Towards Instance-Optimality in Online PAC Reinforcement Learning
This work addresses a fundamental gap in reinforcement learning theory by providing lower bounds for instance-optimality, which is crucial for researchers in RL and optimization, though it is incremental as it builds on prior upper bounds.
The paper establishes the first instance-dependent lower bound on the sample complexity for PAC identification of near-optimal policies in tabular episodic MDPs, showing that the PEDEL algorithm nearly matches this bound but is computationally intractable.
Several recent works have proposed instance-dependent upper bounds on the number of episodes needed to identify, with probability $1-δ$, an $\varepsilon$-optimal policy in finite-horizon tabular Markov Decision Processes (MDPs). These upper bounds feature various complexity measures for the MDP, which are defined based on different notions of sub-optimality gaps. However, as of now, no lower bound has been established to assess the optimality of any of these complexity measures, except for the special case of MDPs with deterministic transitions. In this paper, we propose the first instance-dependent lower bound on the sample complexity required for the PAC identification of a near-optimal policy in any tabular episodic MDP. Additionally, we demonstrate that the sample complexity of the PEDEL algorithm of \cite{Wagenmaker22linearMDP} closely approaches this lower bound. Considering the intractability of PEDEL, we formulate an open question regarding the possibility of achieving our lower bound using a computationally-efficient algorithm.