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We consider learning Nash equilibria in two-player zero-sum Markov Games withnonlinear function approximation, where the action-value function isapproximated by a function in a Reproducing Kernel Hilbert Space (RKHS). Thekey challenge is how to do exploration in the high-dimensional function space.We propose a novel online learning algorithm to find a Nash equilibrium byminimizing the duality gap. At the core of our algorithms are upper and lowerconfidence bounds that are derived based on the principle of optimism in theface of uncertainty. We prove that our algorithm is able to attain an$O(\sqrt{T})$ regret with polynomial computational complexity, under very mildassumptions on the reward function and the underlying dynamic of the MarkovGames. We also propose several extensions of our algorithm, including analgorithm with Bernstein-type bonus that can achieve a tighter regret bound,and another algorithm for model misspecification that can be applied to neuralfunction approximation.

Learning Two-Player Mixture Markov Games: Kernel Function Approximation and Correlated Equilibrium