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Second-order mirror descent: exact convergence beyond strictly stable equilibria in concave games

We propose a second-order extension of the continuous-time game-theoretic mirror descent (MD) dynamics, which we refer to as MD2, and show that MD2 can converge to merely (not necessarily strictly) variationally stable Nash equilibria while enjoying no-regret. MD2 also overcomes the inexact convergence problem of discounted MD. We then derive the rate of convergence of an augmented MD2 whenever the equilibrium is strongly variationally stable. Selected simulations are provided to illustrate and affirm our results.