $T = \mathsf{polylog}(n)$ rounds; this is an exponential improvement compared to the super-linear number of rounds required by the state-of-the-art algorithm, and resolves the main open problem of [Blum and Mansour 2007]. Our algorithm has an exponential dependence on $\varepsilon$, but we prove a new, matching lower bound. Our algorithm for swap regret implies faster convergence to $\varepsilon$-Correlated Equilibrium ($\varepsilon$-CE) in several regimes: For normal form two-player games with $n$ actions, it implies the first uncoupled dynamics that converges to the set of $\varepsilon$-CE in polylogarithmic rounds; a $\mathsf{polylog}(n)$-bit communication protocol for $\varepsilon$-CE in two-player games (resolving an open problem mentioned by [Babichenko-Rubinstein'2017, Goos-Rubinstein'2018, Ganor-CS'2018]); and an $\tilde{O}(n)$-query algorithm for $\varepsilon$-CE (resolving an open problem of [Babichenko'2020] and obtaining the first separation between $\varepsilon$-CE and $\varepsilon$-Nash equilibrium in the query complexity model). For extensive-form games, our algorithm implies a PTAS for $\mathit{normal}$ $\mathit{form}$ $\mathit{correlated}$ $\mathit{equilibria}$, a solution concept often conjectured to be computationally intractable (e.g. [Stengel-Forges'08, Fujii'23]).

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