  This paper proposes a payoff perturbation technique for the Mirror Descent
(MD) algorithm in games where the gradient of the payoff functions is monotone
in the strategy profile space, potentially containing additive noise. The
optimistic family of learning algorithms, exemplified by optimistic MD,
successfully achieves {\it last-iterate} convergence in scenarios devoid of
noise, leading the dynamics to a Nash equilibrium. A recent re-emerging trend
underscores the promise of the perturbation approach, where payoff functions
are perturbed based on the distance from an anchoring, or {\it slingshot},
strategy. In response, we propose {\it Adaptively Perturbed MD} (APMD), which
adjusts the magnitude of the perturbation by repeatedly updating the slingshot
strategy at a predefined interval. This innovation empowers us to find a Nash
equilibrium of the underlying game with guaranteed rates. Empirical
demonstrations affirm that our algorithm exhibits significantly accelerated
convergence.
