A Statistical Analysis of Polyak-Ruppert Averaged Q-Learning
Xiang Li, Wenhao Yang, Jiadong Liang, Zhihua Zhang, Michael I. Jordan
Abstract
We study Q-learning with Polyak-Ruppert averaging (a.k.a., averaged Q-learning) in a discounted markov decision process in synchronous and tabular settings. Under a Lipschitz condition, we establish a functional central limit theorem for the averaged iteration $\bar{\mathbf{Q}}_T$ and show that its standardized partial-sum process converges weakly to a rescaled Brownian motion. The FCLT implies a fully online inference method for reinforcement learning. Furthermore, we show that $\bar{\mathbf{Q}}_T$ is the regular asymptotically linear (RAL) estimator for the optimal Q-value function $\mathbf{Q}^*$ that has the most efficient influence function. We present a nonasymptotic analysis for the $\ell_{\infty}$ error, $\mathbb{E}\|\bar{\mathbf{Q}}_T-\mathbf{Q}^*\|_{\infty}$, showing that it matches the instance-dependent lower bound for polynomial step sizes. Similar results are provided for entropy-regularized Q-Learning without the Lipschitz condition.
BibTeX
@InProceedings{pmlr-v206-li23b,
title = {A Statistical Analysis of Polyak-Ruppert Averaged Q-Learning},
author = {Li, Xiang and Yang, Wenhao and Liang, Jiadong and Zhang, Zhihua and Jordan, Michael I.},
booktitle = {Proceedings of The 26th International Conference on Artificial Intelligence and Statistics},
pages = {2207--2261},
year = {2023},
editor = {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem},
volume = {206},
series = {Proceedings of Machine Learning Research},
month = {25--27 Apr},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v206/li23b/li23b.pdf},
url = {https://proceedings.mlr.press/v206/li23b.html},
abstract = {We study Q-learning with Polyak-Ruppert averaging (a.k.a., averaged Q-learning) in a discounted markov decision process in synchronous and tabular settings. Under a Lipschitz condition, we establish a functional central limit theorem for the averaged iteration $\bar{\mathbf{Q}}_T$ and show that its standardized partial-sum process converges weakly to a rescaled Brownian motion. The FCLT implies a fully online inference method for reinforcement learning. Furthermore, we show that $\bar{\mathbf{Q}}_T$ is the regular asymptotically linear (RAL) estimator for the optimal Q-value function $\mathbf{Q}^*$ that has the most efficient influence function. We present a nonasymptotic analysis for the $\ell_{\infty}$ error, $\mathbb{E}\|\bar{\mathbf{Q}}_T-\mathbf{Q}^*\|_{\infty}$, showing that it matches the instance-dependent lower bound for polynomial step sizes. Similar results are provided for entropy-regularized Q-Learning without the Lipschitz condition.}
}