Optimistic Q-learning for average reward and episodic reinforcement learning

TL;DR

This paper introduces an optimistic Q-learning algorithm for average reward reinforcement learning that generalizes episodic settings, achieves regret bounds, and employs a novel operator with contraction properties.

Contribution

It presents a new optimistic Q-learning method for average reward RL under a relaxed assumption, introducing the arL operator with contraction properties, unifying episodic and non-episodic analysis.

Findings

Regret bound of  H^5 S  T

arL operator has strict contraction in span

Algorithm generalizes episodic and average reward settings

Abstract

We present an optimistic Q-learning algorithm for regret minimization in average reward reinforcement learning under an additional assumption on the underlying MDP that for all policies, the time to visit some frequent state $s_0$ is finite and upper bounded by $H$, either in expectation or with constant probability. Our setting strictly generalizes the episodic setting and is significantly less restrictive than the assumption of bounded hitting time \textit{for all states} made by most previous literature on model-free algorithms in average reward settings. We demonstrate a regret bound of $\tilde{O}(H^5 S\sqrt{AT})$, where $S$ and $A$ are the numbers of states and actions, and $T$ is the horizon. A key technical novelty of our work is the introduction of an $\overline{L}$ operator defined as $\overline{L} v = \frac{1}{H} \sum_{h=1}^H L^h v$ where $L$ denotes the Bellman operator. Under the given assumption, we show that the $\overline{L}$ operator has a strict contraction (in span) even in the average-reward setting where the discount factor is $1$. Our algorithm design uses ideas from episodic Q-learning to estimate and apply this operator iteratively. Thus, we provide a unified view of regret minimization in episodic and non-episodic settings, which may be of independent interest.