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.

翻译：本文针对博弈中收益函数梯度在策略轮廓空间上单调（可能包含加性噪声）的场景，提出了一种面向镜像下降（MD）算法的收益扰动技术。以乐观MD为代表的乐观学习算法系列，在无噪声场景中成功实现了{\it 末轮迭代}收敛，使动态过程趋近纳什均衡。近期新兴研究趋势凸显了扰动方法的潜力——该方法基于与锚定策略（即{\it 弹弓}策略）的距离对收益函数进行扰动。为此，我们提出{\it 自适应扰动镜像下降法}（APMD），通过按预设时间间隔反复更新弹弓策略来动态调整扰动幅度。这一创新使我们能够以可保证的收敛速率找到潜在博弈的纳什均衡。实验证明，我们的算法展现出显著加速的收敛特性。