We study the global convergence of a Fisher-Rao policy gradient flow for infinite-horizon entropy-regularised Markov decision processes with Polish state and action space. The flow is a continuous-time analogue of a policy mirror descent method. We establish the global well-posedness of the gradient flow and demonstrate its exponential convergence to the optimal policy. Moreover, we prove the flow is stable with respect to gradient evaluation, offering insights into the performance of a natural policy gradient flow with log-linear policy parameterisation. To overcome challenges stemming from the lack of the convexity of the objective function and the discontinuity arising from the entropy regulariser, we leverage the performance difference lemma and the duality relationship between the gradient and mirror descent flows.

翻译：研究无穷时域下具有Polish状态与动作空间的熵正则化马尔可夫决策过程中Fisher-Rao策略梯度流的全局收敛性。该流是策略镜像下降法的连续时间类比。我们建立了梯度流的全局适定性，并证明了其以指数速度收敛至最优策略。进一步，我们证明了该流对梯度评估具有稳定性，为对数线性策略参数化下的自然策略梯度流性能提供了理论洞见。为克服目标函数缺乏凸性及熵正则化项导致的非连续性带来的挑战，我们借助性能差异引理以及梯度流与镜像下降流之间的对偶关系展开研究。