Variance-Aware Regret Bounds for Undiscounted Reinforcement Learning in MDPs

By Mohammad Sadegh Talebi, Odalric-Ambrym Maillard. 2018

79 citations recorded by the source corpus.

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 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 $\widetilde {\mathcal O}(DS\sqrt{AT})$ 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.

Subfields: Artificial Intelligence, Reinforcement Learning

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Citations in the corpus

Dueling RL: Reinforcement Learning with Trajectory Preferences (2021, background): “, 2020a; Lattimore and Hutter, 2012) and minimizing regret via balancing exploration and exploitation (Zhang and Ji, 2019; Azar et al., 2017; Ortner, 2020; Talebi and Maillard, 2018; Efroni et al., 2020; Domingues et al., 2020b).”
Tighter Problem-Dependent Regret Bounds in Reinforcement Learning without Domain Knowledge using Value Function Bounds (2019, background): “• Bounds that depend on MDP mixing: (Auer & Ortner, 2006; Talebi & Maillard, 2018) have bounds on the mix- ing time but hold for MDPs that are unichain or ergodic.”