In this paper, we study reinforcement learning in Markov Decision Processes with Probabilistic Reward Machines (PRMs), a form of non-Markovian reward commonly found in robotics tasks. We design an algorithm for PRMs that achieves a regret bound of $\widetilde{O}(\sqrt{HOAT} + H^2O^2A^{3/2} + H\sqrt{T})$, where $H$ is the time horizon, $O$ is the number of observations, $A$ is the number of actions, and $T$ is the number of time-steps. This result improves over the best-known bound, $\widetilde{O}(H\sqrt{OAT})$ of \citet{pmlr-v206-bourel23a} for MDPs with Deterministic Reward Machines (DRMs), a special case of PRMs. When $T \geq H^3O^3A^2$ and $OA \geq H$, our regret bound leads to a regret of $\widetilde{O}(\sqrt{HOAT})$, which matches the established lower bound of $\Omega(\sqrt{HOAT})$ for MDPs with DRMs up to a logarithmic factor. To the best of our knowledge, this is the first efficient algorithm for PRMs. Additionally, we present a new simulation lemma for non-Markovian rewards, which enables reward-free exploration for any non-Markovian reward given access to an approximate planner. Complementing our theoretical findings, we show through extensive experiment evaluations that our algorithm indeed outperforms prior methods in various PRM environments.

本研究针对在带有概率奖励机器的马尔可夫决策过程中强化学习的挑战，设计了一种新算法，显著改善了已知的后悔界限。我们的方法首次在理论和实验层面上展示了针对非马尔可夫奖励的高效学习能力，具有广泛的潜在应用价值。

在概率奖励机器中的高效强化学习