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# Variance-Aware Regret Bounds for Undiscounted Reinforcement Learning in MDPs

*Proceedings of Algorithmic Learning Theory*, PMLR 83:770-805, 2018.

#### Abstract

The problem of reinforcement learning in an unknown and discrete Markov Decision Process (MDP) under the average-reward criterion is considered, when
the learner interacts with the system in a single stream of observations, starting from an initial state without any reset.
We revisit the minimax lower bound for that problem by making appear the local variance of the bias function in place of the diameter of the MDP.
Furthermore, we provide a novel analysis of the \texttt{\textsc{KL-Ucrl}} algorithm establishing a high-probability regret bound scaling as
$\widetilde {\mathcal O}\Bigl({\textstyle \sqrt{S\sum_{s,a}{\bf V}^\star_{s,a}T}}\Big)$ for this algorithm for ergodic MDPs, where $S$ denotes the number of states and
where ${\bf V}^\star_{s,a}$ is the variance of the bias function with respect to the next-state distribution following action $a$ in state $s$.
The resulting bound improves upon the best previously known regret bound $\widetilde {\Ocal}(DS\sqrt{AT})$ for that algorithm, where $A$ and $D$ respectively denote
the maximum number of actions (per state) and the diameter of MDP. We finally compare the leading terms of the two bounds in some benchmark MDPs indicating
that the derived bound can provide an order of magnitude improvement in some cases. Our analysis leverages novel variations of the transportation lemma
combined with Kullback-Leibler concentration inequalities, that we believe to be of independent interest.