Bellman Eluder Dimension: New Rich Classes of RL Problems, and Sample-Efficient Algorithms

TL;DR

This paper introduces the Bellman Eluder (BE) dimension as a new complexity measure for RL problems, demonstrating its broad applicability and designing algorithms with sample-efficient learning guarantees that are independent of state-action space size.

Contribution

The paper defines the BE dimension, shows its applicability to many RL problem classes, and develops algorithms with provably efficient learning guarantees for low BE dimension problems.

Findings

BE dimension encompasses many existing RL problem classes.

Algorithms GOLF and OLIVE achieve polynomial sample complexity for low BE dimension problems.

Sample complexity is independent of state-action space size.

Abstract

Finding the minimal structural assumptions that empower sample-efficient learning is one of the most important research directions in Reinforcement Learning (RL). This paper advances our understanding of this fundamental question by introducing a new complexity measure -- Bellman Eluder (BE) dimension. We show that the family of RL problems of low BE dimension is remarkably rich, which subsumes a vast majority of existing tractable RL problems including but not limited to tabular MDPs, linear MDPs, reactive POMDPs, low Bellman rank problems as well as low Eluder dimension problems. This paper further designs a new optimization-based algorithm -- GOLF, and reanalyzes a hypothesis elimination-based algorithm -- OLIVE (proposed in Jiang et al., 2017). We prove that both algorithms learn the near-optimal policies of low BE dimension problems in a number of samples that is polynomial in all relevant parameters, but independent of the size of state-action space. Our regret and sample complexity results match or improve the best existing results for several well-known subclasses of low BE dimension problems.