On Lower Bounds for Regret in Reinforcement Learning

TL;DR

This paper clarifies existing lower bounds on regret in reinforcement learning, reproduces key results, questions previous conjectures, and suggests potential improvements to upper bounds.

Contribution

It reproduces a known lower bound, critiques a previous conjecture, and proposes that upper bounds can be improved to match weaker lower bounds.

Findings

Reproduces a regret lower bound similar to UCRL2 results

Questions the validity of a previous conjecture on lower bounds

Suggests upper bounds can be tightened to match weaker lower bounds

Abstract

This is a brief technical note to clarify the state of lower bounds on regret for reinforcement learning. In particular, this paper: - Reproduces a lower bound on regret for reinforcement learning, similar to the result of Theorem 5 in the journal UCRL2 paper (Jaksch et al 2010). - Clarifies that the proposed proof of Theorem 6 in the REGAL paper (Bartlett and Tewari 2009) does not hold using the standard techniques without further work. We suggest that this result should instead be considered a conjecture as it has no rigorous proof. - Suggests that the conjectured lower bound given by (Bartlett and Tewari 2009) is incorrect and, in fact, it is possible to improve the scaling of the upper bound to match the weaker lower bounds presented in this paper. We hope that this note serves to clarify existing results in the field of reinforcement learning and provides interesting motivation for future work.