We introduce and study constrained Markov Decision Processes (cMDPs) with anytime constraints. An anytime constraint requires the agent to never violate its budget at any point in time, almost surely. Although Markovian policies are no longer sufficient, we show that there exist optimal deterministic policies augmented with cumulative costs. In fact, we present a fixed-parameter tractable reduction from anytime-constrained cMDPs to unconstrained MDPs. Our reduction yields planning and learning algorithms that are time and sample-efficient for tabular cMDPs so long as the precision of the costs is logarithmic in the size of the cMDP. However, we also show that computing non-trivial approximately optimal policies is NP-hard in general. To circumvent this bottleneck, we design provable approximation algorithms that efficiently compute or learn an approximately feasible policy with optimal value so long as the maximum supported cost is bounded by a polynomial in the cMDP or by the absolute budget. Given our hardness results, our approximation guarantees are the best possible in terms of tractability under worst-case analysis.

我们引入并研究了具有任意时间限制的受限马尔可夫决策过程（cMDPs）。我们提出了一种固定参数可处理的方法，将具有任意时间限制的cMDPs转化为无约束的MDPs。我们设计出了适用于大表cMDPs的计划和学习算法，并设计了近似算法，可以高效地计算或学习一个近似可行策略。

有时间限制的强化学习