Continuous-Time Convergence Rates in Potential and Monotone Games
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In this paper, we provide exponential rates of convergence to the Nash equilibrium of continuous-time game dynamics such as mirror descent (MD) and actor-critic (AC) in N-player continuous games that are either potential games or monotone games but possibly potential-free. In the first part of this paper, under the assumption the game admits a relatively strongly concave potential, we show that MD and AC converge in 𝒪(e^-β t). In the second part of this paper, using relative concavity, we provide a novel relative characterization of monotone games and show that MD and its discounted version converge with 𝒪(e^-β t) in relatively strongly and relatively hypo-monotone games. Moreover, these rates extend their known convergence conditions and also improve the results in the potential game setup. Simulations are performed which empirically back up our results.
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