A Note on Target Q-learning For Solving Finite MDPs with A Generative Oracle

TL;DR

This paper analyzes the sample complexity of target Q-learning with a generative oracle in finite MDPs, correcting previous claims and showing that target networks do not increase sample complexity.

Contribution

It provides a tight analysis of target Q-learning's sample complexity, correcting prior misconceptions and comparing it to vanilla Q-learning.

Findings

Sample complexity is $ ilde{O}(| ext{S}|^2| ext{A}|^2 (1- ext{γ})^{-5} ext{ε}^{-2})$ in prior work.

Improved sample complexity to $ ilde{O}(| ext{S}|| ext{A}| (1- ext{γ})^{-5} ext{ε}^{-2})$ with sequential updates.

Target networks do not increase sample complexity compared to vanilla Q-learning.

Abstract

Q-learning with function approximation could diverge in the off-policy setting and the target network is a powerful technique to address this issue. In this manuscript, we examine the sample complexity of the associated target Q-learning algorithm in the tabular case with a generative oracle. We point out a misleading claim in [Lee and He, 2020] and establish a tight analysis. In particular, we demonstrate that the sample complexity of the target Q-learning algorithm in [Lee and He, 2020] is $\widetilde{\mathcal O}(|\mathcal S|^2|\mathcal A|^2 (1-\gamma)^{-5}\varepsilon^{-2})$. Furthermore, we show that this sample complexity is improved to $\widetilde{\mathcal O}(|\mathcal S||\mathcal A| (1-\gamma)^{-5}\varepsilon^{-2})$ if we can sequentially update all state-action pairs and $\widetilde{\mathcal O}(|\mathcal S||\mathcal A| (1-\gamma)^{-4}\varepsilon^{-2})$ if $\gamma$ is further in $(1/2, 1)$. Compared with the vanilla Q-learning, our results conclude that the introduction of a periodically-frozen target Q-function does not sacrifice the sample complexity.