In many sequential decision-making problems, the goal is to optimize a utility function while satisfying a set of constraints on different utilities. This learning problem is formalized through Constrained Markov Decision Processes (CMDPs). In this paper, we investigate the exploration-exploitation dilemma in CMDPs. While learning in an unknown CMDP, an agent should trade-off exploration to discover new information about the MDP, and exploitation of the current knowledge to maximize the reward while satisfying the constraints. While the agent will eventually learn a good or optimal policy, we do not want the agent to violate the constraints too often during the learning process. In this work, we analyze two approaches for learning in CMDPs. The first approach leverages the linear formulation of CMDP to perform optimistic planning at each episode. The second approach leverages the dual formulation (or saddle-point formulation) of CMDP to perform incremental, optimistic updates of the primal and dual variables. We show that both achieves sublinear regret w.r.t.\ the main utility while having a sublinear regret on the constraint violations. That being said, we highlight a crucial difference between the two approaches; the linear programming approach results in stronger guarantees than in the dual formulation based approach.

本文研究了Constrained Markov Decision Processes下的exploration-exploitation trade-off问题，提出了两种方法：基于线性规划和基于对偶变量逐步更新的方法。研究结果表明，这两种方法都可以实现sublinear regret，但是线性规划方法具有更强的保障性。

有限马尔可夫决策问题中的勘探-利用