Finding good policies in average-reward Markov Decision Processes without prior knowledge

TL;DR

This paper introduces a new algorithm for identifying near-optimal policies in average-reward MDPs without prior knowledge, achieving near-optimal sample complexity in the PAC setting and establishing fundamental limits in the online setting.

Contribution

It presents the first H-agnostic PAC algorithm for average-reward MDPs and provides lower bounds and improved online algorithms with data-dependent stopping rules.

Findings

The PAC algorithm has sample complexity scaling as SAD/ε^2, near the theoretical lower bound.

A lower bound shows polynomial dependence on H is impossible in the online setting.

An online algorithm with sample complexity SAD^2/ε^2 is proposed, with potential for further improvement.

Abstract

We revisit the identification of an $\varepsilon$-optimal policy in average-reward Markov Decision Processes (MDP). In such MDPs, two measures of complexity have appeared in the literature: the diameter, $D$, and the optimal bias span, $H$, which satisfy $H\leq D$. Prior work have studied the complexity of $\varepsilon$-optimal policy identification only when a generative model is available. In this case, it is known that there exists an MDP with $D \simeq H$ for which the sample complexity to output an $\varepsilon$-optimal policy is $\Omega(SAD/\varepsilon^2)$ where $S$ and $A$ are the sizes of the state and action spaces. Recently, an algorithm with a sample complexity of order $SAH/\varepsilon^2$ has been proposed, but it requires the knowledge of $H$. We first show that the sample complexity required to estimate $H$ is not bounded by any function of $S,A$ and $H$, ruling out the possibility to easily make the previous algorithm agnostic to $H$. By relying instead on a diameter estimation procedure, we propose the first algorithm for $(\varepsilon,\delta)$-PAC policy identification that does not need any form of prior knowledge on the MDP. Its sample complexity scales in $SAD/\varepsilon^2$ in the regime of small $\varepsilon$, which is near-optimal. In the online setting, our first contribution is a lower bound which implies that a sample complexity polynomial in $H$ cannot be achieved in this setting. Then, we propose an online algorithm with a sample complexity in $SAD^2/\varepsilon^2$, as well as a novel approach based on a data-dependent stopping rule that we believe is promising to further reduce this bound.