Forward Looking Best-Response Multiplicative Weights Update Methods for Bilinear Zero-sum Games
This addresses convergence issues in game theory for researchers, offering a novel algorithmic approach with practical improvements.
The paper tackles the problem of finding Nash equilibria in bilinear zero-sum games by proposing a variant of Optimistic Mirror Descent with a large intermediate learning rate, proving last-iterate convergence and showing accelerated convergence in experiments compared to existing methods.
Our work focuses on extra gradient learning algorithms for finding Nash equilibria in bilinear zero-sum games. The proposed method, which can be formally considered as a variant of Optimistic Mirror Descent \cite{DBLP:conf/iclr/MertikopoulosLZ19}, uses a large learning rate for the intermediate gradient step which essentially leads to computing (approximate) best response strategies against the profile of the previous iteration. Although counter-intuitive at first sight due to the irrationally large, for an iterative algorithm, intermediate learning step, we prove that the method guarantees last-iterate convergence to an equilibrium. Particularly, we show that the algorithm reaches first an $η^{1/ρ}$-approximate Nash equilibrium, with $ρ> 1$, by decreasing the Kullback-Leibler divergence of each iterate by at least $Ω(η^{1+\frac{1}ρ})$, for sufficiently small learning rate, $η$, until the method becomes a contracting map, and converges to the exact equilibrium. Furthermore, we perform experimental comparisons with the optimistic variant of the multiplicative weights update method, by \cite{Daskalakis2019LastIterateCZ} and show that our algorithm has significant practical potential since it offers substantial gains in terms of accelerated convergence.