Near Instance-Optimal PAC Reinforcement Learning for Deterministic MDPs

TL;DR

This paper establishes nearly optimal sample complexity bounds for PAC reinforcement learning in deterministic episodic MDPs, introducing the deterministic return gap and novel exploration strategies.

Contribution

It provides the first nearly matching upper and lower bounds on sample complexity for PAC RL in deterministic MDPs, with new theoretical insights and simple algorithms.

Findings

Established nearly matching upper and lower bounds on sample complexity.

Introduced the deterministic return gap as a new measure of sub-optimality.

Developed simple algorithms using graph theory and maximum-coverage exploration.

Abstract

In probably approximately correct (PAC) reinforcement learning (RL), an agent is required to identify an $\epsilon$-optimal policy with probability $1-\delta$. While minimax optimal algorithms exist for this problem, its instance-dependent complexity remains elusive in episodic Markov decision processes (MDPs). In this paper, we propose the first nearly matching (up to a horizon squared factor and logarithmic terms) upper and lower bounds on the sample complexity of PAC RL in deterministic episodic MDPs with finite state and action spaces. In particular, our bounds feature a new notion of sub-optimality gap for state-action pairs that we call the deterministic return gap. While our instance-dependent lower bound is written as a linear program, our algorithms are very simple and do not require solving such an optimization problem during learning. Their design and analyses employ novel ideas, including graph-theoretical concepts (minimum flows) and a new maximum-coverage exploration strategy.